s_cbrt.c revision 153382
1193323Sed/* @(#)s_cbrt.c 5.1 93/09/24 */ 2193323Sed/* 3193323Sed * ==================================================== 4193323Sed * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5193323Sed * 6193323Sed * Developed at SunPro, a Sun Microsystems, Inc. business. 7193323Sed * Permission to use, copy, modify, and distribute this 8193323Sed * software is freely granted, provided that this notice 9193323Sed * is preserved. 10193323Sed * ==================================================== 11193323Sed */ 12193323Sed 13193323Sed#ifndef lint 14193323Sedstatic char rcsid[] = "$FreeBSD: head/lib/msun/src/s_cbrt.c 153382 2005-12-13 18:22:00Z bde $"; 15193323Sed#endif 16193323Sed 17193323Sed#include "math.h" 18193323Sed#include "math_private.h" 19193323Sed 20193323Sed/* cbrt(x) 21193323Sed * Return cube root of x 22193323Sed */ 23198090Srdivackystatic const u_int32_t 24198090Srdivacky B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */ 25193323Sed B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ 26193323Sed 27193323Sedstatic const double 28193323SedC = 5.42857142857142815906e-01, /* 19/35 = 0x3FE15F15, 0xF15F15F1 */ 29193323SedD = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */ 30193323SedE = 1.41428571428571436819e+00, /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */ 31193323SedF = 1.60714285714285720630e+00, /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */ 32193323SedG = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */ 33193323Sed 34193323Seddouble 35193323Sedcbrt(double x) 36193323Sed{ 37207618Srdivacky int32_t hx; 38193323Sed double r,s,t=0.0,w; 39193323Sed u_int32_t sign; 40193323Sed u_int32_t high,low; 41193323Sed 42193323Sed GET_HIGH_WORD(hx,x); 43249423Sdim sign=hx&0x80000000; /* sign= sign(x) */ 44193323Sed hx ^=sign; 45193323Sed if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */ 46193323Sed GET_LOW_WORD(low,x); 47193323Sed if((hx|low)==0) 48193323Sed return(x); /* cbrt(0) is itself */ 49193323Sed 50193323Sed SET_HIGH_WORD(x,hx); /* x <- |x| */ 51193323Sed /* 52193323Sed * Rough cbrt to 5 bits: 53193323Sed * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) 54193323Sed * where e is integral and >= 0, m is real and in [0, 1), and "/" and 55193323Sed * "%" are integer division and modulus with rounding towards minus 56193323Sed * infinity. The RHS is always >= the LHS and has a maximum relative 57193323Sed * error of about 1 in 16. Adding a bias of -0.03306235651 to the 58193323Sed * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE 59193323Sed * floating point representation, for finite positive normal values, 60193323Sed * ordinary integer divison of the value in bits magically gives 61221345Sdim * almost exactly the RHS of the above provided we first subtract the 62221345Sdim * exponent bias (1023 for doubles) and later add it back. We do the 63221345Sdim * subtraction virtually to keep e >= 0 so that ordinary integer 64249423Sdim * division rounds towards minus infinity; this is also efficient. 65249423Sdim */ 66226633Sdim if(hx<0x00100000) { /* subnormal number */ 67226633Sdim SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */ 68226633Sdim t*=x; 69243830Sdim GET_HIGH_WORD(high,t); 70234353Sdim SET_HIGH_WORD(t,high/3+B2); 71234353Sdim } else 72193323Sed SET_HIGH_WORD(t,hx/3+B1); 73193323Sed 74193323Sed /* new cbrt to 23 bits; may be implemented in single precision */ 75193323Sed r=t*t/x; 76193323Sed s=C+r*t; 77239462Sdim t*=G+F/(s+E+D/s); 78239462Sdim 79239462Sdim /* chop t to 20 bits and make it larger than cbrt(x) */ 80193323Sed GET_HIGH_WORD(high,t); 81193323Sed INSERT_WORDS(t,high+0x00000001,0); 82193323Sed 83193323Sed /* one step Newton iteration to 53 bits with error less than 0.667 ulps */ 84193323Sed s=t*t; /* t*t is exact */ 85193323Sed r=x/s; 86193323Sed w=t+t; 87193323Sed r=(r-t)/(w+r); /* r-t is exact */ 88249423Sdim t=t+t*r; 89249423Sdim 90249423Sdim /* restore the sign bit */ 91249423Sdim GET_HIGH_WORD(high,t); 92249423Sdim SET_HIGH_WORD(t,high|sign); 93249423Sdim return(t); 94249423Sdim} 95249423Sdim