diff options
Diffstat (limited to 'modules/fdlibm/src')
37 files changed, 4590 insertions, 0 deletions
diff --git a/modules/fdlibm/src/e_acos.cpp b/modules/fdlibm/src/e_acos.cpp new file mode 100644 index 000000000..12be296cb --- /dev/null +++ b/modules/fdlibm/src/e_acos.cpp @@ -0,0 +1,106 @@ + +/* @(#)e_acos.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* __ieee754_acos(x) + * Method : + * acos(x) = pi/2 - asin(x) + * acos(-x) = pi/2 + asin(x) + * For |x|<=0.5 + * acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c) + * For x>0.5 + * acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2))) + * = 2asin(sqrt((1-x)/2)) + * = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z) + * = 2f + (2c + 2s*z*R(z)) + * where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term + * for f so that f+c ~ sqrt(z). + * For x<-0.5 + * acos(x) = pi - 2asin(sqrt((1-|x|)/2)) + * = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z) + * + * Special cases: + * if x is NaN, return x itself; + * if |x|>1, return NaN with invalid signal. + * + * Function needed: sqrt + */ + +#include <float.h> + +#include "math_private.h" + +static const double +one= 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ +pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ +pio2_hi = 1.57079632679489655800e+00; /* 0x3FF921FB, 0x54442D18 */ +static volatile double +pio2_lo = 6.12323399573676603587e-17; /* 0x3C91A626, 0x33145C07 */ +static const double +pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ +pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ +pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ +pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ +pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ +pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ +qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ +qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ +qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ +qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ + +double +__ieee754_acos(double x) +{ + double z,p,q,r,w,s,c,df; + int32_t hx,ix; + GET_HIGH_WORD(hx,x); + ix = hx&0x7fffffff; + if(ix>=0x3ff00000) { /* |x| >= 1 */ + u_int32_t lx; + GET_LOW_WORD(lx,x); + if(((ix-0x3ff00000)|lx)==0) { /* |x|==1 */ + if(hx>0) return 0.0; /* acos(1) = 0 */ + else return pi+2.0*pio2_lo; /* acos(-1)= pi */ + } + return (x-x)/(x-x); /* acos(|x|>1) is NaN */ + } + if(ix<0x3fe00000) { /* |x| < 0.5 */ + if(ix<=0x3c600000) return pio2_hi+pio2_lo;/*if|x|<2**-57*/ + z = x*x; + p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5))))); + q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4))); + r = p/q; + return pio2_hi - (x - (pio2_lo-x*r)); + } else if (hx<0) { /* x < -0.5 */ + z = (one+x)*0.5; + p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5))))); + q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4))); + s = sqrt(z); + r = p/q; + w = r*s-pio2_lo; + return pi - 2.0*(s+w); + } else { /* x > 0.5 */ + z = (one-x)*0.5; + s = sqrt(z); + df = s; + SET_LOW_WORD(df,0); + c = (z-df*df)/(s+df); + p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5))))); + q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4))); + r = p/q; + w = r*s+c; + return 2.0*(df+w); + } +} diff --git a/modules/fdlibm/src/e_acosh.cpp b/modules/fdlibm/src/e_acosh.cpp new file mode 100644 index 000000000..bdabcec3e --- /dev/null +++ b/modules/fdlibm/src/e_acosh.cpp @@ -0,0 +1,63 @@ + +/* @(#)e_acosh.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* __ieee754_acosh(x) + * Method : + * Based on + * acosh(x) = log [ x + sqrt(x*x-1) ] + * we have + * acosh(x) := log(x)+ln2, if x is large; else + * acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else + * acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1. + * + * Special cases: + * acosh(x) is NaN with signal if x<1. + * acosh(NaN) is NaN without signal. + */ + +#include <float.h> + +#include "math_private.h" + +static const double +one = 1.0, +ln2 = 6.93147180559945286227e-01; /* 0x3FE62E42, 0xFEFA39EF */ + +double +__ieee754_acosh(double x) +{ + double t; + int32_t hx; + u_int32_t lx; + EXTRACT_WORDS(hx,lx,x); + if(hx<0x3ff00000) { /* x < 1 */ + return (x-x)/(x-x); + } else if(hx >=0x41b00000) { /* x > 2**28 */ + if(hx >=0x7ff00000) { /* x is inf of NaN */ + return x+x; + } else + return __ieee754_log(x)+ln2; /* acosh(huge)=log(2x) */ + } else if(((hx-0x3ff00000)|lx)==0) { + return 0.0; /* acosh(1) = 0 */ + } else if (hx > 0x40000000) { /* 2**28 > x > 2 */ + t=x*x; + return __ieee754_log(2.0*x-one/(x+sqrt(t-one))); + } else { /* 1<x<2 */ + t = x-one; + return log1p(t+sqrt(2.0*t+t*t)); + } +} diff --git a/modules/fdlibm/src/e_asin.cpp b/modules/fdlibm/src/e_asin.cpp new file mode 100644 index 000000000..396f49449 --- /dev/null +++ b/modules/fdlibm/src/e_asin.cpp @@ -0,0 +1,112 @@ + +/* @(#)e_asin.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* __ieee754_asin(x) + * Method : + * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... + * we approximate asin(x) on [0,0.5] by + * asin(x) = x + x*x^2*R(x^2) + * where + * R(x^2) is a rational approximation of (asin(x)-x)/x^3 + * and its remez error is bounded by + * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) + * + * For x in [0.5,1] + * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) + * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; + * then for x>0.98 + * asin(x) = pi/2 - 2*(s+s*z*R(z)) + * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) + * For x<=0.98, let pio4_hi = pio2_hi/2, then + * f = hi part of s; + * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) + * and + * asin(x) = pi/2 - 2*(s+s*z*R(z)) + * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) + * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) + * + * Special cases: + * if x is NaN, return x itself; + * if |x|>1, return NaN with invalid signal. + * + */ + +#include <float.h> + +#include "math_private.h" + +static const double +one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ +huge = 1.000e+300, +pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ +pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ +pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ + /* coefficient for R(x^2) */ +pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ +pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ +pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ +pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ +pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ +pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ +qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ +qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ +qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ +qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ + +double +__ieee754_asin(double x) +{ + double t=0.0,w,p,q,c,r,s; + int32_t hx,ix; + GET_HIGH_WORD(hx,x); + ix = hx&0x7fffffff; + if(ix>= 0x3ff00000) { /* |x|>= 1 */ + u_int32_t lx; + GET_LOW_WORD(lx,x); + if(((ix-0x3ff00000)|lx)==0) + /* asin(1)=+-pi/2 with inexact */ + return x*pio2_hi+x*pio2_lo; + return (x-x)/(x-x); /* asin(|x|>1) is NaN */ + } else if (ix<0x3fe00000) { /* |x|<0.5 */ + if(ix<0x3e500000) { /* if |x| < 2**-26 */ + if(huge+x>one) return x;/* return x with inexact if x!=0*/ + } + t = x*x; + p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); + q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); + w = p/q; + return x+x*w; + } + /* 1> |x|>= 0.5 */ + w = one-fabs(x); + t = w*0.5; + p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); + q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); + s = sqrt(t); + if(ix>=0x3FEF3333) { /* if |x| > 0.975 */ + w = p/q; + t = pio2_hi-(2.0*(s+s*w)-pio2_lo); + } else { + w = s; + SET_LOW_WORD(w,0); + c = (t-w*w)/(s+w); + r = p/q; + p = 2.0*s*r-(pio2_lo-2.0*c); + q = pio4_hi-2.0*w; + t = pio4_hi-(p-q); + } + if(hx>0) return t; else return -t; +} diff --git a/modules/fdlibm/src/e_atan2.cpp b/modules/fdlibm/src/e_atan2.cpp new file mode 100644 index 000000000..9990072cf --- /dev/null +++ b/modules/fdlibm/src/e_atan2.cpp @@ -0,0 +1,124 @@ + +/* @(#)e_atan2.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* __ieee754_atan2(y,x) + * Method : + * 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x). + * 2. Reduce x to positive by (if x and y are unexceptional): + * ARG (x+iy) = arctan(y/x) ... if x > 0, + * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0, + * + * Special cases: + * + * ATAN2((anything), NaN ) is NaN; + * ATAN2(NAN , (anything) ) is NaN; + * ATAN2(+-0, +(anything but NaN)) is +-0 ; + * ATAN2(+-0, -(anything but NaN)) is +-pi ; + * ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2; + * ATAN2(+-(anything but INF and NaN), +INF) is +-0 ; + * ATAN2(+-(anything but INF and NaN), -INF) is +-pi; + * ATAN2(+-INF,+INF ) is +-pi/4 ; + * ATAN2(+-INF,-INF ) is +-3pi/4; + * ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2; + * + * Constants: + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough + * to produce the hexadecimal values shown. + */ + +#include <float.h> + +#include "math_private.h" + +static volatile double +tiny = 1.0e-300; +static const double +zero = 0.0, +pi_o_4 = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */ +pi_o_2 = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */ +pi = 3.1415926535897931160E+00; /* 0x400921FB, 0x54442D18 */ +static volatile double +pi_lo = 1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */ + +double +__ieee754_atan2(double y, double x) +{ + double z; + int32_t k,m,hx,hy,ix,iy; + u_int32_t lx,ly; + + EXTRACT_WORDS(hx,lx,x); + ix = hx&0x7fffffff; + EXTRACT_WORDS(hy,ly,y); + iy = hy&0x7fffffff; + if(((ix|((lx|-lx)>>31))>0x7ff00000)|| + ((iy|((ly|-ly)>>31))>0x7ff00000)) /* x or y is NaN */ + return x+y; + if((hx-0x3ff00000|lx)==0) return atan(y); /* x=1.0 */ + m = ((hy>>31)&1)|((hx>>30)&2); /* 2*sign(x)+sign(y) */ + + /* when y = 0 */ + if((iy|ly)==0) { + switch(m) { + case 0: + case 1: return y; /* atan(+-0,+anything)=+-0 */ + case 2: return pi+tiny;/* atan(+0,-anything) = pi */ + case 3: return -pi-tiny;/* atan(-0,-anything) =-pi */ + } + } + /* when x = 0 */ + if((ix|lx)==0) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny; + + /* when x is INF */ + if(ix==0x7ff00000) { + if(iy==0x7ff00000) { + switch(m) { + case 0: return pi_o_4+tiny;/* atan(+INF,+INF) */ + case 1: return -pi_o_4-tiny;/* atan(-INF,+INF) */ + case 2: return 3.0*pi_o_4+tiny;/*atan(+INF,-INF)*/ + case 3: return -3.0*pi_o_4-tiny;/*atan(-INF,-INF)*/ + } + } else { + switch(m) { + case 0: return zero ; /* atan(+...,+INF) */ + case 1: return -zero ; /* atan(-...,+INF) */ + case 2: return pi+tiny ; /* atan(+...,-INF) */ + case 3: return -pi-tiny ; /* atan(-...,-INF) */ + } + } + } + /* when y is INF */ + if(iy==0x7ff00000) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny; + + /* compute y/x */ + k = (iy-ix)>>20; + if(k > 60) { /* |y/x| > 2**60 */ + z=pi_o_2+0.5*pi_lo; + m&=1; + } + else if(hx<0&&k<-60) z=0.0; /* 0 > |y|/x > -2**-60 */ + else z=atan(fabs(y/x)); /* safe to do y/x */ + switch (m) { + case 0: return z ; /* atan(+,+) */ + case 1: return -z ; /* atan(-,+) */ + case 2: return pi-(z-pi_lo);/* atan(+,-) */ + default: /* case 3 */ + return (z-pi_lo)-pi;/* atan(-,-) */ + } +} diff --git a/modules/fdlibm/src/e_atanh.cpp b/modules/fdlibm/src/e_atanh.cpp new file mode 100644 index 000000000..a8f0f8deb --- /dev/null +++ b/modules/fdlibm/src/e_atanh.cpp @@ -0,0 +1,63 @@ + +/* @(#)e_atanh.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* __ieee754_atanh(x) + * Method : + * 1.Reduced x to positive by atanh(-x) = -atanh(x) + * 2.For x>=0.5 + * 1 2x x + * atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------) + * 2 1 - x 1 - x + * + * For x<0.5 + * atanh(x) = 0.5*log1p(2x+2x*x/(1-x)) + * + * Special cases: + * atanh(x) is NaN if |x| > 1 with signal; + * atanh(NaN) is that NaN with no signal; + * atanh(+-1) is +-INF with signal. + * + */ + +#include <float.h> + +#include "math_private.h" + +static const double one = 1.0, huge = 1e300; +static const double zero = 0.0; + +double +__ieee754_atanh(double x) +{ + double t; + int32_t hx,ix; + u_int32_t lx; + EXTRACT_WORDS(hx,lx,x); + ix = hx&0x7fffffff; + if ((ix|((lx|(-lx))>>31))>0x3ff00000) /* |x|>1 */ + return (x-x)/(x-x); + if(ix==0x3ff00000) + return x/zero; + if(ix<0x3e300000&&(huge+x)>zero) return x; /* x<2**-28 */ + SET_HIGH_WORD(x,ix); + if(ix<0x3fe00000) { /* x < 0.5 */ + t = x+x; + t = 0.5*log1p(t+t*x/(one-x)); + } else + t = 0.5*log1p((x+x)/(one-x)); + if(hx>=0) return t; else return -t; +} diff --git a/modules/fdlibm/src/e_cosh.cpp b/modules/fdlibm/src/e_cosh.cpp new file mode 100644 index 000000000..42cb277d4 --- /dev/null +++ b/modules/fdlibm/src/e_cosh.cpp @@ -0,0 +1,80 @@ + +/* @(#)e_cosh.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* __ieee754_cosh(x) + * Method : + * mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2 + * 1. Replace x by |x| (cosh(x) = cosh(-x)). + * 2. + * [ exp(x) - 1 ]^2 + * 0 <= x <= ln2/2 : cosh(x) := 1 + ------------------- + * 2*exp(x) + * + * exp(x) + 1/exp(x) + * ln2/2 <= x <= 22 : cosh(x) := ------------------- + * 2 + * 22 <= x <= lnovft : cosh(x) := exp(x)/2 + * lnovft <= x <= ln2ovft: cosh(x) := exp(x/2)/2 * exp(x/2) + * ln2ovft < x : cosh(x) := huge*huge (overflow) + * + * Special cases: + * cosh(x) is |x| if x is +INF, -INF, or NaN. + * only cosh(0)=1 is exact for finite x. + */ + +#include <float.h> + +#include "math_private.h" + +static const double one = 1.0, half=0.5, huge = 1.0e300; + +double +__ieee754_cosh(double x) +{ + double t,w; + int32_t ix; + + /* High word of |x|. */ + GET_HIGH_WORD(ix,x); + ix &= 0x7fffffff; + + /* x is INF or NaN */ + if(ix>=0x7ff00000) return x*x; + + /* |x| in [0,0.5*ln2], return 1+expm1(|x|)^2/(2*exp(|x|)) */ + if(ix<0x3fd62e43) { + t = expm1(fabs(x)); + w = one+t; + if (ix<0x3c800000) return w; /* cosh(tiny) = 1 */ + return one+(t*t)/(w+w); + } + + /* |x| in [0.5*ln2,22], return (exp(|x|)+1/exp(|x|)/2; */ + if (ix < 0x40360000) { + t = __ieee754_exp(fabs(x)); + return half*t+half/t; + } + + /* |x| in [22, log(maxdouble)] return half*exp(|x|) */ + if (ix < 0x40862E42) return half*__ieee754_exp(fabs(x)); + + /* |x| in [log(maxdouble), overflowthresold] */ + if (ix<=0x408633CE) + return __ldexp_exp(fabs(x), -1); + + /* |x| > overflowthresold, cosh(x) overflow */ + return huge*huge; +} diff --git a/modules/fdlibm/src/e_exp.cpp b/modules/fdlibm/src/e_exp.cpp new file mode 100644 index 000000000..b31979134 --- /dev/null +++ b/modules/fdlibm/src/e_exp.cpp @@ -0,0 +1,162 @@ + +/* @(#)e_exp.c 1.6 04/04/22 */ +/* + * ==================================================== + * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. + * + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* __ieee754_exp(x) + * Returns the exponential of x. + * + * Method + * 1. Argument reduction: + * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. + * Given x, find r and integer k such that + * + * x = k*ln2 + r, |r| <= 0.5*ln2. + * + * Here r will be represented as r = hi-lo for better + * accuracy. + * + * 2. Approximation of exp(r) by a special rational function on + * the interval [0,0.34658]: + * Write + * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... + * We use a special Remes algorithm on [0,0.34658] to generate + * a polynomial of degree 5 to approximate R. The maximum error + * of this polynomial approximation is bounded by 2**-59. In + * other words, + * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 + * (where z=r*r, and the values of P1 to P5 are listed below) + * and + * | 5 | -59 + * | 2.0+P1*z+...