High-quality sqrt implementation for Alpha.
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Ulrich Drepper
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87967acd39
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/* Copyright (C) 1996, 1997 Free Software Foundation, Inc.
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Contributed by David Mosberger (davidm@cs.arizona.edu).
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This file is part of the GNU C Library.
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The GNU C Library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Library General Public License as
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published by the Free Software Foundation; either version 2 of the
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License, or (at your option) any later version.
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The GNU C Library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Library General Public License for more details.
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You should have received a copy of the GNU Library General Public
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License along with the GNU C Library; see the file COPYING.LIB. If not,
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write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330,
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Boston, MA 02111-1307, USA. */
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/*
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* We have three versions, depending on how exact we need the results.
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*/
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#if defined(_IEEE_FP) && defined(_IEEE_FP_INEXACT)
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/* Most demanding: go to the original source. */
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#include <libm-ieee754/e_sqrt.c>
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#else
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/* Careful with rearranging this without consulting the assembly below. */
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const static struct sqrt_data_struct {
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unsigned long dn, up, half, almost_three_half;
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unsigned long one_and_a_half, two_to_minus_30, one, nan;
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const int T2[64];
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} sqrt_data = {
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0x3fefffffffffffff, /* __dn = nextafter(1,-Inf) */
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0x3ff0000000000001, /* __up = nextafter(1,+Inf) */
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0x3fe0000000000000, /* half */
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0x3ff7ffffffc00000, /* almost_three_half = 1.5-2^-30 */
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0x3ff8000000000000, /* one_and_a_half */
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0x3e10000000000000, /* two_to_minus_30 */
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0x3ff0000000000000, /* one */
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0xffffffffffffffff, /* nan */
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{ 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
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0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
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0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
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0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
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0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
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0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
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0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
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0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd }
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};
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#ifdef _IEEE_FP
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/*
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* This version is much faster than the standard one included above,
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* but it doesn't maintain the inexact flag.
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*/
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#define lobits(x) (((unsigned int *)&x)[0])
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#define hibits(x) (((unsigned int *)&x)[1])
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static inline double initial_guess(double x, unsigned int k,
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const struct sqrt_data_struct * const ptr)
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{
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double ret = 0.0;
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k = 0x5fe80000 - (k >> 1);
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k = k - ptr->T2[63&(k>>14)];
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hibits(ret) = k;
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return ret;
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}
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/* up = nextafter(1,+Inf), dn = nextafter(1,-Inf) */
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#define __half (ptr->half)
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#define __one_and_a_half (ptr->one_and_a_half)
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#define __two_to_minus_30 (ptr->two_to_minus_30)
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#define __one (ptr->one)
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#define __up (ptr->up)
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#define __dn (ptr->dn)
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#define __Nan (ptr->nan)
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#define Double(x) (*(double *)&x)
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/* Multiply with chopping rounding.. */
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#define choppedmul(a,b,c) \
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__asm__("multc %1,%2,%0":"=&f" (c):"f" (a), "f" (b))
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double
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__ieee754_sqrt(double x)
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{
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const struct sqrt_data_struct * const ptr = &sqrt_data;
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unsigned long k, bits;
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double y, z, zp, zn;
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double dn, up, low, high;
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double half, one_and_a_half, one, two_to_minus_30;
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*(double *)&bits = x;
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k = bits;
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/* Negative or NaN or Inf */
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if ((k >> 52) >= 0x7ff)
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goto special;
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y = initial_guess(x, k >> 32, ptr);
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half = Double(__half);
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one_and_a_half = Double(__one_and_a_half);
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y = y*(one_and_a_half - half*x*y*y);
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dn = Double(__dn);
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two_to_minus_30 = Double(__two_to_minus_30);
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y = y*((one_and_a_half - two_to_minus_30) - half*x*y*y);
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up = Double(__up);
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z = x*y;
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one = Double(__one);
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z = z + half*z*(one-z*y);
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choppedmul(z,dn,zp);
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choppedmul(z,up,zn);
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choppedmul(z,zp,low);
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low = low - x;
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choppedmul(z,zn,high);
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high = high - x;
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/* I can't get gcc to use fcmov's.. */
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__asm__("fcmovge %2,%3,%0"
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:"=f" (z)
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:"0" (z), "f" (low), "f" (zp));
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__asm__("fcmovlt %2,%3,%0"
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:"=f" (z)
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:"0" (z), "f" (high), "f" (zn));
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return z; /* Argh! gcc jumps to end here */
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special:
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/* throw away sign bit */
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k <<= 1;
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/* -0 */
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if (!k)
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return x;
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/* special? */
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if ((k >> 53) == 0x7ff) {
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/* NaN? */
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if (k << 11)
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return x;
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/* sqrt(+Inf) = +Inf */
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if (x > 0)
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return x;
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}
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x = Double(__Nan);
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return x;
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}
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#else
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/*
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* This version is much faster than generic sqrt implementation, but
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* it doesn't handle exceptional values or the inexact flag.
