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acos-td.c
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acos-td.c
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/*
* Correctly rounded arccosine
*
* Author : Christoph Lauter (ENS Lyon)
*
* This file is part of the crlibm library developed by the Arenaire
* project at Ecole Normale Superieure de Lyon
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU Lesser General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
*/
#include <stdio.h>
#include <stdlib.h>
#include "crlibm.h"
#include "crlibm_private.h"
#include "triple-double.h"
#include "acos-td.h"
#define AVOID_FMA 1
void acos_accurate_lower(double *acosh, double *acosm, double *acosl, double x, double xSqh, double xSql, double sign) {
double highPoly;
double t1h, t1l, t2h, t2l, t3h, t3l, t4h, t4l, t5h, t5l, t6h, t6l, t7h, t7l;
double tt1h, tt1l;
double t8h, t8m, t8l, t9h, t9m, t9l, t10h, t10m, t10l, t11h, t11m, t11l, t12h, t12m, t12l;
double tt8h, tt8m, tt8l, tt9h, tt9m, tt9l, tt10h, tt10m, tt10l, tt11h, tt11m, tt11l, tt12h, tt12m, tt12l;
double xCubeh, xCubem, xCubel, tt13h, tt13m, tt13l, t13h, t13m, t13l, polyh, polym, polyl;
double tt11hover, tt11mover, tt11lover;
double zw1h, zw1m, zw1l, acoshover, acosmover, acoslover;
#if EVAL_PERF
crlibm_second_step_taken++;
#endif
/* Evaluate the polynomial of degree 37
Its coefficients start at tbl[0]
p(x) = x + x * x^2 * (c3 + x^2 * (c5 + ...
We receive x^2 as xSqh + xSql = x * x (exactly)
in argument
|x| <= 0.185 = 2^(-2.43)
Compute monomials 27 to 37 in double precision
monomials 13 to 25 in double-double and
1 to 11 in triple-double precision in a
modified Horner form
*/
/* Double computations */
#if defined(PROCESSOR_HAS_FMA) && !defined(AVOID_FMA)
highPoly = FMA(FMA(FMA(FMA(FMA(tbl[33],xSqh,tbl[32]),xSqh,tbl[31]),xSqh,tbl[30]),xSqh,tbl[29]),xSqh,tbl[28]);
#else
highPoly = tbl[28] + xSqh * (tbl[29] + xSqh * (tbl[30] + xSqh * (tbl[31] + xSqh * (tbl[32] + xSqh * tbl[33]))));
#endif
/* Double-double computations */
Mul12(&tt1h,&tt1l,xSqh,highPoly);
Add22(&t1h,&t1l,tbl[27],0,tt1h,tt1l);
MulAdd22(&t2h,&t2l,tbl[25],tbl[26],xSqh,xSql,t1h,t1l);
MulAdd22(&t3h,&t3l,tbl[23],tbl[24],xSqh,xSql,t2h,t2l);
MulAdd22(&t4h,&t4l,tbl[21],tbl[22],xSqh,xSql,t3h,t3l);
MulAdd22(&t5h,&t5l,tbl[19],tbl[20],xSqh,xSql,t4h,t4l);
MulAdd22(&t6h,&t6l,tbl[17],tbl[18],xSqh,xSql,t5h,t5l);
MulAdd22(&t7h,&t7l,tbl[15],tbl[16],xSqh,xSql,t6h,t6l);