+P5*z - R(z) | <= 2 + * | | + * The computation of exp(r) thus becomes + * 2*r + * exp(r) = 1 + ------- + * R - r + * r*R1(r) + * = 1 + r + ----------- (for better accuracy) + * 2 - R1(r) + * where + * 2 4 10 + * R1(r) = r - (P1*r + P2*r + ... + P5*r ). + * + * 3. Scale back to obtain exp(x): + * From step 1, we have + * exp(x) = 2^k * exp(r) + * + * Special cases: + * exp(INF) is INF, exp(NaN) is NaN; + * exp(-INF) is 0, and + * for finite argument, only exp(0)=1 is exact. + * + * Accuracy: + * according to an error analysis, the error is always less than + * 1 ulp (unit in the last place). + * + * Misc. info. + * For IEEE double + * if x > 7.09782712893383973096e+02 then exp(x) overflow + * if x < -7.45133219101941108420e+02 then exp(x) underflow + * + * Constants: + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough + * to produce the hexadecimal values shown. + */ + +#include <float.h> + +#include "math_private.h" + +static const double +one = 1.0, +halF[2] = {0.5,-0.5,}, +o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ +u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ +ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ + -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */ +ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ + -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */ +invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ +P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ +P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ +P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ +P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ +P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ + +static volatile double +huge = 1.0e+300, +twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/ + +double +__ieee754_exp(double x) /* default IEEE double exp */ +{ + double y,hi=0.0,lo=0.0,c,t,twopk; + int32_t k=0,xsb; + u_int32_t hx; + + GET_HIGH_WORD(hx,x); + xsb = (hx>>31)&1; /* sign bit of x */ + hx &= 0x7fffffff; /* high word of |x| */ + + /* filter out non-finite argument */ + if(hx >= 0x40862E42) { /* if |x|>=709.78... */ + if(hx>=0x7ff00000) { + u_int32_t lx; + GET_LOW_WORD(lx,x); + if(((hx&0xfffff)|lx)!=0) + return x+x; /* NaN */ + else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ + } + if(x > o_threshold) return huge*huge; /* overflow */ + if(x < u_threshold) return twom1000*twom1000; /* underflow */ + } + + /* argument reduction */ + if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ + if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ + hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; + } else { + k = (int)(invln2*x+halF[xsb]); + t = k; + hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ + lo = t*ln2LO[0]; + } + STRICT_ASSIGN(double, x, hi - lo); + } + else if(hx < 0x3e300000) { /* when |x|<2**-28 */ + if(huge+x>one) return one+x;/* trigger inexact */ + } + else k = 0; + + /* x is now in primary range */ + t = x*x; + if(k >= -1021) + INSERT_WORDS(twopk,0x3ff00000+(k<<20), 0); + else + INSERT_WORDS(twopk,0x3ff00000+((k+1000)<<20), 0); + c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); + if(k==0) return one-((x*c)/(c-2.0)-x); + else y = one-((lo-(x*c)/(2.0-c))-hi); + if(k >= -1021) { + if (k==1024) { + double const_0x1p1023 = pow(2, 1023); + return y*2.0*const_0x1p1023; + } + return y*twopk; + } else { + return y*twopk*twom1000; + } +} diff --git a/modules/fdlibm/src/e_hypot.cpp b/modules/fdlibm/src/e_hypot.cpp new file mode 100644 index 000000000..f5c7037bb --- /dev/null +++ b/modules/fdlibm/src/e_hypot.cpp @@ -0,0 +1,126 @@ + +/* @(#)e_hypot.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* __ieee754_hypot(x,y) + * + * Method : + * If (assume round-to-nearest) z=x*x+y*y + * has error less than sqrt(2)/2 ulp, than + * sqrt(z) has error less than 1 ulp (exercise). + * + * So, compute sqrt(x*x+y*y) with some care as + * follows to get the error below 1 ulp: + * + * Assume x>y>0; + * (if possible, set rounding to round-to-nearest) + * 1. if x > 2y use + * x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y + * where x1 = x with lower 32 bits cleared, x2 = x-x1; else + * 2. if x <= 2y use + * t1*y1+((x-y)*(x-y)+(t1*y2+t2*y)) + * where t1 = 2x with lower 32 bits cleared, t2 = 2x-t1, + * y1= y with lower 32 bits chopped, y2 = y-y1. + * + * NOTE: scaling may be necessary if some argument is too + * large or too tiny + * + * Special cases: + * hypot(x,y) is INF if x or y is +INF or -INF; else + * hypot(x,y) is NAN if x or y is NAN. + * + * Accuracy: + * hypot(x,y) returns sqrt(x^2+y^2) with error less + * than 1 ulps (units in the last place) + */ + +#include <float.h> + +#include "math_private.h" + +double +__ieee754_hypot(double x, double y) +{ + double a,b,t1,t2,y1,y2,w; + int32_t j,k,ha,hb; + + GET_HIGH_WORD(ha,x); + ha &= 0x7fffffff; + GET_HIGH_WORD(hb,y); + hb &= 0x7fffffff; + if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;} + a = fabs(a); + b = fabs(b); + if((ha-hb)>0x3c00000) {return a+b;} /* x/y > 2**60 */ + k=0; + if(ha > 0x5f300000) { /* a>2**500 */ + if(ha >= 0x7ff00000) { /* Inf or NaN */ + u_int32_t low; + /* Use original arg order iff result is NaN; quieten sNaNs. */ + w = fabs(x+0.0)-fabs(y+0.0); + GET_LOW_WORD(low,a); + if(((ha&0xfffff)|low)==0) w = a; + GET_LOW_WORD(low,b); + if(((hb^0x7ff00000)|low)==0) w = b; + return w; + } + /* scale a and b by 2**-600 */ + ha -= 0x25800000; hb -= 0x25800000; k += 600; + SET_HIGH_WORD(a,ha); + SET_HIGH_WORD(b,hb); + } + if(hb < 0x20b00000) { /* b < 2**-500 */ + if(hb <= 0x000fffff) { /* subnormal b or 0 */ + u_int32_t low; + GET_LOW_WORD(low,b); + if((hb|low)==0) return a; + t1=0; + SET_HIGH_WORD(t1,0x7fd00000); /* t1=2^1022 */ + b *= t1; + a *= t1; + k -= 1022; + } else { /* scale a and b by 2^600 */ + ha += 0x25800000; /* a *= 2^600 */ + hb += 0x25800000; /* b *= 2^600 */ + k -= 600; + SET_HIGH_WORD(a,ha); + SET_HIGH_WORD(b,hb); + } + } + /* medium size a and b */ + w = a-b; + if (w>b) { + t1 = 0; + SET_HIGH_WORD(t1,ha); + t2 = a-t1; + w = sqrt(t1*t1-(b*(-b)-t2*(a+t1))); + } else { + a = a+a; + y1 = 0; + SET_HIGH_WORD(y1,hb); + y2 = b - y1; + t1 = 0; + SET_HIGH_WORD(t1,ha+0x00100000); + t2 = a - t1; + w = sqrt(t1*y1-(w*(-w)-(t1*y2+t2*b))); + } + if(k!=0) { + u_int32_t high; + t1 = 1.0; + GET_HIGH_WORD(high,t1); + SET_HIGH_WORD(t1,high+(k<<20)); + return t1*w; + } else return w; +} diff --git a/modules/fdlibm/src/e_log.cpp b/modules/fdlibm/src/e_log.cpp new file mode 100644 index 000000000..fa2da8fcb --- /dev/null +++ b/modules/fdlibm/src/e_log.cpp @@ -0,0 +1,142 @@ + +/* @(#)e_log.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* __ieee754_log(x) + * Return the logrithm of x + * + * Method : + * 1. Argument Reduction: find k and f such that + * x = 2^k * (1+f), + * where sqrt(2)/2 < 1+f < sqrt(2) . + * + * 2. Approximation of log(1+f). + * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) + * = 2s + 2/3 s**3 + 2/5 s**5 + ....., + * = 2s + s*R + * We use a special Reme algorithm on [0,0.1716] to generate + * a polynomial of degree 14 to approximate R The maximum error + * of this polynomial approximation is bounded by 2**-58.45. In + * other words, + * 2 4 6 8 10 12 14 + * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s + * (the values of Lg1 to Lg7 are listed in the program) + * and + * | 2 14 | -58.45 + * | Lg1*s +...+Lg7*s - R(z) | <= 2 + * | | + * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. + * In order to guarantee error in log below 1ulp, we compute log + * by + * log(1+f) = f - s*(f - R) (if f is not too large) + * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) + * + * 3. Finally, log(x) = k*ln2 + log(1+f). + * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) + * Here ln2 is split into two floating point number: + * ln2_hi + ln2_lo, + * where n*ln2_hi is always exact for |n| < 2000. + * + * Special cases: + * log(x) is NaN with signal if x < 0 (including -INF) ; + * log(+INF) is +INF; log(0) is -INF with signal; + * log(NaN) is that NaN with no signal. + * + * Accuracy: + * according to an error analysis, the error is always less than + * 1 ulp (unit in the last place). + * + * Constants: + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough + * to produce the hexadecimal values shown. + */ + +#include <float.h> + +#include "math_private.h" + +static const double +ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ +ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ +two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ +Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ +Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ +Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ +Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ +Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ +Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ +Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ + +static const double zero = 0.0; +static volatile double vzero = 0.0; + +double +__ieee754_log(double x) +{ + double hfsq,f,s,z,R,w,t1,t2,dk; + int32_t k,hx,i,j; + u_int32_t lx; + + EXTRACT_WORDS(hx,lx,x); + + k=0; + if (hx < 0x00100000) { /* x < 2**-1022 */ + if (((hx&0x7fffffff)|lx)==0) + return -two54/vzero; /* log(+-0)=-inf */ + if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ + k -= 54; x *= two54; /* subnormal number, scale up x */ + GET_HIGH_WORD(hx,x); + } + if (hx >= 0x7ff00000) return x+x; + k += (hx>>20)-1023; + hx &= 0x000fffff; + i = (hx+0x95f64)&0x100000; + SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ + k += (i>>20); + f = x-1.0; + if((0x000fffff&(2+hx))<3) { /* -2**-20 <= f < 2**-20 */ + if(f==zero) { + if(k==0) { + return zero; + } else { + dk=(double)k; + return dk*ln2_hi+dk*ln2_lo; + } + } + R = f*f*(0.5-0.33333333333333333*f); + if(k==0) return f-R; else {dk=(double)k; + return dk*ln2_hi-((R-dk*ln2_lo)-f);} + } + s = f/(2.0+f); + dk = (double)k; + z = s*s; + i = hx-0x6147a; + w = z*z; + j = 0x6b851-hx; + t1= w*(Lg2+w*(Lg4+w*Lg6)); + t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); + i |= j; + R = t2+t1; + if(i>0) { + hfsq=0.5*f*f; + if(k==0) return f-(hfsq-s*(hfsq+R)); else + return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); + } else { + if(k==0) return f-s*(f-R); else + return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); + } +} diff --git a/modules/fdlibm/src/e_log10.cpp b/modules/fdlibm/src/e_log10.cpp new file mode 100644 index 000000000..ed6879885 --- /dev/null +++ b/modules/fdlibm/src/e_log10.cpp @@ -0,0 +1,89 @@ + +/* @(#)e_log10.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* + * Return the base 10 logarithm of x. See e_log.c and k_log.h for most + * comments. + * + * log10(x) = (f - 0.5*f*f + k_log1p(f)) / ln10 + k * log10(2) + * in not-quite-routine extra precision. + */ + +#include <float.h> + +#include "math_private.h" +#include "k_log.h" + +static const double +two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ +ivln10hi = 4.34294481878168880939e-01, /* 0x3fdbcb7b, 0x15200000 */ +ivln10lo = 2.50829467116452752298e-11, /* 0x3dbb9438, 0xca9aadd5 */ +log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */ +log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */ + +static const double zero = 0.0; +static volatile double vzero = 0.0; + +double +__ieee754_log10(double x) +{ + double f,hfsq,hi,lo,r,val_hi,val_lo,w,y,y2; + int32_t i,k,hx; + u_int32_t lx; + + EXTRACT_WORDS(hx,lx,x); + + k=0; + if (hx < 0x00100000) { /* x < 2**-1022 */ + if (((hx&0x7fffffff)|lx)==0) + return -two54/vzero; /* log(+-0)=-inf */ + if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ + k -= 54; x *= two54; /* subnormal number, scale up x */ + GET_HIGH_WORD(hx,x); + } + if (hx >= 0x7ff00000) return x+x; + if (hx == 0x3ff00000 && lx == 0) + return zero; /* log(1) = +0 */ + k += (hx>>20)-1023; + hx &= 0x000fffff; + i = (hx+0x95f64)&0x100000; + SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ + k += (i>>20); + y = (double)k; + f = x - 1.0; + hfsq = 0.5*f*f; + r = k_log1p(f); + + /* See e_log2.c for most details. */ + hi = f - hfsq; + SET_LOW_WORD(hi,0); + lo = (f - hi) - hfsq + r; + val_hi = hi*ivln10hi; + y2 = y*log10_2hi; + val_lo = y*log10_2lo + (lo+hi)*ivln10lo + lo*ivln10hi; + + /* + * Extra precision in for adding y*log10_2hi is not strictly needed + * since there is no very large cancellation near x = sqrt(2) or + * x = 1/sqrt(2), but we do it anyway since it costs little on CPUs + * with some parallelism and it reduces the error for many args. + */ + w = y2 + val_hi; + val_lo += (y2 - w) + val_hi; + val_hi = w; + + return val_lo + val_hi; +} diff --git a/modules/fdlibm/src/e_log2.cpp b/modules/fdlibm/src/e_log2.cpp new file mode 100644 index 000000000..5649fec44 --- /dev/null +++ b/modules/fdlibm/src/e_log2.cpp @@ -0,0 +1,112 @@ + +/* @(#)e_log10.