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*/
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asm ("\
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/* Define offsets into the structure defined in C above. */
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$DN = 0*8
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$UP = 1*8
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$HALF = 2*8
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$ALMOST_THREE_HALF = 3*8
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$NAN = 7*8
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$T2 = 8*8
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/* Stack variables. */
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$K = 0
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$Y = 8
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.text
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.align 3
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.globl __ieee754_sqrt
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.ent __ieee754_sqrt
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__ieee754_sqrt:
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ldgp $29, 0($27)
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subq $sp, 16, $sp
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.frame $sp, 16, $26, 0\n"
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#ifdef PROF
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" lda $28, _mcount
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jsr $28, ($28), _mcount\n"
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#endif
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" .prologue 1
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stt $f16, $K($sp)
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lda $4, sqrt_data # load base address into t3
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fblt $f16, $negative
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/* Compute initial guess. */
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.align 3
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ldah $2, 0x5fe8 # e0 :
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ldq $3, $K($sp) # .. e1 :
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ldt $f12, $HALF($4) # e0 :
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ldt $f18, $ALMOST_THREE_HALF($4) # .. e1 :
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srl $3, 33, $1 # e0 :
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mult $f16, $f12, $f11 # .. fm : $f11 = x * 0.5
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subl $2, $1, $2 # e0 :
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addt $f12, $f12, $f17 # .. fa : $f17 = 1.0
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srl $2, 12, $1 # e0 :
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and $1, 0xfc, $1 # .. e1 :
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addq $1, $4, $1 # e0 :
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ldl $1, $T2($1) # .. e1 :
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addt $f12, $f17, $f15 # fa : $f15 = 1.5
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subl $2, $1, $2 # .. e1 :
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sll $2, 32, $2 # e0 :
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ldt $f14, $DN($4) # .. e1 :
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stq $2, $Y($sp) # e0 :
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ldt $f13, $Y($sp) # e1 :
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mult $f11, $f13, $f10 # fm : $f10 = (x * 0.5) * y
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mult $f10, $f13, $f10 # fm : $f10 = ((x * 0.5) * y) * y
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subt $f15, $f10, $f1 # fa : $f1 = (1.5 - 0.5*x*y*y)
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mult $f13, $f1, $f13 # fm : yp = y*(1.5 - 0.5*x*y*y)
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mult $f11, $f13, $f11 # fm : $f11 = x * 0.5 * yp
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mult $f11, $f13, $f11 # fm : $f11 = (x * 0.5 * yp) * yp
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subt $f18, $f11, $f1 # fa : $f1= (1.5-2^-30) - 0.5*x*yp*yp
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mult $f13, $f1, $f13 # fm : ypp = $f13 = yp*$f1
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subt $f15, $f12, $f1 # fa : $f1 = (1.5 - 0.5)
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ldt $f15, $UP($4) # .. e1 :
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mult $f16, $f13, $f10 # fm : z = $f10 = x * ypp
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mult $f10, $f13, $f11 # fm : $f11 = z*ypp
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mult $f10, $f12, $f12 # fm : $f12 = z*0.5
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subt $f1, $f11, $f1 # .. fa : $f1 = 1 - z*ypp
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mult $f12, $f1, $f12 # fm : $f12 = z*0.5*(1 - z*ypp)
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addt $f10, $f12, $f0 # fa : zp=res=$f0= z + z*0.5*(1 - z*ypp)
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mult/c $f0, $f14, $f12 # fm : zmi = zp * DN
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mult/c $f0, $f15, $f11 # fm : zpl = zp * UP
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mult/c $f0, $f12, $f1 # fm : $f1 = zp * zmi
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mult/c $f0, $f11, $f15 # fm : $f15 = zp * zpl
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subt $f1, $f16, $f13 # fa : y1 = zp*zmi - x
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subt $f15, $f16, $f15 # fa : y2 = zp*zpl - x
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fcmovge $f13, $f12, $f0 # res = (y1 >= 0) ? zmi : res
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fcmovlt $f15, $f11, $f0 # res = (y2 < 0) ? zpl : res
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addq $sp, 16, $sp # e0 :
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ret # .. e1 :
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$negative:
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ldt $f0, $NAN($4)
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addq $sp, 16, $sp
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ret
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.end __ieee754_sqrt");
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#endif /* _IEEE_FP */
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#endif /* _IEEE_FP && _IEEE_FP_INEXACT */
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