| The Android Open Source Project | 1dc9e47 | 2009-03-03 19:28:35 -0800 | [diff] [blame^] | 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 | } |