/* @(#)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); }