/* Triple-double computations */
Mul23(&tt8h,&tt8m,&tt8l,xSqh,xSql,t7h,t7l); /* 149 - 48/53 */
Add33(&t8h,&t8m,&t8l,tbl[12],tbl[13],tbl[14],tt8h,tt8m,tt8l); /* 145 - 43/53 */
Mul233(&tt9h,&tt9m,&tt9l,xSqh,xSql,t8h,t8m,t8l); /* 139 - 39/53 */
Add33(&t9h,&t9m,&t9l,tbl[9],tbl[10],tbl[11],tt9h,tt9m,tt9l); /* 136 - 34/53 */
Mul233(&tt10h,&tt10m,&tt10l,xSqh,xSql,t9h,t9m,t9l); /* 130 - 30/53 */
Add33(&t10h,&t10m,&t10l,tbl[6],tbl[7],tbl[8],tt10h,tt10m,tt10l); /* 127 - 25/53 */
Mul233(&tt11hover,&tt11mover,&tt11lover,xSqh,xSql,t10h,t10m,t10l); /* 121 - 21/53 */
Renormalize3(&tt11h,&tt11m,&tt11l,tt11hover,tt11mover,tt11lover); /* infty - 52/53 */
Add33(&t11h,&t11m,&t11l,tbl[3],tbl[4],tbl[5],tt11h,tt11m,tt11l); /* 149 - 47/53 */
Mul233(&tt12h,&tt12m,&tt12l,xSqh,xSql,t11h,t11m,t11l); /* 143 - 43/53 */
Add33(&t12h,&t12m,&t12l,tbl[0],tbl[1],tbl[2],tt12h,tt12m,tt12l); /* 140 - 38/53 */
Mul123(&xCubeh,&xCubem,&xCubel,x,xSqh,xSql); /* 154 - 47/53 */
Mul33(&tt13h,&tt13m,&tt13l,xCubeh,xCubem,xCubel,t12h,t12m,t12l); /* 136 - 34/53 */
Add133(&t13h,&t13m,&t13l,x,tt13h,tt13m,tt13l); /* 138 - 32/53 */
Renormalize3(&polyh,&polym,&polyl,t13h,t13m,t13l); /* infty - 52/53 */
/* Reconstruction:
- Multiply by the inverted sign
- Add Pi/2 in triple-double
- Renormalize
*/
zw1h = -sign * polyh;
zw1m = -sign * polym;
zw1l = -sign * polyl;
Add33(&acoshover,&acosmover,&acoslover,PIHALFH,PIHALFM,PIHALFL,zw1h,zw1m,zw1l);
Renormalize3(acosh,acosm,acosl,acoshover,acosmover,acoslover);
}
void acos_accurate_middle(double *acosh, double *acosm, double *acosl, double z, int i, double sign) {
double highPoly;
double t1h, t1l, t2h, t2l, t3h, t3l, t4h, t4l, t5h, t5l, t6h, t6l, t7h, t7l, t8h, t8l, t9h, t9l;
double t10h, t10m, t10l, t11h, t11m, t11l, t12h, t12m, t12l, t13h, t13m, t13l, t14h, t14m, t14l;
double t15h, t15m, t15l, t16h, t16m, t16l;
double tt1h, tt1l;
double tt10h, tt10m, tt10l, tt11h, tt11m, tt11l, tt12h, tt12m, tt12l;
double tt13h, tt13m, tt13l, tt14h, tt14m, tt14l, tt15h, tt15m, tt15l, tt16h, tt16m, tt16l;
double polyh, polym, polyl, tt13hover, tt13mover, tt13lover;
double zw1h, zw1m, zw1l, acoshover, acosmover, acoslover;
#if EVAL_PERF
crlibm_second_step_taken++;
#endif
/* Evaluate the polynomial of degree 35
Its coefficients start at tbl[i+1]
Evaluate degrees 35 to 20 in double precision,
degrees 20 to 7 in double-double precision and
finally degrees 6 to 1 in triple-double.
