1 /* @(#)s_cbrt.c 5.1 93/09/24 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 * 12 * Optimized by Bruce D. Evans. 13 */ 14 15 #ifndef lint 16 static char rcsid[] = "$FreeBSD: src/lib/msun/src/s_cbrt.c,v 1.10 2005/12/13 20:17:23 bde Exp $"; 17 #endif 18 19 #include "math.h" 20 #include "math_private.h" 21 22 /* cbrt(x) 23 * Return cube root of x 24 */ 25 static const u_int32_t 26 B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */ 27 B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ 28 29 static const double 30 C = 5.42857142857142815906e-01, /* 19/35 = 0x3FE15F15, 0xF15F15F1 */ 31 D = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */ 32 E = 1.41428571428571436819e+00, /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */ 33 F = 1.60714285714285720630e+00, /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */ 34 G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */ 35 36 double 37 cbrt(double x) 38 { 39 int32_t hx; 40 double r,s,t=0.0,w; 41 u_int32_t sign; 42 u_int32_t high,low; 43 44 GET_HIGH_WORD(hx,x); 45 sign=hx&0x80000000; /* sign= sign(x) */ 46 hx ^=sign; 47 if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */ 48 GET_LOW_WORD(low,x); 49 if((hx|low)==0) 50 return(x); /* cbrt(0) is itself */ 51 52 /* 53 * Rough cbrt to 5 bits: 54 * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) 55 * where e is integral and >= 0, m is real and in [0, 1), and "/" and 56 * "%" are integer division and modulus with rounding towards minus 57 * infinity. The RHS is always >= the LHS and has a maximum relative 58 * error of about 1 in 16. Adding a bias of -0.03306235651 to the 59 * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE 60 * floating point representation, for finite positive normal values, 61 * ordinary integer divison of the value in bits magically gives 62 * almost exactly the RHS of the above provided we first subtract the 63 * exponent bias (1023 for doubles) and later add it back. We do the 64 * subtraction virtually to keep e >= 0 so that ordinary integer 65 * division rounds towards minus infinity; this is also efficient. 66 */ 67 if(hx<0x00100000) { /* subnormal number */ 68 SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */ 69 t*=x; 70 GET_HIGH_WORD(high,t); 71 SET_HIGH_WORD(t,sign|((high&0x7fffffff)/3+B2)); 72 } else 73 SET_HIGH_WORD(t,sign|(hx/3+B1)); 74 75 /* new cbrt to 23 bits; may be implemented in single precision */ 76 r=t*t/x; 77 s=C+r*t; 78 t*=G+F/(s+E+D/s); 79 80 /* chop t to 20 bits and make it larger in magnitude than cbrt(x) */ 81 GET_HIGH_WORD(high,t); 82 INSERT_WORDS(t,high+0x00000001,0); 83 84 /* one step Newton iteration to 53 bits with error less than 0.667 ulps */ 85 s=t*t; /* t*t is exact */ 86 r=x/s; 87 w=t+t; 88 r=(r-t)/(w+r); /* r-t is exact */ 89 t=t+t*r; 90 91 return(t); 92 } 93
