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      1 /*-
      2  * ====================================================
      3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
      4  * Copyright (c) 2009-2011, Bruce D. Evans, Steven G. Kargl, David Schultz.
      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  * The argument reduction and testing for exceptional cases was
     13  * written by Steven G. Kargl with input from Bruce D. Evans
     14  * and David A. Schultz.
     15  */
     16 #include <sys/cdefs.h>
     17 __RCSID("$NetBSD: s_cbrtl.c,v 1.4 2024/04/03 18:53:42 christos Exp $");
     18 
     19 
     20 #include "namespace.h"
     21 #include <machine/ieee.h>
     22 #include <float.h>
     23 
     24 #include "math.h"
     25 #include "math_private.h"
     26 
     27 #ifdef __HAVE_LONG_DOUBLE
     28 __weak_alias(cbrtl, _cbrtl)
     29 
     30 #define	BIAS	(LDBL_MAX_EXP - 1)
     31 
     32 static const unsigned
     33     B1 = 709958130;	/* B1 = (127-127.0/3-0.03306235651)*2**23 */
     34 
     35 long double
     36 cbrtl(long double x)
     37 {
     38 	union ieee_ext_u u, v;
     39 	long double r, s, t, w;
     40 	double dr, dt, dx;
     41 	float ft, fx;
     42 	uint32_t hx;
     43 	uint16_t expsign;
     44 	int k;
     45 
     46 	u.extu_ld = x;
     47 	expsign = GET_EXPSIGN(&u);
     48 	k = expsign & 0x7fff;
     49 
     50 	/*
     51 	 * If x = +-Inf, then cbrt(x) = +-Inf.
     52 	 * If x = NaN, then cbrt(x) = NaN.
     53 	 */
     54 	if (k == BIAS + LDBL_MAX_EXP)
     55 		return (x + x);
     56 
     57 	ENTERI();
     58 	if (k == 0) {
     59 		/* If x = +-0, then cbrt(x) = +-0. */
     60 		if ((u.extu_frach | u.extu_fracl) == 0)
     61 			RETURNI(x);
     62 		/* Adjust subnormal numbers. */
     63 		u.extu_ld *= 0x1.0p514;
     64 		k = u.extu_exp;
     65 		k -= BIAS + 514;
     66  	} else
     67 		k -= BIAS;
     68 	SET_EXPSIGN(&u, BIAS);
     69 	v.extu_ld = 1;
     70 
     71 	x = u.extu_ld;
     72 	switch (k % 3) {
     73 	case 1:
     74 	case -2:
     75 		x = 2*x;
     76 		k--;
     77 		break;
     78 	case 2:
     79 	case -1:
     80 		x = 4*x;
     81 		k -= 2;
     82 		break;
     83 	}
     84 	SET_EXPSIGN(&v, (expsign & 0x8000) | (BIAS + k / 3));
     85 
     86 	/*
     87 	 * The following is the guts of s_cbrtf, with the handling of
     88 	 * special values removed and extra care for accuracy not taken,
     89 	 * but with most of the extra accuracy not discarded.
     90 	 */
     91 
     92 	/* ~5-bit estimate: */
     93 	fx = x;
     94 	GET_FLOAT_WORD(hx, fx);
     95 	SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
     96 
     97 	/* ~16-bit estimate: */
     98 	dx = x;
     99 	dt = ft;
    100 	dr = dt * dt * dt;
    101 	dt = dt * (dx + dx + dr) / (dx + dr + dr);
    102 
    103 	/* ~47-bit estimate: */
    104 	dr = dt * dt * dt;
    105 	dt = dt * (dx + dx + dr) / (dx + dr + dr);
    106 
    107 #if LDBL_MANT_DIG == 64
    108 	/*
    109 	 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
    110 	 * Round it away from zero to 32 bits (32 so that t*t is exact, and
    111 	 * away from zero for technical reasons).
    112 	 */
    113 	volatile double vd2 = 0x1.0p32;
    114 	volatile double vd1 = 0x1.0p-31;
    115 	#define vd ((long double)vd2 + vd1)
    116 
    117 	t = dt + vd - 0x1.0p32;
    118 #elif LDBL_MANT_DIG == 113
    119 	/*
    120 	 * Round dt away from zero to 47 bits.  Since we don't trust the 47,
    121 	 * add 2 47-bit ulps instead of 1 to round up.  Rounding is slow and
    122 	 * might be avoidable in this case, since on most machines dt will
    123 	 * have been evaluated in 53-bit precision and the technical reasons
    124 	 * for rounding up might not apply to either case in cbrtl() since
    125 	 * dt is much more accurate than needed.
    126 	 */
    127 	t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
    128 #else
    129 #error "Unsupported long double format"
    130 #endif
    131 
    132 	/*
    133      	 * Final step Newton iteration to 64 or 113 bits with
    134 	 * error < 0.667 ulps
    135 	 */
    136 	s=t*t;				/* t*t is exact */
    137 	r=x/s;				/* error <= 0.5 ulps; |r| < |t| */
    138 	w=t+t;				/* t+t is exact */
    139 	r=(r-t)/(w+r);			/* r-t is exact; w+r ~= 3*t */
    140 	t=t+t*r;			/* error <= (0.5 + 0.5/3) * ulp */
    141 
    142 	t *= v.extu_ld;
    143 	RETURNI(t);
    144 }
    145 #endif /* __HAVE_LONG_DOUBLE */
    146