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* + * Return the base 2 logarithm of x. See e_log.c and k_log.h for most + * comments. + * + * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel, + * then does the combining and scaling steps + * log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k + * in not-quite-routine extra precision. + */ + +#include <float.h> + +#include "math_private.h" +#include "k_log.h" + +static const double +two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ +ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */ +ivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */ + +static const double zero = 0.0; +static volatile double vzero = 0.0; + +double +__ieee754_log2(double x) +{ + double f,hfsq,hi,lo,r,val_hi,val_lo,w,y; + int32_t i,k,hx; + u_int32_t lx; + + EXTRACT_WORDS(hx,lx,x); + + k=0; + if (hx < 0x00100000) { /* x < 2**-1022 */ + if (((hx&0x7fffffff)|lx)==0) + return -two54/vzero; /* log(+-0)=-inf */ + if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ + k -= 54; x *= two54; /* subnormal number, scale up x */ + GET_HIGH_WORD(hx,x); + } + if (hx >= 0x7ff00000) return x+x; + if (hx == 0x3ff00000 && lx == 0) + return zero; /* log(1) = +0 */ + k += (hx>>20)-1023; + hx &= 0x000fffff; + i = (hx+0x95f64)&0x100000; + SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ + k += (i>>20); + y = (double)k; + f = x - 1.0; + hfsq = 0.5*f*f; + r = k_log1p(f); + + /* + * f-hfsq must (for args near 1) be evaluated in extra precision + * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2). + * This is fairly efficient since f-hfsq only depends on f, so can + * be evaluated in parallel with R. Not combining hfsq with R also + * keeps R small (though not as small as a true `lo' term would be), + * so that extra precision is not needed for terms involving R. + * + * Compiler bugs involving extra precision used to break Dekker's + * theorem for spitting f-hfsq as hi+lo, unless double_t was used + * or the multi-precision calculations were avoided when double_t + * has extra precision. These problems are now automatically + * avoided as a side effect of the optimization of combining the + * Dekker splitting step with the clear-low-bits step. + * + * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra + * precision to avoid a very large cancellation when x is very near + * these values. Unlike the above cancellations, this problem is + * specific to base 2. It is strange that adding +-1 is so much + * harder than adding +-ln2 or +-log10_2. + * + * This uses Dekker's theorem to normalize y+val_hi, so the + * compiler bugs are back in some configurations, sigh. And I + * don't want to used double_t to avoid them, since that gives a + * pessimization and the support for avoiding the pessimization + * is not yet available. + * + * The multi-precision calculations for the multiplications are + * routine. + */ + hi = f - hfsq; + SET_LOW_WORD(hi,0); + lo = (f - hi) - hfsq + r; + val_hi = hi*ivln2hi; + val_lo = (lo+hi)*ivln2lo + lo*ivln2hi; + + /* spadd(val_hi, val_lo, y), except for not using double_t: */ + w = y + val_hi; + val_lo += (y - w) + val_hi; + val_hi = w; + + return val_lo + val_hi; +} diff --git a/modules/fdlibm/src/e_pow.cpp b/modules/fdlibm/src/e_pow.cpp new file mode 100644 index 000000000..366e3933b --- /dev/null +++ b/modules/fdlibm/src/e_pow.cpp @@ -0,0 +1,305 @@ +/* @(#)e_pow.c 1.5 04/04/22 SMI */ +/* + * ==================================================== + * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. + * + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* __ieee754_pow(x,y) return x**y + * + * n + * Method: Let x = 2 * (1+f) + * 1. Compute and return log2(x) in two pieces: + * log2(x) = w1 + w2, + * where w1 has 53-24 = 29 bit trailing zeros. + * 2. Perform y*log2(x) = n+y' by simulating multi-precision + * arithmetic, where |y'|<=0.5. + * 3. Return x**y = 2**n*exp(y'*log2) + * + * Special cases: + * 1. (anything) ** 0 is 1 + * 2. (anything) ** 1 is itself + * 3. (anything) ** NAN is NAN except 1 ** NAN = 1 + * 4. NAN ** (anything except 0) is NAN + * 5. +-(|x| > 1) ** +INF is +INF + * 6. +-(|x| > 1) ** -INF is +0 + * 7. +-(|x| < 1) ** +INF is +0 + * 8. +-(|x| < 1) ** -INF is +INF + * 9. +-1 ** +-INF is 1 + * 10. +0 ** (+anything except 0, NAN) is +0 + * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 + * 12. +0 ** (-anything except 0, NAN) is +INF + * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF + * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) + * 15. +INF ** (+anything except 0,NAN) is +INF + * 16. +INF ** (-anything except 0,NAN) is +0 + * 17. -INF ** (anything) = -0 ** (-anything) + * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) + * 19. (-anything except 0 and inf) ** (non-integer) is NAN + * + * Accuracy: + * pow(x,y) returns x**y nearly rounded. In particular + * pow(integer,integer) + * always returns the correct integer provided it is + * representable. + * + * Constants : + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough + * to produce the hexadecimal values shown. + */ + +#include "math_private.h" + +static const double +bp[] = {1.0, 1.5,}, +dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ +dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ +zero = 0.0, +one = 1.0, +two = 2.0, +two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ +huge = 1.0e300, +tiny = 1.0e-300, + /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ +L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ +L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ +L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ +L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ +L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ +L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ +P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ +P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ +P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ +P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ +P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ +lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ +lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ +lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ +ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ +cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ +cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ +cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ +ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ +ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ +ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ + +double +__ieee754_pow(double x, double y) +{ + double z,ax,z_h,z_l,p_h,p_l; + double y1,t1,t2,r,s,t,u,v,w; + int32_t i,j,k,yisint,n; + int32_t hx,hy,ix,iy; + u_int32_t lx,ly; + + EXTRACT_WORDS(hx,lx,x); + EXTRACT_WORDS(hy,ly,y); + ix = hx&0x7fffffff; iy = hy&0x7fffffff; + + /* y==zero: x**0 = 1 */ + if((iy|ly)==0) return one; + + /* x==1: 1**y = 1, even if y is NaN */ + if (hx==0x3ff00000 && lx == 0) return one; + + /* y!=zero: result is NaN if either arg is NaN */ + if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || + iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) + return (x+0.0)+(y+0.0); + + /* determine if y is an odd int when x < 0 + * yisint = 0 ... y is not an integer + * yisint = 1 ... y is an odd int + * yisint = 2 ... y is an even int + */ + yisint = 0; + if(hx<0) { + if(iy>=0x43400000) yisint = 2; /* even integer y */ + else if(iy>=0x3ff00000) { + k = (iy>>20)-0x3ff; /* exponent */ + if(k>20) { + j = ly>>(52-k); + if((j<<(52-k))==ly) yisint = 2-(j&1); + } else if(ly==0) { + j = iy>>(20-k); + if((j<<(20-k))==iy) yisint = 2-(j&1); + } + } + } + + /* special value of y */ + if(ly==0) { + if (iy==0x7ff00000) { /* y is +-inf */ + if(((ix-0x3ff00000)|lx)==0) + return one; /* (-1)**+-inf is 1 */ + else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ + return (hy>=0)? y: zero; + else /* (|x|<1)**-,+inf = inf,0 */ + return (hy<0)?-y: zero; + } + if(iy==0x3ff00000) { /* y is +-1 */ + if(hy<0) return one/x; else return x; + } + if(hy==0x40000000) return x*x; /* y is 2 */ + if(hy==0x3fe00000) { /* y is 0.5 */ + if(hx>=0) /* x >= +0 */ + return sqrt(x); + } + } + + ax = fabs(x); + /* special value of x */ + if(lx==0) { + if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ + z = ax; /*x is +-0,+-inf,+-1*/ + if(hy<0) z = one/z; /* z = (1/|x|) */ + if(hx<0) { + if(((ix-0x3ff00000)|yisint)==0) { + z = (z-z)/(z-z); /* (-1)**non-int is NaN */ + } else if(yisint==1) + z = -z; /* (x<0)**odd = -(|x|**odd) */ + } + return z; + } + } + + /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be + n = (hx>>31)+1; + but ANSI C says a right shift of a signed negative quantity is + implementation defined. */ + n = ((u_int32_t)hx>>31)-1; + + /* (x<0)**(non-int) is NaN */ + if((n|yisint)==0) return (x-x)/(x-x); + + s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ + if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ + + /* |y| is huge */ + if(iy>0x41e00000) { /* if |y| > 2**31 */ + if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ + if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; + if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; + } + /* over/underflow if x is not close to one */ + if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny; + if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny; + /* now |1-x| is tiny <= 2**-20, suffice to compute + log(x) by x-x^2/2+x^3/3-x^4/4 */ + t = ax-one; /* t has 20 trailing zeros */ + w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); + u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ + v = t*ivln2_l-w*ivln2; + t1 = u+v; + SET_LOW_WORD(t1,0); + t2 = v-(t1-u); + } else { + double ss,s2,s_h,s_l,t_h,t_l; + n = 0; + /* take care subnormal number */ + if(ix<0x00100000) + {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } + n += ((ix)>>20)-0x3ff; + j = ix&0x000fffff; + /* determine interval */ + ix = j|0x3ff00000; /* normalize ix */ + if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ + else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ + else {k=0;n+=1;ix -= 0x00100000;} + SET_HIGH_WORD(ax,ix); + + /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ + u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ + v = one/(ax+bp[k]); + ss = u*v; + s_h = ss; + SET_LOW_WORD(s_h,0); + /* t_h=ax+bp[k] High */ + t_h = zero; + SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); + t_l = ax - (t_h-bp[k]); + s_l = v*((u-s_h*t_h)-s_h*t_l); + /* compute log(ax) */ + s2 = ss*ss; + r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); + r += s_l*(s_h+ss); + s2 = s_h*s_h; + t_h = 3.0+s2+r; + SET_LOW_WORD(t_h,0); + t_l = r-((t_h-3.0)-s2); + /* u+v = ss*(1+...) */ + u = s_h*t_h; + v = s_l*t_h+t_l*ss; + /* 2/(3log2)*(ss+...) */ + p_h = u+v; + SET_LOW_WORD(p_h,0); + p_l = v-(p_h-u); + z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ + z_l = cp_l*p_h+p_l*cp+dp_l[k]; + /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ + t = (double)n; + t1 = (((z_h+z_l)+dp_h[k])+t); + SET_LOW_WORD(t1,0); + t2 = z_l-(((t1-t)-dp_h[k])-z_h); + } + + /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ + y1 = y; + SET_LOW_WORD(y1,0); + p_l = (y-y1)*t1+y*t2; + p_h = y1*t1; + z = p_l+p_h; + EXTRACT_WORDS(j,i,z); + if (j>=0x40900000) { /* z >= 1024 */ + if(((j-0x40900000)|i)!=0) /* if z > 1024 */ + return s*huge*huge; /* overflow */ + else { + if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ + } + } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ + if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ + return s*tiny*tiny; /* underflow */ + else { + if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ + } + } + /* + * compute 2**(p_h+p_l) + */ + i = j&0x7fffffff; + k = (i>>20)-0x3ff; + n = 0; + if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ + n = j+(0x00100000>>(k+1)); + k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ + t = zero; + SET_HIGH_WORD(t,n&~(0x000fffff>>k)); + n = ((n&0x000fffff)|0x00100000)>>(20-k); + if(j<0) n = -n; + p_h -= t; + } + t = p_l+p_h; + SET_LOW_WORD(t,0); + u = t*lg2_h; + v = (p_l-(t-p_h))*lg2+t*lg2_l; + z = u+v; + w = v-(z-u); + t = z*z; + t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); + r = (z*t1)/(t1-two)-(w+z*w); + z = one-(r-z); + GET_HIGH_WORD(j,z); + j += (n<<20); + if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ + else SET_HIGH_WORD(z,j); + return s*z; +} diff --git a/modules/fdlibm/src/e_sinh.cpp b/modules/fdlibm/src/e_sinh.cpp new file mode 100644 index 000000000..c3418e687 --- /dev/null +++ b/modules/fdlibm/src/e_sinh.cpp @@ -0,0 +1,74 @@ + +/* @(#)e_sinh.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* __ieee754_sinh(x) + * Method : + * mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2 + * 1. Replace x by |x| (sinh(-x) = -sinh(x)). + * 2. + * E + E/(E+1) + * 0 <= x <= 22 : sinh(x) := --------------, E=expm1(x) + * 2 + * + * 22 <= x <= lnovft : sinh(x) := exp(x)/2 + * lnovft <= x <= ln2ovft: sinh(x) := exp(x/2)/2 * exp(x/2) + * ln2ovft < x : sinh(x) := x*shuge (overflow) + * + * Special cases: + * sinh(x) is |x| if x is +INF, -INF, or NaN. + * only sinh(0)=0 is exact for finite x. + */ + +#include <float.h> + +#include "math_private.h" + +static const double one = 1.0, shuge = 1.0e307; + +double +__ieee754_sinh(double x) +{ + double t,h; + int32_t ix,jx; + + /* High word of |x|. */ + GET_HIGH_WORD(jx,x); + ix = jx&0x7fffffff; + + /* x is INF or NaN */ + if(ix>=0x7ff00000) return x+x; + + h = 0.5; + if (jx<0) h = -h; + /* |x| in [0,22], return sign(x)*0.5*(E+E/(E+1))) */ + if (ix < 0x40360000) { /* |x|<22 */ + if (ix<0x3e300000) /* |x|<2**-28 */ + if(shuge+x>one) return x;/* sinh(tiny) = tiny with inexact */ + t = expm1(fabs(x)); + if(ix<0x3ff00000) return h*(2.0*t-t*t/(t+one)); + return h*(t+t/(t+one)); + } + + /* |x| in [22, log(maxdouble)] return 0.5*exp(|x|) */ + if (ix < 0x40862E42) return h*__ieee754_exp(fabs(x)); + + /* |x| in [log(maxdouble), overflowthresold] */ + if (ix<=0x408633CE) + return h*2.0*__ldexp_exp(fabs(x), -1); + + /* |x| > overflowthresold, sinh(x) overflow */ + return x*shuge; +} diff --git a/modules/fdlibm/src/e_sqrt.cpp b/modules/fdlibm/src/e_sqrt.cpp new file mode 100644 index 000000000..681505390 --- /dev/null +++ b/modules/fdlibm/src/e_sqrt.cpp @@ -0,0 +1,446 @@ + +/* @(#)e_sqrt.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* __ieee754_sqrt(x) + * Return correctly rounded sqrt. + * ------------------------------------------ + * | Use the hardware sqrt if you have one | + * ------------------------------------------ + * Method: + * Bit by bit method using integer arithmetic. (Slow, but portable) + * 1. Normalization + * Scale x to y in [1,4) with even powers of 2: + * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then + * sqrt(x) = 2^k * sqrt(y) + * 2. Bit by bit computation + * Let q = sqrt(y) truncated to i bit after binary point (q = 1), + * i 0 + * i+1 2 + * s = 2*q , and y = 2 * ( y - q ). (1) + * i i i i + * + * To compute q from q , one checks whether + * i+1 i + * + * -(i+1) 2 + * (q + 2 ) <= y. (2) + * i + * -(i+1) + * If (2) is false, then q = q ; otherwise q = q + 2 . + * i+1 i i+1 i + * + * With some algebric manipulation, it is not difficult to see + * that (2) is equivalent to + * -(i+1) + * s + 2 <= y (3) + * i i + * + * The advantage of (3) is that s and y can be computed by + * i i + * the following recurrence formula: + * if (3) is false + * + * s = s , y = y ; (4) + * i+1 i i+1 i + * + * otherwise, + * -i -(i+1) + * s = s + 2 , y = y - s - 2 (5) + * i+1 i i+1 i i + * + * One may easily use induction to prove (4) and (5). + * Note. Since the left hand side of (3) contain only i+2 bits, + * it does not necessary to do a full (53-bit) comparison + * in (3). + * 3. Final rounding + * After generating the 53 bits result, we compute one more bit. + * Together with the remainder, we can decide whether the + * result is exact, bigger than 1/2ulp, or less than 1/2ulp + * (it will never equal to 1/2ulp). + * The rounding mode can be detected by checking whether + * huge + tiny is equal to huge, and whether huge - tiny is + * equal to huge for some floating point number "huge" and "tiny". + * + * Special cases: + * sqrt(+-0) = +-0 ... exact + * sqrt(inf) = inf + * sqrt(-ve) = NaN ... with invalid signal + * sqrt(NaN) = NaN ... with invalid signal for signaling NaN + * + * Other methods : see the appended file at the end of the program below. + *--------------- + */ + +#include <float.h> + +#include "math_private.h" + +static const double one = 1.0, tiny=1.0e-300; + +double +__ieee754_sqrt(double x) +{ + double z; + int32_t sign = (int)0x80000000; + int32_t ix0,s0,q,m,t,i; + u_int32_t r,t1,s1,ix1,q1; + + EXTRACT_WORDS(ix0,ix1,x); + + /* take care of Inf and NaN */ + if((ix0&0x7ff00000)==0x7ff00000) { + return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf + sqrt(-inf)=sNaN */ + } + /* take care of zero */ + if(ix0<=0) { + if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */ + else if(ix0<0) + return (x-x)/(x-x); /* sqrt(-ve) = sNaN */ + } + /* normalize x */ + m = (ix0>>20); + if(m==0) { /* subnormal x */ + while(ix0==0) { + m -= 21; + ix0 |= (ix1>>11); ix1 <<= 21; + } + for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1; + m -= i-1; + ix0 |= (ix1>>(32-i)); + ix1 <<= i; + } + m -= 1023; /* unbias exponent */ + ix0 = (ix0&0x000fffff)|0x00100000; + if(m&1){ /* odd m, double x to make it even */ + ix0 += ix0 + ((ix1&sign)>>31); + ix1 += ix1; + } + m >>= 1; /* m = [m/2] */ + + /* generate sqrt(x) bit by bit */ + ix0 += ix0 + ((ix1&sign)>>31); + ix1 += ix1; + q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */ + r = 0x00200000; /* r = moving bit from right to left */ + + while(r!