| /* @(#)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. |
| */ |
| |
| #ifndef lint |
| static char rcsid[] = "$FreeBSD: src/lib/msun/src/s_cbrt.c,v 1.10 2005/12/13 20:17:23 bde Exp $"; |
| #endif |
| |
| #include "math.h" |
| #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 */ |
| |
| static const double |
| C = 5.42857142857142815906e-01, /* 19/35 = 0x3FE15F15, 0xF15F15F1 */ |
| D = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */ |
| E = 1.41428571428571436819e+00, /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */ |
| F = 1.60714285714285720630e+00, /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */ |
| G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */ |
| |
| double |
| cbrt(double x) |
| { |
| int32_t hx; |
| double r,s,t=0.0,w; |
| u_int32_t sign; |
| u_int32_t high,low; |
| |
| GET_HIGH_WORD(hx,x); |
| sign=hx&0x80000000; /* sign= sign(x) */ |
| hx ^=sign; |
| if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */ |
| GET_LOW_WORD(low,x); |
| if((hx|low)==0) |
| return(x); /* cbrt(0) 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 divison 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) { /* subnormal number */ |
| SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */ |
| t*=x; |
| GET_HIGH_WORD(high,t); |
| SET_HIGH_WORD(t,sign|((high&0x7fffffff)/3+B2)); |
| } else |
| SET_HIGH_WORD(t,sign|(hx/3+B1)); |
| |
| /* new cbrt to 23 bits; may be implemented in single precision */ |
| r=t*t/x; |
| s=C+r*t; |
| t*=G+F/(s+E+D/s); |
| |
| /* chop t to 20 bits and make it larger in magnitude than cbrt(x) */ |
| GET_HIGH_WORD(high,t); |
| INSERT_WORDS(t,high+0x00000001,0); |
| |
| /* one step Newton iteration to 53 bits with error less than 0.667 ulps */ |
| s=t*t; /* t*t is exact */ |
| r=x/s; |
| w=t+t; |
| r=(r-t)/(w+r); /* r-t is exact */ |
| t=t+t*r; |
| |
| return(t); |
| } |