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      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 #include <sys/cdefs.h>
     14 #if defined(LIBM_SCCS) && !defined(lint)
     15 __RCSID("$NetBSD: s_cbrt.c,v 1.14 2026/02/06 22:40:47 andvar Exp $");
     16 #endif
     17 
     18 #include "namespace.h"
     19 #include "math.h"
     20 #include "math_private.h"
     21 
     22 #ifndef __HAVE_LONG_DOUBLE
     23 __strong_alias(_cbrtl, cbrt)
     24 __weak_alias(cbrtl, _cbrtl)
     25 #endif
     26 
     27 /* cbrt(x)
     28  * Return cube root of x
     29  */
     30 static const u_int32_t
     31 	B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */
     32 	B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
     33 
     34 static const double
     35 C =  5.42857142857142815906e-01, /* 19/35     = 0x3FE15F15, 0xF15F15F1 */
     36 D = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */
     37 E =  1.41428571428571436819e+00, /* 99/70     = 0x3FF6A0EA, 0x0EA0EA0F */
     38 F =  1.60714285714285720630e+00, /* 45/28     = 0x3FF9B6DB, 0x6DB6DB6E */
     39 G =  3.57142857142857150787e-01; /* 5/14      = 0x3FD6DB6D, 0xB6DB6DB7 */
     40 
     41 double
     42 cbrt(double x)
     43 {
     44 	int32_t	hx;
     45 	double r,s,t=0.0,w;
     46 	u_int32_t sign;
     47 	u_int32_t high,low;
     48 
     49 	GET_HIGH_WORD(hx,x);
     50 	sign=hx&0x80000000; 		/* sign= sign(x) */
     51 	hx  ^=sign;
     52 	if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
     53 	GET_LOW_WORD(low,x);
     54 	if((hx|low)==0)
     55 	    return(x);		/* cbrt(0) is itself */
     56 
     57 	SET_HIGH_WORD(x,hx);	/* x <- |x| */
     58     /* rough cbrt to 5 bits */
     59 	if(hx<0x00100000) 		/* subnormal number */
     60 	  {SET_HIGH_WORD(t,0x43500000);	/* set t= 2**54 */
     61 	   t*=x; GET_HIGH_WORD(high,t); SET_HIGH_WORD(t,high/3+B2);
     62 	  }
     63 	else
     64 	  SET_HIGH_WORD(t,hx/3+B1);
     65 
     66 
     67     /* new cbrt to 23 bits, may be implemented in single precision */
     68 	r=t*t/x;
     69 	s=C+r*t;
     70 	t*=G+F/(s+E+D/s);
     71 
     72     /* chopped to 20 bits and make it larger than cbrt(x) */
     73 	GET_HIGH_WORD(high,t);
     74 	INSERT_WORDS(t,high+0x00000001,0);
     75 
     76 
     77     /* one step newton iteration to 53 bits with error less than 0.667 ulps */
     78 	s=t*t;		/* t*t is exact */
     79 	r=x/s;
     80 	w=t+t;
     81 	r=(r-t)/(w+r);	/* r-s is exact */
     82 	t=t+t*r;
     83 
     84     /* restore the sign bit */
     85 	GET_HIGH_WORD(high,t);
     86 	SET_HIGH_WORD(t,high|sign);
     87 	return(t);
     88 }
     89