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 13 #ifndef lint 14 static char rcsid[] = "$FreeBSD$"; 15 #endif 16 17 #include "math.h" 18 #include "math_private.h" 19 20 /* cbrt(x) 21 * Return cube root of x 22 */ 23 static const u_int32_t 24 B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */ 25 B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ 26 27 static const double 28 C = 5.42857142857142815906e-01, /* 19/35 = 0x3FE15F15, 0xF15F15F1 */ 29 D = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */ 30 E = 1.41428571428571436819e+00, /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */ 31 F = 1.60714285714285720630e+00, /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */ 32 G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */ 33 34 double 35 cbrt(double x) 36 { 37 int32_t hx; 38 double r,s,t=0.0,w; 39 u_int32_t sign; 40 u_int32_t high,low; 41 42 GET_HIGH_WORD(hx,x); 43 sign=hx&0x80000000; /* sign= sign(x) */ 44 hx ^=sign; 45 if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */ 46 GET_LOW_WORD(low,x); 47 if((hx|low)==0) 48 return(x); /* cbrt(0) is itself */ 49 50 SET_HIGH_WORD(x,hx); /* x <- |x| */ 51 /* 52 * Rough cbrt to 5 bits: 53 * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) 54 * where e is integral and >= 0, m is real and in [0, 1), and "/" and 55 * "%" are integer division and modulus with rounding towards minus 56 * infinity. The RHS is always >= the LHS and has a maximum relative 57 * error of about 1 in 16. Adding a bias of -0.03306235651 to the 58 * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE 59 * floating point representation, for finite positive normal values, 60 * ordinary integer divison of the value in bits magically gives 61 * almost exactly the RHS of the above provided we first subtract the 62 * exponent bias (1023 for doubles) and later add it back. We do the 63 * subtraction virtually to keep e >= 0 so that ordinary integer 64 * division rounds towards minus infinity; this is also efficient. 65 */ 66 if(hx<0x00100000) { /* subnormal number */ 67 SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */ 68 t*=x; 69 GET_HIGH_WORD(high,t); 70 SET_HIGH_WORD(t,high/3+B2); 71 } else 72 SET_HIGH_WORD(t,hx/3+B1); 73 74 /* new cbrt to 23 bits; may be implemented in single precision */ 75 r=t*t/x; 76 s=C+r*t; 77 t*=G+F/(s+E+D/s); 78 79 /* chop t to 20 bits and make it larger than cbrt(x) */ 80 GET_HIGH_WORD(high,t); 81 INSERT_WORDS(t,high+0x00000001,0); 82 83 /* one step Newton iteration to 53 bits with error less than 0.667 ulps */ 84 s=t*t; /* t*t is exact */ 85 r=x/s; 86 w=t+t; 87 r=(r-t)/(w+r); /* r-t is exact */ 88 t=t+t*r; 89 90 /* restore the sign bit */ 91 GET_HIGH_WORD(high,t); 92 SET_HIGH_WORD(t,high|sign); 93 return(t); 94 } 95