=0) { + t = s0+r; + if(t<=ix0) { + s0 = t+r; + ix0 -= t; + q += r; + } + ix0 += ix0 + ((ix1&sign)>>31); + ix1 += ix1; + r>>=1; + } + + r = sign; + while(r!=0) { + t1 = s1+r; + t = s0; + if((t<ix0)||((t==ix0)&&(t1<=ix1))) { + s1 = t1+r; + if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1; + ix0 -= t; + if (ix1 < t1) ix0 -= 1; + ix1 -= t1; + q1 += r; + } + ix0 += ix0 + ((ix1&sign)>>31); + ix1 += ix1; + r>>=1; + } + + /* use floating add to find out rounding direction */ + if((ix0|ix1)!=0) { + z = one-tiny; /* trigger inexact flag */ + if (z>=one) { + z = one+tiny; + if (q1==(u_int32_t)0xffffffff) { q1=0; q += 1;} + else if (z>one) { + if (q1==(u_int32_t)0xfffffffe) q+=1; + q1+=2; + } else + q1 += (q1&1); + } + } + ix0 = (q>>1)+0x3fe00000; + ix1 = q1>>1; + if ((q&1)==1) ix1 |= sign; + ix0 += (m <<20); + INSERT_WORDS(z,ix0,ix1); + return z; +} + +/* +Other methods (use floating-point arithmetic) +------------- +(This is a copy of a drafted paper by Prof W. Kahan +and K.C. Ng, written in May, 1986) + + Two algorithms are given here to implement sqrt(x) + (IEEE double precision arithmetic) in software. + Both supply sqrt(x) correctly rounded. The first algorithm (in + Section A) uses newton iterations and involves four divisions. + The second one uses reciproot iterations to avoid division, but + requires more multiplications. Both algorithms need the ability + to chop results of arithmetic operations instead of round them, + and the INEXACT flag to indicate when an arithmetic operation + is executed exactly with no roundoff error, all part of the + standard (IEEE 754-1985). The ability to perform shift, add, + subtract and logical AND operations upon 32-bit words is needed + too, though not part of the standard. + +A. sqrt(x) by Newton Iteration + + (1) Initial approximation + + Let x0 and x1 be the leading and the trailing 32-bit words of + a floating point number x (in IEEE double format) respectively + + 1 11 52 ...widths + ------------------------------------------------------ + x: |s| e | f | + ------------------------------------------------------ + msb lsb msb lsb ...order + + + ------------------------ ------------------------ + x0: |s| e | f1 | x1: | f2 | + ------------------------ ------------------------ + + By performing shifts and subtracts on x0 and x1 (both regarded + as integers), we obtain an 8-bit approximation of sqrt(x) as + follows. + + k := (x0>>1) + 0x1ff80000; + y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits + Here k is a 32-bit integer and T1[] is an integer array containing + correction terms. Now magically the floating value of y (y's + leading 32-bit word is y0, the value of its trailing word is 0) + approximates sqrt(x) to almost 8-bit. + + Value of T1: + static int T1[32]= { + 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592, + 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215, + 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581, + 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,}; + + (2) Iterative refinement + + Apply Heron's rule three times to y, we have y approximates + sqrt(x) to within 1 ulp (Unit in the Last Place): + + y := (y+x/y)/2 ... almost 17 sig. bits + y := (y+x/y)/2 ... almost 35 sig. bits + y := y-(y-x/y)/2 ... within 1 ulp + + + Remark 1. + Another way to improve y to within 1 ulp is: + + y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x) + y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x) + + 2 + (x-y )*y + y := y + 2* ---------- ...within 1 ulp + 2 + 3y + x + + + This formula has one division fewer than the one above; however, + it requires more multiplications and additions. Also x must be + scaled in advance to avoid spurious overflow in evaluating the + expression 3y*y+x. Hence it is not recommended uless division + is slow. If division is very slow, then one should use the + reciproot algorithm given in section B. + + (3) Final adjustment + + By twiddling y's last bit it is possible to force y to be + correctly rounded according to the prevailing rounding mode + as follows. Let r and i be copies of the rounding mode and + inexact flag before entering the square root program. Also we + use the expression y+-ulp for the next representable floating + numbers (up and down) of y. Note that y+-ulp = either fixed + point y+-1, or multiply y by nextafter(1,+-inf) in chopped + mode. + + I := FALSE; ... reset INEXACT flag I + R := RZ; ... set rounding mode to round-toward-zero + z := x/y; ... chopped quotient, possibly inexact + If(not I) then { ... if the quotient is exact + if(z=y) { + I := i; ... restore inexact flag + R := r; ... restore rounded mode + return sqrt(x):=y. + } else { + z := z - ulp; ... special rounding + } + } + i := TRUE; ... sqrt(x) is inexact + If (r=RN) then z=z+ulp ... rounded-to-nearest + If (r=RP) then { ... round-toward-+inf + y = y+ulp; z=z+ulp; + } + y := y+z; ... chopped sum + y0:=y0-0x00100000; ... y := y/2 is correctly rounded. + I := i; ... restore inexact flag + R := r; ... restore rounded mode + return sqrt(x):=y. + + (4) Special cases + + Square root of +inf, +-0, or NaN is itself; + Square root of a negative number is NaN with invalid signal. + + +B. sqrt(x) by Reciproot Iteration + + (1) Initial approximation + + Let x0 and x1 be the leading and the trailing 32-bit words of + a floating point number x (in IEEE double format) respectively + (see section A). By performing shifs and subtracts on x0 and y0, + we obtain a 7.8-bit approximation of 1/sqrt(x) as follows. + + k := 0x5fe80000 - (x0>>1); + y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits + + Here k is a 32-bit integer and T2[] is an integer array + containing correction terms. Now magically the floating + value of y (y's leading 32-bit word is y0, the value of + its trailing word y1 is set to zero) approximates 1/sqrt(x) + to almost 7.8-bit. + + Value of T2: + static int T2[64]= { + 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866, + 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f, + 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d, + 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0, + 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989, + 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd, + 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e, + 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,}; + + (2) Iterative refinement + + Apply Reciproot iteration three times to y and multiply the + result by x to get an approximation z that matches sqrt(x) + to about 1 ulp. To be exact, we will have + -1ulp < sqrt(x)-z<1.0625ulp. + + ... set rounding mode to Round-to-nearest + y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x) + y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x) + ... special arrangement for better accuracy + z := x*y ... 29 bits to sqrt(x), with z*y<1 + z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x) + + Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that + (a) the term z*y in the final iteration is always less than 1; + (b) the error in the final result is biased upward so that + -1 ulp < sqrt(x) - z < 1.0625 ulp + instead of |sqrt(x)-z|<1.03125ulp. + + (3) Final adjustment + + By twiddling y's last bit it is possible to force y to be + correctly rounded according to the prevailing rounding mode + as follows. Let r and i be copies of the rounding mode and + inexact flag before entering the square root program. Also we + use the expression y+-ulp for the next representable floating + numbers (up and down) of y. Note that y+-ulp = either fixed + point y+-1, or multiply y by nextafter(1,+-inf) in chopped + mode. + + R := RZ; ... set rounding mode to round-toward-zero + switch(r) { + case RN: ... round-to-nearest + if(x<= z*(z-ulp)...chopped) z = z - ulp; else + if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp; + break; + case RZ:case RM: ... round-to-zero or round-to--inf + R:=RP; ... reset rounding mod to round-to-+inf + if(x<z*z ... rounded up) z = z - ulp; else + if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp; + break; + case RP: ... round-to-+inf + if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else + if(x>z*z ...chopped) z = z+ulp; + break; + } + + Remark 3. The above comparisons can be done in fixed point. For + example, to compare x and w=z*z chopped, it suffices to compare + x1 and w1 (the trailing parts of x and w), regarding them as + two's complement integers. + + ...Is z an exact square root? + To determine whether z is an exact square root of x, let z1 be the + trailing part of z, and also let x0 and x1 be the leading and + trailing parts of x. + + If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0 + I := 1; ... Raise Inexact flag: z is not exact + else { + j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2 + k := z1 >> 26; ... get z's 25-th and 26-th + fraction bits + I := i or (k&j) or ((k&(j+j+1))!=(x1&3)); + } + R:= r ... restore rounded mode + return sqrt(x):=z. + + If multiplication is cheaper then the foregoing red tape, the + Inexact flag can be evaluated by + + I := i; + I := (z*z!=x) or I. + + Note that z*z can overwrite I; this value must be sensed if it is + True. + + Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be + zero. + + -------------------- + z1: | f2 | + -------------------- + bit 31 bit 0 + + Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd + or even of logb(x) have the following relations: + + ------------------------------------------------- + bit 27,26 of z1 bit 1,0 of x1 logb(x) + ------------------------------------------------- + 00 00 odd and even + 01 01 even + 10 10 odd + 10 00 even + 11 01 even + ------------------------------------------------- + + (4) Special cases (see (4) of Section A). + + */ + diff --git a/modules/fdlibm/src/fdlibm.h b/modules/fdlibm/src/fdlibm.h new file mode 100644 index 000000000..0ad215911 --- /dev/null +++ b/modules/fdlibm/src/fdlibm.h @@ -0,0 +1,65 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* + * from: @(#)fdlibm.h 5.1 93/09/24 + * $FreeBSD$ + */ + +#ifndef mozilla_imported_fdlibm_h +#define mozilla_imported_fdlibm_h + +namespace fdlibm { + +double acos(double); +double asin(double); +double atan(double); +double atan2(double, double); + +double cosh(double); +double sinh(double); +double tanh(double); + +double exp(double); +double log(double); +double log10(double); + +double pow(double, double); +double sqrt(double); +double fabs(double); + +double floor(double); +double trunc(double); +double ceil(double); + +double acosh(double); +double asinh(double); +double atanh(double); +double cbrt(double); +double expm1(double); +double hypot(double, double); +double log1p(double); +double log2(double); +double rint(double); +double copysign(double, double); +double nearbyint(double); +double scalbn(double, int); + +float ceilf(float); +float floorf(float); + +float nearbyintf(float); +float rintf(float); +float truncf(float); + +} /* namespace fdlibm */ + +#endif /* mozilla_imported_fdlibm_h */ diff --git a/modules/fdlibm/src/k_exp.cpp b/modules/fdlibm/src/k_exp.cpp new file mode 100644 index 000000000..a0699fa4a --- /dev/null +++ b/modules/fdlibm/src/k_exp.cpp @@ -0,0 +1,81 @@ +/*- + * Copyright (c) 2011 David Schultz <das@FreeBSD.ORG> + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + + #include "math_private.h" + +static const uint32_t k = 1799; /* constant for reduction */ +static const double kln2 = 1246.97177782734161156; /* k * ln2 */ + +/* + * Compute exp(x), scaled to avoid spurious overflow. An exponent is + * returned separately in 'expt'. + * + * Input: ln(DBL_MAX) <= x < ln(2 * DBL_MAX / DBL_MIN_DENORM) ~= 1454.91 + * Output: 2**1023 <= y < 2**1024 + */ +static double +__frexp_exp(double x, int *expt) +{ + double exp_x; + uint32_t hx; + + /* + * We use exp(x) = exp(x - kln2) * 2**k, carefully chosen to + * minimize |exp(kln2) - 2**k|. We also scale the exponent of + * exp_x to MAX_EXP so that the result can be multiplied by + * a tiny number without losing accuracy due to denormalization. + */ + exp_x = exp(x - kln2); + GET_HIGH_WORD(hx, exp_x); + *expt = (hx >> 20) - (0x3ff + 1023) + k; + SET_HIGH_WORD(exp_x, (hx & 0xfffff) | ((0x3ff + 1023) << 20)); + return (exp_x); +} + +/* + * __ldexp_exp(x, expt) and __ldexp_cexp(x, expt) compute exp(x) * 2**expt. + * They are intended for large arguments (real part >= ln(DBL_MAX)) + * where care is needed to avoid overflow. + * + * The present implementation is narrowly tailored for our hyperbolic and + * exponential functions. We assume expt is small (0 or -1), and the caller + * has filtered out very large x, for which overflow would be inevitable. + */ + +double +__ldexp_exp(double x, int expt) +{ + double exp_x, scale; + int ex_expt; + + exp_x = __frexp_exp(x, &ex_expt); + expt += ex_expt; + INSERT_WORDS(scale, (0x3ff + expt) << 20, 0); + return (exp_x * scale); +} diff --git a/modules/fdlibm/src/k_log.h b/modules/fdlibm/src/k_log.h new file mode 100644 index 000000000..0efa020f6 --- /dev/null +++ b/modules/fdlibm/src/k_log.h @@ -0,0 +1,100 @@ + +/* @(#)e_log.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* + * k_log1p(f): + * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)]. + * + * The following describes the overall strategy for computing + * logarithms in base e. The argument reduction and adding the final + * term of the polynomial are done by the caller for increased accuracy + * when different bases are used. + * + * Method : + * 1. Argument Reduction: find k and f such that + * x = 2^k * (1+f), + * where sqrt(2)/2 < 1+f < sqrt(2) . + * + * 2. Approximation of log(1+f). + * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) + * = 2s + 2/3 s**3 + 2/5 s**5 + ....., + * = 2s + s*R + * We use a special Reme algorithm on [0,0.1716] to generate + * a polynomial of degree 14 to approximate R The maximum error + * of this polynomial approximation is bounded by 2**-58.45. In + * other words, + * 2 4 6 8 10 12 14 + * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s + * (the values of Lg1 to Lg7 are listed in the program) + * and + * | 2 14 | -58.45 + * | Lg1*s +...