/* @(#)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); | |

} |