+Lg7*s - R(z) | <= 2 + * | | + * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. + * In order to guarantee error in log below 1ulp, we compute log + * by + * log(1+f) = f - s*(f - R) (if f is not too large) + * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) + * + * 3. Finally, log(x) = k*ln2 + log(1+f). + * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) + * Here ln2 is split into two floating point number: + * ln2_hi + ln2_lo, + * where n*ln2_hi is always exact for |n| < 2000. + * + * Special cases: + * log(x) is NaN with signal if x < 0 (including -INF) ; + * log(+INF) is +INF; log(0) is -INF with signal; + * log(NaN) is that NaN with no signal. + * + * Accuracy: + * according to an error analysis, the error is always less than + * 1 ulp (unit in the last place). + * + * Constants: + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough + * to produce the hexadecimal values shown. + */ + +static const double +Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ +Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ +Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ +Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ +Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ +Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ +Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ + +/* + * We always inline k_log1p(), since doing so produces a + * substantial performance improvement (~40% on amd64). + */ +static inline double +k_log1p(double f) +{ + double hfsq,s,z,R,w,t1,t2; + + s = f/(2.0+f); + z = s*s; + w = z*z; + t1= w*(Lg2+w*(Lg4+w*Lg6)); + t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); + R = t2+t1; + hfsq=0.5*f*f; + return s*(hfsq+R); +} diff --git a/modules/fdlibm/src/math_private.h b/modules/fdlibm/src/math_private.h new file mode 100644 index 000000000..6947cecc0 --- /dev/null +++ b/modules/fdlibm/src/math_private.h @@ -0,0 +1,822 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* + * from: @(#)fdlibm.h 5.1 93/09/24 + * $FreeBSD$ + */ + +#ifndef _MATH_PRIVATE_H_ +#define _MATH_PRIVATE_H_ + +#include <cfloat> +#include <stdint.h> +#include <sys/types.h> + +#include "fdlibm.h" + +#include "mozilla/EndianUtils.h" + +/* + * The original fdlibm code used statements like: + * n0 = ((*(int*)&one)>>29)^1; * index of high word * + * ix0 = *(n0+(int*)&x); * high word of x * + * ix1 = *((1-n0)+(int*)&x); * low word of x * + * to dig two 32 bit words out of the 64 bit IEEE floating point + * value. That is non-ANSI, and, moreover, the gcc instruction + * scheduler gets it wrong. We instead use the following macros. + * Unlike the original code, we determine the endianness at compile + * time, not at run time; I don't see much benefit to selecting + * endianness at run time. + */ + +#ifdef WIN32 +#define u_int32_t uint32_t +#define u_int64_t uint64_t +#endif + +/* + * A union which permits us to convert between a double and two 32 bit + * ints. + */ + +#if MOZ_BIG_ENDIAN + +typedef union +{ + double value; + struct + { + u_int32_t msw; + u_int32_t lsw; + } parts; + struct + { + u_int64_t w; + } xparts; +} ieee_double_shape_type; + +#endif + +#if MOZ_LITTLE_ENDIAN + +typedef union +{ + double value; + struct + { + u_int32_t lsw; + u_int32_t msw; + } parts; + struct + { + u_int64_t w; + } xparts; +} ieee_double_shape_type; + +#endif + +/* Get two 32 bit ints from a double. */ + +#define EXTRACT_WORDS(ix0,ix1,d) \ +do { \ + ieee_double_shape_type ew_u; \ + ew_u.value = (d); \ + (ix0) = ew_u.parts.msw; \ + (ix1) = ew_u.parts.lsw; \ +} while (0) + +/* Get a 64-bit int from a double. */ +#define EXTRACT_WORD64(ix,d) \ +do { \ + ieee_double_shape_type ew_u; \ + ew_u.value = (d); \ + (ix) = ew_u.xparts.w; \ +} while (0) + +/* Get the more significant 32 bit int from a double. */ + +#define GET_HIGH_WORD(i,d) \ +do { \ + ieee_double_shape_type gh_u; \ + gh_u.value = (d); \ + (i) = gh_u.parts.msw; \ +} while (0) + +/* Get the less significant 32 bit int from a double. */ + +#define GET_LOW_WORD(i,d) \ +do { \ + ieee_double_shape_type gl_u; \ + gl_u.value = (d); \ + (i) = gl_u.parts.lsw; \ +} while (0) + +/* Set a double from two 32 bit ints. */ + +#define INSERT_WORDS(d,ix0,ix1) \ +do { \ + ieee_double_shape_type iw_u; \ + iw_u.parts.msw = (ix0); \ + iw_u.parts.lsw = (ix1); \ + (d) = iw_u.value; \ +} while (0) + +/* Set a double from a 64-bit int. */ +#define INSERT_WORD64(d,ix) \ +do { \ + ieee_double_shape_type iw_u; \ + iw_u.xparts.w = (ix); \ + (d) = iw_u.value; \ +} while (0) + +/* Set the more significant 32 bits of a double from an int. */ + +#define SET_HIGH_WORD(d,v) \ +do { \ + ieee_double_shape_type sh_u; \ + sh_u.value = (d); \ + sh_u.parts.msw = (v); \ + (d) = sh_u.value; \ +} while (0) + +/* Set the less significant 32 bits of a double from an int. */ + +#define SET_LOW_WORD(d,v) \ +do { \ + ieee_double_shape_type sl_u; \ + sl_u.value = (d); \ + sl_u.parts.lsw = (v); \ + (d) = sl_u.value; \ +} while (0) + +/* + * A union which permits us to convert between a float and a 32 bit + * int. + */ + +typedef union +{ + float value; + /* FIXME: Assumes 32 bit int. */ + unsigned int word; +} ieee_float_shape_type; + +/* Get a 32 bit int from a float. */ + +#define GET_FLOAT_WORD(i,d) \ +do { \ + ieee_float_shape_type gf_u; \ + gf_u.value = (d); \ + (i) = gf_u.word; \ +} while (0) + +/* Set a float from a 32 bit int. */ + +#define SET_FLOAT_WORD(d,i) \ +do { \ + ieee_float_shape_type sf_u; \ + sf_u.word = (i); \ + (d) = sf_u.value; \ +} while (0) + +/* + * Get expsign and mantissa as 16 bit and 64 bit ints from an 80 bit long + * double. + */ + +#define EXTRACT_LDBL80_WORDS(ix0,ix1,d) \ +do { \ + union IEEEl2bits ew_u; \ + ew_u.e = (d); \ + (ix0) = ew_u.xbits.expsign; \ + (ix1) = ew_u.xbits.man; \ +} while (0) + +/* + * Get expsign and mantissa as one 16 bit and two 64 bit ints from a 128 bit + * long double. + */ + +#define EXTRACT_LDBL128_WORDS(ix0,ix1,ix2,d) \ +do { \ + union IEEEl2bits ew_u; \ + ew_u.e = (d); \ + (ix0) = ew_u.xbits.expsign; \ + (ix1) = ew_u.xbits.manh; \ + (ix2) = ew_u.xbits.manl; \ +} while (0) + +/* Get expsign as a 16 bit int from a long double. */ + +#define GET_LDBL_EXPSIGN(i,d) \ +do { \ + union IEEEl2bits ge_u; \ + ge_u.e = (d); \ + (i) = ge_u.xbits.expsign; \ +} while (0) + +/* + * Set an 80 bit long double from a 16 bit int expsign and a 64 bit int + * mantissa. + */ + +#define INSERT_LDBL80_WORDS(d,ix0,ix1) \ +do { \ + union IEEEl2bits iw_u; \ + iw_u.xbits.expsign = (ix0); \ + iw_u.xbits.man = (ix1); \ + (d) = iw_u.e; \ +} while (0) + +/* + * Set a 128 bit long double from a 16 bit int expsign and two 64 bit ints + * comprising the mantissa. + */ + +#define INSERT_LDBL128_WORDS(d,ix0,ix1,ix2) \ +do { \ + union IEEEl2bits iw_u; \ + iw_u.xbits.expsign = (ix0); \ + iw_u.xbits.manh = (ix1); \ + iw_u.xbits.manl = (ix2); \ + (d) = iw_u.e; \ +} while (0) + +/* Set expsign of a long double from a 16 bit int. */ + +#define SET_LDBL_EXPSIGN(d,v) \ +do { \ + union IEEEl2bits se_u; \ + se_u.e = (d); \ + se_u.xbits.expsign = (v); \ + (d) = se_u.e; \ +} while (0) + +#ifdef __i386__ +/* Long double constants are broken on i386. */ +#define LD80C(m, ex, v) { \ + .xbits.man = __CONCAT(m, ULL), \ + .xbits.expsign = (0x3fff + (ex)) | ((v) < 0 ? 0x8000 : 0), \ +} +#else +/* The above works on non-i386 too, but we use this to check v. */ +#define LD80C(m, ex, v) { .e = (v), } +#endif + +#ifdef FLT_EVAL_METHOD +/* + * Attempt to get strict C99 semantics for assignment with non-C99 compilers. + */ +#if !defined(_MSC_VER) && (FLT_EVAL_METHOD == 0 || __GNUC__ == 0) +#define STRICT_ASSIGN(type, lval, rval) ((lval) = (rval)) +#else +#define STRICT_ASSIGN(type, lval, rval) do { \ + volatile type __lval; \ + \ + if (sizeof(type) >= sizeof(long double)) \ + (lval) = (rval); \ + else { \ + __lval = (rval); \ + (lval) = __lval; \ + } \ +} while (0) +#endif +#else +#define STRICT_ASSIGN(type, lval, rval) do { \ + volatile type __lval; \ + \ + if (sizeof(type) >= sizeof(long double)) \ + (lval) = (rval); \ + else { \ + __lval = (rval); \ + (lval) = __lval; \ + } \ +} while (0) +#endif /* FLT_EVAL_METHOD */ + +/* Support switching the mode to FP_PE if necessary. */ +#if defined(__i386__) && !defined(NO_FPSETPREC) +#define ENTERI() \ + long double __retval; \ + fp_prec_t __oprec; \ + \ + if ((__oprec = fpgetprec()) != FP_PE) \ + fpsetprec(FP_PE) +#define RETURNI(x) do { \ + __retval = (x); \ + if (__oprec != FP_PE) \ + fpsetprec(__oprec); \ + RETURNF(__retval); \ +} while (0) +#else +#define ENTERI(x) +#define RETURNI(x) RETURNF(x) +#endif + +/* Default return statement if hack*_t() is not used. */ +#define RETURNF(v) return (v) + +/* + * 2sum gives the same result as 2sumF without requiring |a| >= |b| or + * a == 0, but is slower. + */ +#define _2sum(a, b) do { \ + __typeof(a) __s, __w; \ + \ + __w = (a) + (b); \ + __s = __w - (a); \ + (b) = ((a) - (__w - __s)) + ((b) - __s); \ + (a) = __w; \ +} while (0) + +/* + * 2sumF algorithm. + * + * "Normalize" the terms in the infinite-precision expression a + b for + * the sum of 2 floating point values so that b is as small as possible + * relative to 'a'. (The resulting 'a' is the value of the expression in + * the same precision as 'a' and the resulting b is the rounding error.) + * |a| must be >= |b| or 0, b's type must be no larger than 'a's type, and + * exponent overflow or underflow must not occur. This uses a Theorem of + * Dekker (1971). See Knuth (1981) 4.2.2 Theorem C. The name "TwoSum" + * is apparently due to Skewchuk (1997). + * + * For this to always work, assignment of a + b to 'a' must not retain any + * extra precision in a + b. This is required by C standards but broken + * in many compilers. The brokenness cannot be worked around using + * STRICT_ASSIGN() like we do elsewhere, since the efficiency of this + * algorithm would be destroyed by non-null strict assignments. (The + * compilers are correct to be broken -- the efficiency of all floating + * point code calculations would be destroyed similarly if they forced the + * conversions.) + * + * Fortunately, a case that works well can usually be arranged by building + * any extra precision into the type of 'a' -- 'a' should have type float_t, + * double_t or long double. b's type should be no larger than 'a's type. + * Callers should use these types with scopes as large as possible, to + * reduce their own extra-precision and efficiciency problems. In + * particular, they shouldn't convert back and forth just to call here. + */ +#ifdef DEBUG +#define _2sumF(a, b) do { \ + __typeof(a) __w; \ + volatile __typeof(a) __ia, __ib, __r, __vw; \ + \ + __ia = (a); \ + __ib = (b); \ + assert(__ia == 0 || fabsl(__ia) >= fabsl(__ib)); \ + \ + __w = (a) + (b); \ + (b) = ((a) - __w) + (b); \ + (a) = __w; \ + \ + /* The next 2 assertions are weak if (a) is already long double. */ \ + assert((long double)__ia + __ib == (long double)(a) + (b)); \ + __vw = __ia + __ib; \ + __r = __ia - __vw; \ + __r += __ib; \ + assert(__vw == (a) && __r == (b)); \ +} while (0) +#else /* !DEBUG */ +#define _2sumF(a, b) do { \ + __typeof(a) __w; \ + \ + __w = (a) + (b); \ + (b) = ((a) - __w) + (b); \ + (a) = __w; \ +} while (0) +#endif /* DEBUG */ + +/* + * Set x += c, where x is represented in extra precision as a + b. + * x must be sufficiently normalized and sufficiently larger than c, + * and the result is then sufficiently normalized. + * + * The details of ordering are that |a| must be >= |c| (so that (a, c) + * can be normalized without extra work to swap 'a' with c). The details of + * the normalization are that b must be small relative to the normalized 'a'. + * Normalization of (a, c) makes the normalized c tiny relative to the + * normalized a, so b remains small relative to 'a' in the result. However, + * b need not ever be tiny relative to 'a'. For example, b might be about + * 2**20 times smaller than 'a' to give about 20 extra bits of precision. + * That is usually enough, and adding c (which by normalization is about + * 2**53 times smaller than a) cannot change b significantly. However, + * cancellation of 'a' with c in normalization of (a, c) may reduce 'a' + * significantly relative to b. The caller must ensure that significant + * cancellation doesn't occur, either by having c of the same sign as 'a', + * or by having |c| a few percent smaller than |a|. Pre-normalization of + * (a, b) may help. + * + * This is is a variant of an algorithm of Kahan (see Knuth (1981) 4.2.2 + * exercise 19). We gain considerable efficiency by requiring the terms to + * be sufficiently normalized and sufficiently increasing. + */ +#define _3sumF(a, b, c) do { \ + __typeof(a) __tmp; \ + \ + __tmp = (c); \ + _2sumF(__tmp, (a)); \ + (b) += (a); \ + (a) = __tmp; \ +} while (0) + +/* + * Common routine to process the arguments to nan(), nanf(), and nanl(). + */ +void _scan_nan(uint32_t *__words, int __num_words, const char *__s); + +#ifdef _COMPLEX_H + +/* + * C99 specifies that complex numbers have the same representation as + * an array of two elements, where the first element is the real part + * and the second element is the imaginary part. + */ +typedef union { + float complex f; + float a[2]; +} float_complex; +typedef union { + double complex f; + double a[2]; +} double_complex; +typedef union { + long double complex f; + long double a[2]; +} long_double_complex; +#define REALPART(z) ((z).a[0]) +#define IMAGPART(z) ((z).a[1]) + +/* + * Inline functions that can be used to construct complex values. + * + * The C99 standard intends x+I*y to be used for this, but x+I*y is + * currently unusable in general since gcc introduces many overflow, + * underflow, sign and efficiency bugs by rewriting I*y as + * (0.0+I)*(y+0.0*I) and laboriously computing the full complex product. + * In particular, I*Inf is corrupted to NaN+I*Inf, and I*-0 is corrupted + * to -0.0+I*0.0. + * + * The C11 standard introduced the macros CMPLX(), CMPLXF() and CMPLXL() + * to construct complex values. Compilers that conform to the C99 + * standard require the following functions to avoid the above issues. + */ + +#ifndef CMPLXF +static __inline float complex +CMPLXF(float x, float y) +{ + float_complex z; + + REALPART(z) = x; + IMAGPART(z) = y; + return (z.f); +} +#endif + +#ifndef CMPLX +static __inline double complex +CMPLX(double x, double y) +{ + double_complex z; + + REALPART(z) = x; + IMAGPART(z) = y; + return (z.f); +} +#endif + +#ifndef CMPLXL +static __inline long double complex +CMPLXL(long double x, long double y) +{ + long_double_complex z; + + REALPART(z) = x; + IMAGPART(z) = y; + return (z.f); +} +#endif + +#endif /* _COMPLEX_H */ + +#ifdef __GNUCLIKE_ASM + +/* Asm versions of some functions. */ + +#ifdef __amd64__ +static __inline int +irint(double x) +{ + int n; + + asm("cvtsd2si %1,%0" : "=r" (n) : "x" (x)); + return (n); +} +#define HAVE_EFFICIENT_IRINT +#endif + +#ifdef __i386__ +static __inline int +irint(double x) +{ + int n; + + asm("fistl %0" : "=m" (n) : "t" (x)); + return (n); +} +#define HAVE_EFFICIENT_IRINT +#endif + +#if defined(__amd64__) || defined(__i386__) +static __inline int +irintl(long double x) +{ + int n; + + asm("fistl %0" : "=m" (n) : "t" (x)); + return (n); +} +#define HAVE_EFFICIENT_IRINTL +#endif + +#endif /* __GNUCLIKE_ASM */ + +#ifdef DEBUG +#if defined(__amd64__) || defined(__i386__) +#define breakpoint() asm("int $3") +#else +#include <signal.h> + +#define breakpoint() raise(SIGTRAP) +#endif +#endif + +/* Write a pari script to test things externally. */ +#ifdef DOPRINT +#include <stdio.h> + +#ifndef DOPRINT_SWIZZLE +#define DOPRINT_SWIZZLE 0 +#endif + +#ifdef DOPRINT_LD80 + +#define DOPRINT_START(xp) do { \ + uint64_t __lx; \ + uint16_t __hx; \ + \ + /* Hack to give more-problematic args. */ \ + EXTRACT_LDBL80_WORDS(__hx, __lx, *xp); \ + __lx ^= DOPRINT_SWIZZLE; \ + INSERT_LDBL80_WORDS(*xp, __hx, __lx); \ + printf("x = %.21Lg; ", (long double)*xp); \ +} while (0) +#define DOPRINT_END1(v) \ + printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v)) +#define DOPRINT_END2(hi, lo) \ + printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \ + (long double)(hi), (long double)(lo)) + +#elif defined(DOPRINT_D64) + +#define DOPRINT_START(xp) do { \ + uint32_t __hx, __lx; \ + \ + EXTRACT_WORDS(__hx, __lx, *xp); \ + __lx ^= DOPRINT_SWIZZLE; \ + INSERT_WORDS(*xp, __hx, __lx); \ + printf("x = %.21Lg; ", (long double)*xp); \ +} while (0) +#define DOPRINT_END1(v) \ + printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v)) +#define DOPRINT_END2(hi, lo) \ + printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \ + (long double)(hi), (long double)(lo)) + +#elif defined(DOPRINT_F32) + +#define DOPRINT_START(xp) do { \ + uint32_t __hx; \ + \ + GET_FLOAT_WORD(__hx, *xp); \ + __hx ^= DOPRINT_SWIZZLE; \ + SET_FLOAT_WORD(*xp, __hx); \ + printf("x = %.21Lg; ", (long double)*xp); \ +} while (0) +#define DOPRINT_END1(v) \ + printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v)) +#define DOPRINT_END2(hi, lo) \ + printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \ + (long double)(hi), (long double)(lo)) + +#else /* !DOPRINT_LD80 && !DOPRINT_D64 (LD128 only) */ + +#ifndef DOPRINT_SWIZZLE_HIGH +#define DOPRINT_SWIZZLE_HIGH 0 +#endif + +#define DOPRINT_START(xp) do { \ + uint64_t __lx, __llx; \ + uint16_t __hx; \ + \ + EXTRACT_LDBL128_WORDS(__hx, __lx, __llx, *xp); \ + __llx ^= DOPRINT_SWIZZLE; \ + __lx ^= DOPRINT_SWIZZLE_HIGH; \ + INSERT_LDBL128_WORDS(*xp, __hx, __lx, __llx); \ + printf("x = %.36Lg; ", (long double)*xp); \ +} while (0) +#define DOPRINT_END1(v) \ + printf("y = %.36Lg; z = 0; show(x, y, z);\n", (long double)(v)) +#define DOPRINT_END2(hi, lo) \ + printf("y = %.36Lg; z = %.36Lg; show(x, y, z);\n", \ + (long double)(hi), (long double)(lo)) + +#endif /* DOPRINT_LD80 */ + +#else /* !DOPRINT */ +#define DOPRINT_START(xp) +#define DOPRINT_END1(v) +#define DOPRINT_END2(hi, lo) +#endif /* DOPRINT */ + +#define RETURNP(x) do { \ + DOPRINT_END1(x); \ + RETURNF(x); \ +} while (0) +#define RETURNPI(x) do { \ + DOPRINT_END1(x); \ + RETURNI(x); \ +} while (0) +#define RETURN2P(x, y) do { \ + DOPRINT_END2((x), (y)); \ + RETURNF((x) + (y)); \ +} while (0) +#define RETURN2PI(x, y) do { \ + DOPRINT_END2((x), (y)); \ + RETURNI((x) + (y)); \ +} while (0) +#ifdef STRUCT_RETURN +#define RETURNSP(rp) do { \ + if (!(rp)->lo_set) \ + RETURNP((rp)->hi); \ + RETURN2P((rp)->hi, (rp)->lo); \ +} while (0) +#define RETURNSPI(rp) do { \ + if (!(rp)->lo_set) \ + RETURNPI((rp)->hi); \ + RETURN2PI((rp)->hi, (rp)->lo); \ +} while (0) +#endif +#define SUM2P(x, y) ({ \ + const __typeof (x) __x = (x); \ + const __typeof (y) __y = (y); \ + \ + DOPRINT_END2(__x, __y); \ + __x + __y; \ +}) + +/* + * ieee style elementary functions + * + * We rename functions here to improve other sources' diffability + * against fdlibm. + */ +#define __ieee754_sqrt sqrt +#define __ieee754_acos acos +#define __ieee754_acosh acosh +#define __ieee754_log log +#define __ieee754_log2 log2 +#define __ieee754_atanh atanh +#define __ieee754_asin asin +#define __ieee754_atan2 atan2 +#define __ieee754_exp exp +#define __ieee754_cosh cosh +#define __ieee754_fmod fmod +#define __ieee754_pow pow +#define __ieee754_lgamma lgamma +#define __ieee754_gamma gamma +#define __ieee754_lgamma_r lgamma_r +#define __ieee754_gamma_r gamma_r +#define __ieee754_log10 log10 +#define __ieee754_sinh sinh +#define __ieee754_hypot hypot +#define __ieee754_j0 j0 +#define __ieee754_j1 j1 +#define __ieee754_y0 y0 +#define __ieee754_y1 y1 +#define __ieee754_jn jn +#define __ieee754_yn yn +#define __ieee754_remainder remainder +#define __ieee754_scalb scalb +#define __ieee754_sqrtf sqrtf +#define __ieee754_acosf acosf +#define __ieee754_acoshf acoshf +#define __ieee754_logf logf +#define __ieee754_atanhf atanhf +#define __ieee754_asinf asinf +#define __ieee754_atan2f atan2f +#define __ieee754_expf expf +#define __ieee754_coshf coshf +#define __ieee754_fmodf fmodf +#define __ieee754_powf powf +#define __ieee754_lgammaf lgammaf +#define __ieee754_gammaf gammaf +#define __ieee754_lgammaf_r lgammaf_r +#define __ieee754_gammaf_r gammaf_r +#define __ieee754_log10f log10f +#define __ieee754_log2f log2f +#define __ieee754_sinhf sinhf +#define __ieee754_hypotf hypotf +#define __ieee754_j0f j0f +#define __ieee754_j1f j1f +#define __ieee754_y0f y0f +#define __ieee754_y1f y1f +#define __ieee754_jnf jnf +#define __ieee754_ynf ynf +#define __ieee754_remainderf remainderf +#define __ieee754_scalbf scalbf + +#define acos fdlibm::acos +#define asin fdlibm::asin +#define atan fdlibm::atan +#define atan2 fdlibm::atan2 +#define cosh fdlibm::cosh +#define sinh fdlibm::sinh +#define tanh fdlibm::tanh +#define exp fdlibm::exp +#define log fdlibm::log +#define log10 fdlibm::log10 +#define pow fdlibm::pow +#define sqrt fdlibm::sqrt +#define ceil fdlibm::ceil +#define ceilf fdlibm::ceilf +#define fabs fdlibm::fabs +#define floor fdlibm::floor +#define acosh fdlibm::acosh +#define asinh fdlibm::asinh +#define atanh fdlibm::atanh +#define cbrt fdlibm::cbrt +#define expm1 fdlibm::expm1 +#define hypot fdlibm::hypot +#define log1p fdlibm::log1p +#define log2 fdlibm::log2 +#define scalb fdlibm::scalb +#define copysign fdlibm::copysign +#define scalbn fdlibm::scalbn +#define trunc fdlibm::trunc +#define truncf fdlibm::truncf +#define floorf fdlibm::floorf +#define nearbyint fdlibm::nearbyint +#define nearbyintf fdlibm::nearbyintf +#define rint fdlibm::rint +#define rintf fdlibm::rintf + +/* fdlibm kernel function */ +int __kernel_rem_pio2(double*,double*,int,int,int); + +/* double precision kernel functions */ +#ifndef INLINE_REM_PIO2 +int __ieee754_rem_pio2(double,double*); +#endif +double __kernel_sin(double,double,int); +double __kernel_cos(double,double); +double __kernel_tan(double,double,int); +double __ldexp_exp(double,int); +#ifdef _COMPLEX_H +double complex __ldexp_cexp(double complex,int); +#endif + +/* float precision kernel functions */ +#ifndef INLINE_REM_PIO2F +int __ieee754_rem_pio2f(float,double*); +#endif +#ifndef INLINE_KERNEL_SINDF +float __kernel_sindf(double); +#endif +#ifndef INLINE_KERNEL_COSDF +float __kernel_cosdf(double); +#endif +#ifndef INLINE_KERNEL_TANDF +float __kernel_tandf(double,int); +#endif +float __ldexp_expf(float,int); +#ifdef _COMPLEX_H +float complex __ldexp_cexpf(float complex,int); +#endif + +/* long double precision kernel functions */ +long double __kernel_sinl(long double, long double, int); +long double __kernel_cosl(long double, long double); +long double __kernel_tanl(long double, long double, int); + +#endif /* !_MATH_PRIVATE_H_ */ diff --git a/modules/fdlibm/src/moz.build b/modules/fdlibm/src/moz.build new file mode 100644 index 000000000..b197881ac --- /dev/null +++ b/modules/fdlibm/src/moz.build @@ -0,0 +1,67 @@ +# -*- Mode: python; indent-tabs-mode: nil; tab-width: 40 -*- +# vim: set filetype=python: +# This Source Code Form is subject to the terms of the Mozilla Public +# License, v. 2.0. If a copy of the MPL was not distributed with this +# file, You can obtain one at http://mozilla.org/MPL/2.0/. + +EXPORTS += [ + 'fdlibm.h', +] + +FINAL_LIBRARY = 'js' + +if CONFIG['GNU_CXX']: + CXXFLAGS += [ + '-Wno-parentheses', + '-Wno-sign-compare', + ] + +if CONFIG['CLANG_CXX']: + CXXFLAGS += [ + '-Wno-dangling-else', + ] + +if CONFIG['_MSC_VER']: + CXXFLAGS += [ + '-wd4018', # signed/unsigned mismatch + '-wd4146', # unary minus operator applied to unsigned type + '-wd4305', # truncation from 'double' to 'const float' + '-wd4723', # potential divide by 0 + '-wd4756', # overflow in constant arithmetic + ] + +SOURCES += [ + 'e_acos.cpp', + 'e_acosh.cpp', + 'e_asin.cpp', + 'e_atan2.cpp', + 'e_atanh.cpp', + 'e_cosh.cpp', + 'e_exp.cpp', + 'e_hypot.cpp', + 'e_log.cpp', + 'e_log10.cpp', + 'e_log2.cpp', + 'e_pow.cpp', + 'e_sinh.cpp', + 'e_sqrt.cpp', + 'k_exp.cpp', + 's_asinh.cpp', + 's_atan.cpp', + 's_cbrt.cpp', + 's_ceil.cpp', + 's_ceilf.cpp', + 's_copysign.cpp', + 's_expm1.cpp', + 's_fabs.cpp', + 's_floor.cpp', + 's_floorf.cpp', + 's_log1p.cpp', + 's_nearbyint.cpp', + 's_rint.cpp', + 's_rintf.cpp', + 's_scalbn.cpp', + 's_tanh.cpp', + 's_trunc.cpp', + 's_truncf.cpp', +] diff --git a/modules/fdlibm/src/s_asinh.cpp b/modules/fdlibm/src/s_asinh.cpp new file mode 100644 index 000000000..400fd89c1 --- /dev/null +++ b/modules/fdlibm/src/s_asinh.cpp @@ -0,0 +1,57 @@ +/* @(#)s_asinh.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* asinh(x) + * Method : + * Based on + * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ] + * we have + * asinh(x) := x if 1+x*x=1, + * := sign(x)*(log(x)+ln2)) for large |x|, else + * := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else + * := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2))) + */ + +#include <float.h> + +#include "math_private.h" + +static const double +one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ +ln2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ +huge= 1.00000000000000000000e+300; + +double +asinh(double x) +{ + double t,w; + int32_t hx,ix; + GET_HIGH_WORD(hx,x); + ix = hx&0x7fffffff; + if(ix>=0x7ff00000) return x+x; /* x is inf or NaN */ + if(ix< 0x3e300000) { /* |x|<2**-28 */ + if(huge+x>one) return x; /* return x inexact except 0 */ + } + if(ix>0x41b00000) { /* |x| > 2**28 */ + w = __ieee754_log(fabs(x))+ln2; + } else if (ix>0x40000000) { /* 2**28 > |x| > 2.0 */ + t = fabs(x); + w = __ieee754_log(2.0*t+one/(__ieee754_sqrt(x*x+one)+t)); + } else { /* 2.0 > |x| > 2**-28 */ + t = x*x; + w =log1p(fabs(x)+t/(one+__ieee754_sqrt(one+t))); + } + if(hx>0) return w; else return -w; +} diff --git a/modules/fdlibm/src/s_atan.cpp b/modules/fdlibm/src/s_atan.cpp new file mode 100644 index 000000000..21bc0d820 --- /dev/null +++ b/modules/fdlibm/src/s_atan.cpp @@ -0,0 +1,119 @@ +/* @(#)s_atan.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* atan(x) + * Method + * 1. Reduce x to positive by atan(x) = -atan(-x). + * 2. According to the integer k=4t+0.25 chopped, t=x, the argument + * is further reduced to one of the following intervals and the + * arctangent of t is evaluated by the corresponding formula: + * + * [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...) + * [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) ) + * [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) ) + * [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) ) + * [39/16,INF] atan(x) = atan(INF) + atan( -1/t ) + * + * Constants: + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough + * to produce the hexadecimal values shown. + */ + +#include <float.h> + +#include "math_private.h" + +static const double atanhi[] = { + 4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */ + 7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */ + 9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */ + 1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */ +}; + +static const double atanlo[] = { + 2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */ + 3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */ + 1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */ + 6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */ +}; + +static const double aT[] = { + 3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */ + -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */ + 1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */ + -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */ + 9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */ + -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */ + 6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */ + -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */ + 4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */ + -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */ + 1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */ +}; + + static const double +one = 1.0, +huge = 1.0e300; + +double +atan(double x) +{ + double w,s1,s2,z; + int32_t ix,hx,id; + + GET_HIGH_WORD(hx,x); + ix = hx&0x7fffffff; + if(ix>=0x44100000) { /* if |x| >= 2^66 */ + u_int32_t low; + GET_LOW_WORD(low,x); + if(ix>0x7ff00000|| + (ix==0x7ff00000&&(low!=0))) + return x+x; /* NaN */ + if(hx>0) return atanhi[3]+*(volatile double *)&atanlo[3]; + else return -atanhi[3]-*(volatile double *)&atanlo[3]; + } if (ix < 0x3fdc0000) { /* |x| < 0.4375 */ + if (ix < 0x3e400000) { /* |x| < 2^-27 */ + if(huge+x>one) return x; /* raise inexact */ + } + id = -1; + } else { + x = fabs(x); + if (ix < 0x3ff30000) { /* |x| < 1.1875 */ + if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */ + id = 0; x = (2.0*x-one)/(2.0+x); + } else { /* 11/16<=|x|< 19/16 */ + id = 1; x = (x-one)/(x+one); + } + } else { + if (ix < 0x40038000) { /* |x| < 2.4375 */ + id = 2; x = (x-1.5)/(one+1.5*x); + } else { /* 2.4375 <= |x| < 2^66 */ + id = 3; x = -1.0/x; + } + }} + /* end of argument reduction */ + z = x*x; + w = z*z; + /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */ + s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10]))))); + s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9])))); + if (id<0) return x - x*(s1+s2); + else { + z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x); + return (hx<0)? -z:z; + } +} diff --git a/modules/fdlibm/src/s_cbrt.cpp b/modules/fdlibm/src/s_cbrt.cpp new file mode 100644 index 000000000..a2de24af7 --- /dev/null +++ b/modules/fdlibm/src/s_cbrt.cpp @@ -0,0 +1,112 @@ +/* @(#)s_cbrt.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + * Optimized by Bruce D. Evans. + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +#include "math_private.h" + +/* cbrt(x) + * Return cube root of x + */ +static const u_int32_t + B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */ + B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ + +/* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */ +static const double +P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */ +P1 = -1.88497979543377169875, /* 0xbffe28e0, 0x92f02420 */ +P2 = 1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */ +P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */ +P4 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */ + +double +cbrt(double x) +{ + int32_t hx; + union { + double value; + uint64_t bits; + } u; + double r,s,t=0.0,w; + u_int32_t sign; + u_int32_t high,low; + + EXTRACT_WORDS(hx,low,x); + sign=hx&0x80000000; /* sign= sign(x) */ + hx ^=sign; + if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */ + + /* + * Rough cbrt to 5 bits: + * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) + * where e is integral and >= 0, m is real and in [0, 1), and "/" and + * "%" are integer division and modulus with rounding towards minus + * infinity. The RHS is always >= the LHS and has a maximum relative + * error of about 1 in 16. Adding a bias of -0.03306235651 to the + * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE + * floating point representation, for finite positive normal values, + * ordinary integer division of the value in bits magically gives + * almost exactly the RHS of the above provided we first subtract the + * exponent bias (1023 for doubles) and later add it back. We do the + * subtraction virtually to keep e >= 0 so that ordinary integer + * division rounds towards minus infinity; this is also efficient. + */ + if(hx<0x00100000) { /* zero or subnormal? */ + if((hx|low)==0) + return(x); /* cbrt(0) is itself */ + SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */ + t*=x; + GET_HIGH_WORD(high,t); + INSERT_WORDS(t,sign|((high&0x7fffffff)/3+B2),0); + } else + INSERT_WORDS(t,sign|(hx/3+B1),0); + + /* + * New cbrt to 23 bits: + * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x) + * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r) + * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation + * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this + * gives us bounds for r = t**3/x. + * + * Try to optimize for parallel evaluation as in k_tanf.c. + */ + r=(t*t)*(t/x); + t=t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4)); + + /* + * Round t away from zero to 23 bits (sloppily except for ensuring that + * the result is larger in magnitude than cbrt(x) but not much more than + * 2 23-bit ulps larger). With rounding towards zero, the error bound + * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps + * in the rounded t, the infinite-precision error in the Newton + * approximation barely affects third digit in the final error + * 0.667; the error in the rounded t can be up to about 3 23-bit ulps + * before the final error is larger than 0.667 ulps. + */ + u.value=t; + u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL; + t=u.value; + + /* one step Newton iteration to 53 bits with error < 0.667 ulps */ + s=t*t; /* t*t is exact */ + r=x/s; /* error <= 0.5 ulps; |r| < |t| */ + w=t+t; /* t+t is exact */ + r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */ + t=t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */ + + return(t); +} diff --git a/modules/fdlibm/src/s_ceil.cpp b/modules/fdlibm/src/s_ceil.cpp new file mode 100644 index 000000000..67e9c1679 --- /dev/null +++ b/modules/fdlibm/src/s_ceil.cpp @@ -0,0 +1,72 @@ +/* @(#)s_ceil.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* + * ceil(x) + * Return x rounded toward -inf to integral value + * Method: + * Bit twiddling. + * Exception: + * Inexact flag raised if x not equal to ceil(x). + */ + +#include <float.h> + +#include "math_private.h" + +static const double huge = 1.0e300; + +double +ceil(double x) +{ + int32_t i0,i1,j0; + u_int32_t i,j; + EXTRACT_WORDS(i0,i1,x); + j0 = ((i0>>20)&0x7ff)-0x3ff; + if(j0<20) { + if(j0<0) { /* raise inexact if x != 0 */ + if(huge+x>0.0) {/* return 0*sign(x) if |x|<1 */ + if(i0<0) {i0=0x80000000;i1=0;} + else if((i0|i1)!=0) { i0=0x3ff00000;i1=0;} + } + } else { + i = (0x000fffff)>>j0; + if(((i0&i)|i1)==0) return x; /* x is integral */ + if(huge+x>0.0) { /* raise inexact flag */ + if(i0>0) i0 += (0x00100000)>>j0; + i0 &= (~i); i1=0; + } + } + } else if (j0>51) { + if(j0==0x400) return x+x; /* inf or NaN */ + else return x; /* x is integral */ + } else { + i = ((u_int32_t)(0xffffffff))>>(j0-20); + if((i1&i)==0) return x; /* x is integral */ + if(huge+x>0.0) { /* raise inexact flag */ + if(i0>0) { + if(j0==20) i0+=1; + else { + j = i1 + (1<<(52-j0)); + if(j<i1) i0+=1; /* got a carry */ + i1 = j; + } + } + i1 &= (~i); + } + } + INSERT_WORDS(x,i0,i1); + return x; +} diff --git a/modules/fdlibm/src/s_ceilf.cpp b/modules/fdlibm/src/s_ceilf.cpp new file mode 100644 index 000000000..7b52deeed --- /dev/null +++ b/modules/fdlibm/src/s_ceilf.cpp @@ -0,0 +1,51 @@ +/* s_ceilf.c -- float version of s_ceil.c. + * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +#include "math_private.h" + +static const float huge = 1.0e30; + +float +ceilf(float x) +{ + int32_t i0,j0; + u_int32_t i; + + GET_FLOAT_WORD(i0,x); + j0 = ((i0>>23)&0xff)-0x7f; + if(j0<23) { + if(j0<0) { /* raise inexact if x != 0 */ + if(huge+x>(float)0.0) {/* return 0*sign(x) if |x|<1 */ + if(i0<0) {i0=0x80000000;} + else if(i0!=0) { i0=0x3f800000;} + } + } else { + i = (0x007fffff)>>j0; + if((i0&i)==0) return x; /* x is integral */ + if(huge+x>(float)0.0) { /* raise inexact flag */ + if(i0>0) i0 += (0x00800000)>>j0; + i0 &= (~i); + } + } + } else { + if(j0==0x80) return x+x; /* inf or NaN */ + else return x; /* x is integral */ + } + SET_FLOAT_WORD(x,i0); + return x; +} diff --git a/modules/fdlibm/src/s_copysign.cpp b/modules/fdlibm/src/s_copysign.cpp new file mode 100644 index 000000000..b150106fb --- /dev/null +++ b/modules/fdlibm/src/s_copysign.cpp @@ -0,0 +1,32 @@ +/* @(#)s_copysign.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* + * copysign(double x, double y) + * copysign(x,y) returns a value with the magnitude of x and + * with the sign bit of y. + */ + +#include "math_private.h" + +double +copysign(double x, double y) +{ + u_int32_t hx,hy; + GET_HIGH_WORD(hx,x); + GET_HIGH_WORD(hy,y); + SET_HIGH_WORD(x,(hx&0x7fffffff)|(hy&0x80000000)); + return x; +} diff --git a/modules/fdlibm/src/s_expm1.cpp b/modules/fdlibm/src/s_expm1.cpp new file mode 100644 index 000000000..4c19485de --- /dev/null +++ b/modules/fdlibm/src/s_expm1.cpp @@ -0,0 +1,220 @@ +/* @(#)s_expm1.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* expm1(x) + * Returns exp(x)-1, the exponential of x minus 1. + * + * Method + * 1. Argument reduction: + * Given x, find r and integer k such that + * + * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 + * + * Here a correction term c will be computed to compensate + * the error in r when rounded to a floating-point number. + * + * 2. Approximating expm1(r) by a special rational function on + * the interval [0,0.34658]: + * Since + * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... + * we define R1(r*r) by + * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) + * That is, + * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) + * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) + * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... + * We use a special Reme algorithm on [0,0.347] to generate + * a polynomial of degree 5 in r*r to approximate R1. The + * maximum error of this polynomial approximation is bounded + * by 2**-61. In other words, + * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 + * where Q1 = -1.6666666666666567384E-2, + * Q2 = 3.9682539681370365873E-4, + * Q3 = -9.9206344733435987357E-6, + * Q4 = 2.5051361420808517002E-7, + * Q5 = -6.2843505682382617102E-9; + * z = r*r, + * with error bounded by + * | 5 | -61 + * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 + * | | + * + * expm1(r) = exp(r)-1 is then computed by the following + * specific way which minimize the accumulation rounding error: + * 2 3 + * r r [ 3 - (R1 + R1*r/2) ] + * expm1(r) = r + --- + --- * [--------------------] + * 2 2 [ 6 - r*(3 - R1*r/2) ] + * + * To compensate the error in the argument reduction, we use + * expm1(r+c) = expm1(r) + c + expm1(r)*c + * ~ expm1(r) + c + r*c + * Thus c+r*c will be added in as the correction terms for + * expm1(r+c). Now rearrange the term to avoid optimization + * screw up: + * ( 2 2 ) + * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) + * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) + * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) + * ( ) + * + * = r - E + * 3. Scale back to obtain expm1(x): + * From step 1, we have + * expm1(x) = either 2^k*[expm1(r)+1] - 1 + * = or 2^k*[expm1(r) + (1-2^-k)] + * 4. Implementation notes: + * (A). To save one multiplication, we scale the coefficient Qi + * to Qi*2^i, and replace z by (x^2)/2. + * (B). To achieve maximum accuracy, we compute expm1(x) by + * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) + * (ii) if k=0, return r-E + * (iii) if k=-1, return 0.5*(r-E)-0.5 + * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) + * else return 1.0+2.0*(r-E); + * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) + * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else + * (vii) return 2^k(1-((E+2^-k)-r)) + * + * Special cases: + * expm1(INF) is INF, expm1(NaN) is NaN; + * expm1(-INF) is -1, and + * for finite argument, only expm1(0)=0 is exact. + * + * Accuracy: + * according to an error analysis, the error is always less than + * 1 ulp (unit in the last place). + * + * Misc. info. + * For IEEE double + * if x > 7.09782712893383973096e+02 then expm1(x) overflow + * + * Constants: + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough + * to produce the hexadecimal values shown. + */ + +#include <float.h> + +#include "math_private.h" + +static const double +one = 1.0, +tiny = 1.0e-300, +o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ +ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ +ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ +invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ +/* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */ +Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ +Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ +Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ +Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ +Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ + +static volatile double huge = 1.0e+300; + +double +expm1(double x) +{ + double y,hi,lo,c,t,e,hxs,hfx,r1,twopk; + int32_t k,xsb; + u_int32_t hx; + + GET_HIGH_WORD(hx,x); + xsb = hx&0x80000000; /* sign bit of x */ + hx &= 0x7fffffff; /* high word of |x| */ + + /* filter out huge and non-finite argument */ + if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ + if(hx >= 0x40862E42) { /* if |x|>=709.78... */ + if(hx>=0x7ff00000) { + u_int32_t low; + GET_LOW_WORD(low,x); + if(((hx&0xfffff)|low)!=0) + return x+x; /* NaN */ + else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ + } + if(x > o_threshold) return huge*huge; /* overflow */ + } + if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ + if(x+tiny<0.0) /* raise inexact */ + return tiny-one; /* return -1 */ + } + } + + /* argument reduction */ + if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ + if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ + if(xsb==0) + {hi = x - ln2_hi; lo = ln2_lo; k = 1;} + else + {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} + } else { + k = invln2*x+((xsb==0)?0.5:-0.5); + t = k; + hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ + lo = t*ln2_lo; + } + STRICT_ASSIGN(double, x, hi - lo); + c = (hi-x)-lo; + } + else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ + t = huge+x; /* return x with inexact flags when x!=0 */ + return x - (t-(huge+x)); + } + else k = 0; + + /* x is now in primary range */ + hfx = 0.5*x; + hxs = x*hfx; + r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); + t = 3.0-r1*hfx; + e = hxs*((r1-t)/(6.0 - x*t)); + if(k==0) return x - (x*e-hxs); /* c is 0 */ + else { + INSERT_WORDS(twopk,0x3ff00000+(k<<20),0); /* 2^k */ + e = (x*(e-c)-c); + e -= hxs; + if(k== -1) return 0.5*(x-e)-0.5; + if(k==1) { + if(x < -0.25) return -2.0*(e-(x+0.5)); + else return one+2.0*(x-e); + } + if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ + y = one-(e-x); + if (k == 1024) { + double const_0x1p1023 = pow(2, 1023); + y = y*2.0*const_0x1p1023; + } + else y = y*twopk; + return y-one; + } + t = one; + if(k<20) { + SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ + y = t-(e-x); + y = y*twopk; + } else { + SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ + y = x-(e+t); + y += one; + y = y*twopk; + } + } + return y; +} diff --git a/modules/fdlibm/src/s_fabs.cpp b/modules/fdlibm/src/s_fabs.cpp new file mode 100644 index 000000000..3bea0478a --- /dev/null +++ b/modules/fdlibm/src/s_fabs.cpp @@ -0,0 +1,30 @@ + /* @(#)s_fabs.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#ifndef lint + //static char rcsid[] = "$FreeBSD$"; +#endif + +/* + * fabs(x) returns the absolute value of x. + */ + +#include "math_private.h" + +double +fabs(double x) +{ + u_int32_t high; + GET_HIGH_WORD(high,x); + SET_HIGH_WORD(x,high&0x7fffffff); + return x; +} diff --git a/modules/fdlibm/src/s_floor.cpp b/modules/fdlibm/src/s_floor.cpp new file mode 100644 index 000000000..da57fc828 --- /dev/null +++ b/modules/fdlibm/src/s_floor.cpp @@ -0,0 +1,73 @@ +/* @(#)s_floor.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* + * floor(x) + * Return x rounded toward -inf to integral value + * Method: + * Bit twiddling. + * Exception: + * Inexact flag raised if x not equal to floor(x). + */ + +#include <float.h> + +#include "math_private.h" + +static const double huge = 1.0e300; + +double +floor(double x) +{ + int32_t i0,i1,j0; + u_int32_t i,j; + EXTRACT_WORDS(i0,i1,x); + j0 = ((i0>>20)&0x7ff)-0x3ff; + if(j0<20) { + if(j0<0) { /* raise inexact if x != 0 */ + if(huge+x>0.0) {/* return 0*sign(x) if |x|<1 */ + if(i0>=0) {i0=i1=0;} + else if(((i0&0x7fffffff)|i1)!=0) + { i0=0xbff00000;i1=0;} + } + } else { + i = (0x000fffff)>>j0; + if(((i0&i)|i1)==0) return x; /* x is integral */ + if(huge+x>0.0) { /* raise inexact flag */ + if(i0<0) i0 += (0x00100000)>>j0; + i0 &= (~i); i1=0; + } + } + } else if (j0>51) { + if(j0==0x400) return x+x; /* inf or NaN */ + else return x; /* x is integral */ + } else { + i = ((u_int32_t)(0xffffffff))>>(j0-20); + if((i1&i)==0) return x; /* x is integral */ + if(huge+x>0.0) { /* raise inexact flag */ + if(i0<0) { + if(j0==20) i0+=1; + else { + j = i1+(1<<(52-j0)); + if(j<i1) i0 +=1 ; /* got a carry */ + i1=j; + } + } + i1 &= (~i); + } + } + INSERT_WORDS(x,i0,i1); + return x; +} diff --git a/modules/fdlibm/src/s_floorf.cpp b/modules/fdlibm/src/s_floorf.cpp new file mode 100644 index 000000000..88511f209 --- /dev/null +++ b/modules/fdlibm/src/s_floorf.cpp @@ -0,0 +1,60 @@ +/* s_floorf.c -- float version of s_floor.c. + * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* + * floorf(x) + * Return x rounded toward -inf to integral value + * Method: + * Bit twiddling. + * Exception: + * Inexact flag raised if x not equal to floorf(x). + */ + +#include "math_private.h" + +static const float huge = 1.0e30; + +float +floorf(float x) +{ + int32_t i0,j0; + u_int32_t i; + GET_FLOAT_WORD(i0,x); + j0 = ((i0>>23)&0xff)-0x7f; + if(j0<23) { + if(j0<0) { /* raise inexact if x != 0 */ + if(huge+x>(float)0.0) {/* return 0*sign(x) if |x|<1 */ + if(i0>=0) {i0=0;} + else if((i0&0x7fffffff)!=0) + { i0=0xbf800000;} + } + } else { + i = (0x007fffff)>>j0; + if((i0&i)==0) return x; /* x is integral */ + if(huge+x>(float)0.0) { /* raise inexact flag */ + if(i0<0) i0 += (0x00800000)>>j0; + i0 &= (~i); + } + } + } else { + if(j0==0x80) return x+x; /* inf or NaN */ + else return x; /* x is integral */ + } + SET_FLOAT_WORD(x,i0); + return x; +} diff --git a/modules/fdlibm/src/s_log1p.cpp b/modules/fdlibm/src/s_log1p.cpp new file mode 100644 index 000000000..afc6919c6 --- /dev/null +++ b/modules/fdlibm/src/s_log1p.cpp @@ -0,0 +1,175 @@ +/* @(#)s_log1p.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* double log1p(double x) + * + * Method : + * 1. Argument Reduction: find k and f such that + * 1+x = 2^k * (1+f), + * where sqrt(2)/2 < 1+f < sqrt(2) . + * + * Note. If k=0, then f=x is exact. However, if k!=0, then f + * may not be representable exactly. In that case, a correction + * term is need. Let u=1+x rounded. Let c = (1+x)-u, then + * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), + * and add back the correction term c/u. + * (Note: when x > 2**53, one can simply return log(x)) + * + * 2. Approximation of log1p(f). + * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) + * = 2s + 2/3 s**3 + 2/5 s**5 + ....., + * = 2s + s*R + * We use a special Reme algorithm on [0,0.1716] to generate + * a polynomial of degree 14 to approximate R The maximum error + * of this polynomial approximation is bounded by 2**-58.45. In + * other words, + * 2 4 6 8 10 12 14 + * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s + * (the values of Lp1 to Lp7 are listed in the program) + * and + * | 2 14 | -58.45 + * | Lp1*s +...+Lp7*s - R(z) | <= 2 + * | | + * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. + * In order to guarantee error in log below 1ulp, we compute log + * by + * log1p(f) = f - (hfsq - s*(hfsq+R)). + * + * 3. Finally, log1p(x) = k*ln2 + log1p(f). + * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) + * Here ln2 is split into two floating point number: + * ln2_hi + ln2_lo, + * where n*ln2_hi is always exact for |n| < 2000. + * + * Special cases: + * log1p(x) is NaN with signal if x < -1 (including -INF) ; + * log1p(+INF) is +INF; log1p(-1) is -INF with signal; + * log1p(NaN) is that NaN with no signal. + * + * Accuracy: + * according to an error analysis, the error is always less than + * 1 ulp (unit in the last place). + * + * Constants: + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough + * to produce the hexadecimal values shown. + * + * Note: Assuming log() return accurate answer, the following + * algorithm can be used to compute log1p(x) to within a few ULP: + * + * u = 1+x; + * if(u==1.0) return x ; else + * return log(u)*(x/(u-1.0)); + * + * See HP-15C Advanced Functions Handbook, p.193. + */ + +#include <float.h> + +#include "math_private.h" + +static const double +ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ +ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ +two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ +Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ +Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ +Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ +Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ +Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ +Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ +Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ + +static const double zero = 0.0; +static volatile double vzero = 0.0; + +double +log1p(double x) +{ + double hfsq,f,c,s,z,R,u; + int32_t k,hx,hu,ax; + + GET_HIGH_WORD(hx,x); + ax = hx&0x7fffffff; + + k = 1; + if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */ + if(ax>=0x3ff00000) { /* x <= -1.0 */ + if(x==-1.0) return -two54/vzero; /* log1p(-1)=+inf */ + else return (x-x)/(x-x); /* log1p(x<-1)=NaN */ + } + if(ax<0x3e200000) { /* |x| < 2**-29 */ + if(two54+x>zero /* raise inexact */ + &&ax<0x3c900000) /* |x| < 2**-54 */ + return x; + else + return x - x*x*0.5; + } + if(hx>0||hx<=((int32_t)0xbfd2bec4)) { + k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ + } + if (hx >= 0x7ff00000) return x+x; + if(k!=0) { + if(hx<0x43400000) { + STRICT_ASSIGN(double,u,1.0+x); + GET_HIGH_WORD(hu,u); + k = (hu>>20)-1023; + c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */ + c /= u; + } else { + u = x; + GET_HIGH_WORD(hu,u); + k = (hu>>20)-1023; + c = 0; + } + hu &= 0x000fffff; + /* + * The approximation to sqrt(2) used in thresholds is not + * critical. However, the ones used above must give less + * strict bounds than the one here so that the k==0 case is + * never reached from here, since here we have committed to + * using the correction term but don't use it if k==0. + */ + if(hu<0x6a09e) { /* u ~< sqrt(2) */ + SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */ + } else { + k += 1; + SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */ + hu = (0x00100000-hu)>>2; + } + f = u-1.0; + } + hfsq=0.5*f*f; + if(hu==0) { /* |f| < 2**-20 */ + if(f==zero) { + if(k==0) { + return zero; + } else { + c += k*ln2_lo; + return k*ln2_hi+c; + } + } + R = hfsq*(1.0-0.66666666666666666*f); + if(k==0) return f-R; else + return k*ln2_hi-((R-(k*ln2_lo+c))-f); + } + s = f/(2.0+f); + z = s*s; + R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7)))))); + if(k==0) return f-(hfsq-s*(hfsq+R)); else + return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); +} diff --git a/modules/fdlibm/src/s_nearbyint.cpp b/modules/fdlibm/src/s_nearbyint.cpp new file mode 100644 index 000000000..532bb5d8d --- /dev/null +++ b/modules/fdlibm/src/s_nearbyint.cpp @@ -0,0 +1,58 @@ +/*- + * Copyright (c) 2004 David Schultz <das@FreeBSD.ORG> + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +#include <fenv.h> +#include "math_private.h" + +/* + * We save and restore the floating-point environment to avoid raising + * an inexact exception. We can get away with using fesetenv() + * instead of feclearexcept()/feupdateenv() to restore the environment + * because the only exception defined for rint() is overflow, and + * rounding can't overflow as long as emax >= p. + * + * The volatile keyword is needed below because clang incorrectly assumes + * that rint won't raise any floating-point exceptions. Declaring ret volatile + * is sufficient to trick the compiler into doing the right thing. + */ +#define DECL(type, fn, rint) \ +type \ +fn(type x) \ +{ \ + volatile type ret; \ + fenv_t env; \ + \ + fegetenv(&env); \ + ret = rint(x); \ + fesetenv(&env); \ + return (ret); \ +} + +DECL(double, nearbyint, rint) +DECL(float, nearbyintf, rintf) diff --git a/modules/fdlibm/src/s_rint.cpp b/modules/fdlibm/src/s_rint.cpp new file mode 100644 index 000000000..19171f87f --- /dev/null +++ b/modules/fdlibm/src/s_rint.cpp @@ -0,0 +1,87 @@ +/* @(#)s_rint.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* + * rint(x) + * Return x rounded to integral value according to the prevailing + * rounding mode. + * Method: + * Using floating addition. + * Exception: + * Inexact flag raised if x not equal to rint(x). + */ + +#include <float.h> + +#include "math_private.h" + +static const double +TWO52[2]={ + 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */ + -4.50359962737049600000e+15, /* 0xC3300000, 0x00000000 */ +}; + +double +rint(double x) +{ + int32_t i0,j0,sx; + u_int32_t i,i1; + double w,t; + EXTRACT_WORDS(i0,i1,x); + sx = (i0>>31)&1; + j0 = ((i0>>20)&0x7ff)-0x3ff; + if(j0<20) { + if(j0<0) { + if(((i0&0x7fffffff)|i1)==0) return x; + i1 |= (i0&0x0fffff); + i0 &= 0xfffe0000; + i0 |= ((i1|-i1)>>12)&0x80000; + SET_HIGH_WORD(x,i0); + STRICT_ASSIGN(double,w,TWO52[sx]+x); + t = w-TWO52[sx]; + GET_HIGH_WORD(i0,t); + SET_HIGH_WORD(t,(i0&0x7fffffff)|(sx<<31)); + return t; + } else { + i = (0x000fffff)>>j0; + if(((i0&i)|i1)==0) return x; /* x is integral */ + i>>=1; + if(((i0&i)|i1)!=0) { + /* + * Some bit is set after the 0.5 bit. To avoid the + * possibility of errors from double rounding in + * w = TWO52[sx]+x, adjust the 0.25 bit to a lower + * guard bit. We do this for all j0<=51. The + * adjustment is trickiest for j0==18 and j0==19 + * since then it spans the word boundary. + */ + if(j0==19) i1 = 0x40000000; else + if(j0==18) i1 = 0x80000000; else + i0 = (i0&(~i))|((0x20000)>>j0); + } + } + } else if (j0>51) { + if(j0==0x400) return x+x; /* inf or NaN */ + else return x; /* x is integral */ + } else { + i = ((u_int32_t)(0xffffffff))>>(j0-20); + if((i1&i)==0) return x; /* x is integral */ + i>>=1; + if((i1&i)!=0) i1 = (i1&(~i))|((0x40000000)>>(j0-20)); + } + INSERT_WORDS(x,i0,i1); + STRICT_ASSIGN(double,w,TWO52[sx]+x); + return w-TWO52[sx]; +} diff --git a/modules/fdlibm/src/s_rintf.cpp b/modules/fdlibm/src/s_rintf.cpp new file mode 100644 index 000000000..3a729b005 --- /dev/null +++ b/modules/fdlibm/src/s_rintf.cpp @@ -0,0 +1,52 @@ +/* s_rintf.c -- float version of s_rint.c. + * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. + */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +#include <float.h> +#include <stdint.h> + +#include "math_private.h" + +static const float +TWO23[2]={ + 8.3886080000e+06, /* 0x4b000000 */ + -8.3886080000e+06, /* 0xcb000000 */ +}; + +float +rintf(float x) +{ + int32_t i0,j0,sx; + float w,t; + GET_FLOAT_WORD(i0,x); + sx = (i0>>31)&1; + j0 = ((i0>>23)&0xff)-0x7f; + if(j0<23) { + if(j0<0) { + if((i0&0x7fffffff)==0) return x; + STRICT_ASSIGN(float,w,TWO23[sx]+x); + t = w-TWO23[sx]; + GET_FLOAT_WORD(i0,t); + SET_FLOAT_WORD(t,(i0&0x7fffffff)|(sx<<31)); + return t; + } + STRICT_ASSIGN(float,w,TWO23[sx]+x); + return w-TWO23[sx]; + } + if(j0==0x80) return x+x; /* inf or NaN */ + else return x; /* x is integral */ +} diff --git a/modules/fdlibm/src/s_scalbn.cpp b/modules/fdlibm/src/s_scalbn.cpp new file mode 100644 index 000000000..5dbf58c23 --- /dev/null +++ b/modules/fdlibm/src/s_scalbn.cpp @@ -0,0 +1,60 @@ +/* @(#)s_scalbn.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#ifndef lint +//static char rcsid[] = "$FreeBSD$"; +#endif + +/* + * scalbn (double x, int n) + * scalbn(x,n) returns x* 2**n computed by exponent + * manipulation rather than by actually performing an + * exponentiation or a multiplication. + */ + +//#include <sys/cdefs.h> +#include <float.h> + +#include "math_private.h" + +static const double +two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ +twom54 = 5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */ +huge = 1.0e+300, +tiny = 1.0e-300; + +double +scalbn (double x, int n) +{ + int32_t k,hx,lx; + EXTRACT_WORDS(hx,lx,x); + k = (hx&0x7ff00000)>>20; /* extract exponent */ + if (k==0) { /* 0 or subnormal x */ + if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */ + x *= two54; + GET_HIGH_WORD(hx,x); + k = ((hx&0x7ff00000)>>20) - 54; + if (n< -50000) return tiny*x; /*underflow*/ + } + if (k==0x7ff) return x+x; /* NaN or Inf */ + k = k+n; + if (k > 0x7fe) return huge*copysign(huge,x); /* overflow */ + if (k > 0) /* normal result */ + {SET_HIGH_WORD(x,(hx&0x800fffff)|(k<<20)); return x;} + if (k <= -54) + if (n > 50000) /* in case integer overflow in n+k */ + return huge*copysign(huge,x); /*overflow*/ + else return tiny*copysign(tiny,x); /*underflow*/ + k += 54; /* subnormal result */ + SET_HIGH_WORD(x,(hx&0x800fffff)|(k<<20)); + return x*twom54; +} diff --git a/modules/fdlibm/src/s_tanh.cpp b/modules/fdlibm/src/s_tanh.cpp new file mode 100644 index 000000000..238973fce --- /dev/null +++ b/modules/fdlibm/src/s_tanh.cpp @@ -0,0 +1,79 @@ +/* @(#)s_tanh.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* Tanh(x) + * Return the Hyperbolic Tangent of x + * + * Method : + * x -x + * e - e + * 0. tanh(x) is defined to be ----------- + * x -x + * e + e + * 1. reduce x to non-negative by tanh(-x) = -tanh(x). + * 2. 0 <= x < 2**-28 : tanh(x) := x with inexact if x != 0 + * -t + * 2**-28 <= x < 1 : tanh(x) := -----; t = expm1(-2x) + * t + 2 + * 2 + * 1 <= x < 22 : tanh(x) := 1 - -----; t = expm1(2x) + * t + 2 + * 22 <= x <= INF : tanh(x) := 1. + * + * Special cases: + * tanh(NaN) is NaN; + * only tanh(0)=0 is exact for finite argument. + */ + +#include <float.h> + +#include "math_private.h" + +static const volatile double tiny = 1.0e-300; +static const double one = 1.0, two = 2.0, huge = 1.0e300; + +double +tanh(double x) +{ + double t,z; + int32_t jx,ix; + + GET_HIGH_WORD(jx,x); + ix = jx&0x7fffffff; + + /* x is INF or NaN */ + if(ix>=0x7ff00000) { + if (jx>=0) return one/x+one; /* tanh(+-inf)=+-1 */ + else return one/x-one; /* tanh(NaN) = NaN */ + } + + /* |x| < 22 */ + if (ix < 0x40360000) { /* |x|<22 */ + if (ix<0x3e300000) { /* |x|<2**-28 */ + if(huge+x>one) return x; /* tanh(tiny) = tiny with inexact */ + } + if (ix>=0x3ff00000) { /* |x|>=1 */ + t = expm1(two*fabs(x)); + z = one - two/(t+two); + } else { + t = expm1(-two*fabs(x)); + z= -t/(t+two); + } + /* |x| >= 22, return +-1 */ + } else { + z = one - tiny; /* raise inexact flag */ + } + return (jx>=0)? z: -z; +} diff --git a/modules/fdlibm/src/s_trunc.cpp b/modules/fdlibm/src/s_trunc.cpp new file mode 100644 index 000000000..d2294a272 --- /dev/null +++ b/modules/fdlibm/src/s_trunc.cpp @@ -0,0 +1,62 @@ +/* @(#)s_floor.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* + * trunc(x) + * Return x rounded toward 0 to integral value + * Method: + * Bit twiddling. + * Exception: + * Inexact flag raised if x not equal to trunc(x). + */ + +#include <float.h> + +#include "math_private.h" + +static const double huge = 1.0e300; + +double +trunc(double x) +{ + int32_t i0,i1,j0; + u_int32_t i; + EXTRACT_WORDS(i0,i1,x); + j0 = ((i0>>20)&0x7ff)-0x3ff; + if(j0<20) { + if(j0<0) { /* raise inexact if x != 0 */ + if(huge+x>0.0) {/* |x|<1, so return 0*sign(x) */ + i0 &= 0x80000000U; + i1 = 0; + } + } else { + i = (0x000fffff)>>j0; + if(((i0&i)|i1)==0) return x; /* x is integral */ + if(huge+x>0.0) { /* raise inexact flag */ + i0 &= (~i); i1=0; + } + } + } else if (j0>51) { + if(j0==0x400) return x+x; /* inf or NaN */ + else return x; /* x is integral */ + } else { + i = ((u_int32_t)(0xffffffff))>>(j0-20); + if((i1&i)==0) return x; /* x is integral */ + if(huge+x>0.0) /* raise inexact flag */ + i1 &= (~i); + } + INSERT_WORDS(x,i0,i1); + return x; +} diff --git a/modules/fdlibm/src/s_truncf.cpp b/modules/fdlibm/src/s_truncf.cpp new file mode 100644 index 000000000..4853a4450 --- /dev/null +++ b/modules/fdlibm/src/s_truncf.cpp @@ -0,0 +1,52 @@ +/* @(#)s_floor.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +//#include <sys/cdefs.h> +//__FBSDID("$FreeBSD$"); + +/* + * truncf(x) + * Return x rounded toward 0 to integral value + * Method: + * Bit twiddling. + * Exception: + * Inexact flag raised if x not equal to truncf(x). + */ + +#include "math_private.h" + +static const float huge = 1.0e30F; + +float +truncf(float x) +{ + int32_t i0,j0; + u_int32_t i; + GET_FLOAT_WORD(i0,x); + j0 = ((i0>>23)&0xff)-0x7f; + if(j0<23) { + if(j0<0) { /* raise inexact if x != 0 */ + if(huge+x>0.0F) /* |x|<1, so return 0*sign(x) */ + i0 &= 0x80000000; + } else { + i = (0x007fffff)>>j0; + if((i0&i)==0) return x; /* x is integral */ + if(huge+x>0.0F) /* raise inexact flag */ + i0 &= (~i); + } + } else { + if(j0==0x80) return x+x; /* inf or NaN */ + else return x; /* x is integral */ + } + SET_FLOAT_WORD(x,i0); + return x; +} |