The constant coefficient is a double-double, the
computations are nevertheless in triple-double
*/
/* Double computations */
#if defined(PROCESSOR_HAS_FMA) && !defined(AVOID_FMA)
highPoly = FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(
tbl[i+58] ,z,tbl[i+57]),z,tbl[i+56]),z,tbl[i+55]),z,tbl[i+54]),z,
tbl[i+53]),z,tbl[i+52]),z,tbl[i+51]),z,tbl[i+50]),z,tbl[i+49]),z,
tbl[i+48]),z,tbl[i+47]),z,tbl[i+46]),z,tbl[i+45]),z,tbl[i+44]),z,
tbl[i+43]),z,tbl[i+42]),z,tbl[i+41]),z,tbl[i+40]),z,tbl[i+39]);
#else
highPoly = tbl[i+39] + z * (tbl[i+40] + z * (tbl[i+41] + z * (tbl[i+42] + z * (
tbl[i+43] + z * (tbl[i+44] + z * (tbl[i+45] + z * (tbl[i+46] + z * (
tbl[i+47] + z * (tbl[i+48] + z * (tbl[i+49] + z * (tbl[i+50] + z * (
tbl[i+51] + z * (tbl[i+52] + z * (tbl[i+53] + z * (tbl[i+54] + z * (
tbl[i+55] + z * (tbl[i+56] + z * (tbl[i+57] + z * tbl[i+58]))))))))))))))))));
#endif
/* Double-double computations */
Mul12(&tt1h,&tt1l,z,highPoly);
Add22(&t1h,&t1l,tbl[i+37],tbl[i+38],tt1h,tt1l);
MulAdd212(&t2h,&t2l,tbl[i+35],tbl[i+36],z,t1h,t1l);
MulAdd212(&t3h,&t3l,tbl[i+33],tbl[i+34],z,t2h,t2l);
MulAdd212(&t4h,&t4l,tbl[i+31],tbl[i+32],z,t3h,t3l);
MulAdd212(&t5h,&t5l,tbl[i+29],tbl[i+30],z,t4h,t4l);
MulAdd212(&t6h,&t6l,tbl[i+27],tbl[i+28],z,t5h,t5l);
MulAdd212(&t7h,&t7l,tbl[i+25],tbl[i+26],z,t6h,t6l);
MulAdd212(&t8h,&t8l,tbl[i+23],tbl[i+24],z,t7h,t7l);
MulAdd212(&t9h,&t9l,tbl[i+21],tbl[i+22],z,t8h,t8l);
/* Triple-double computations */
Mul123(&tt10h,&tt10m,&tt10l,z,t9h,t9l); /* 154 - 47/53 */
Add33(&t10h,&t10m,&t10l,tbl[i+18],tbl[i+19],tbl[i+20],tt10h,tt10m,tt10l); /* 144 - 42/53 */
Mul133(&tt11h,&tt11m,&tt11l,z,t10h,t10m,t10l); /* 142 - 38/53 */
Add33(&t11h,&t11m,&t11l,tbl[i+15],tbl[i+16],tbl[i+17],tt11h,tt11m,tt11l); /* 136 - 33/53 */
Mul133(&tt12h,&tt12m,&tt12l,z,t11h,t11m,t11l); /* 133 - 28/53 */
Add33(&t12h,&t12m,&t12l,tbl[i+12],tbl[i+13],tbl[i+14],tt12h,tt12m,tt12l); /* 125 - 23/53 */
Mul133(&tt13hover,&tt13mover,&tt13lover,z,t12h,t12m,t12l); /* 123 - 18/53 */
Renormalize3(&tt13h,&tt13m,&tt13l,tt13hover,tt13mover,tt13lover); /* infty - 52/53 */
Add33(&t13h,&t13m,&t13l,tbl[i+9],tbl[i+10],tbl[i+11],tt13h,tt13m,tt13l); /* 149 - 47/53 */
Mul133(&tt14h,&tt14m,&tt14l,z,t13h,t13m,t13l); /* 147 - 42/53 */
Add33(&t14h,&t14m,&t14l,tbl[i+6],tbl[i+7],tbl[i+8],tt14h,tt14m,tt14l); /* 139 - 37/53 */
Mul133(&tt15h,&tt15m,&tt15l,z,t14h,t14m,t14l); /* 137 - 32/53 */
Add33(&t15h,&t15m,&t15l,tbl[i+3],tbl[i+4],tbl[i+5],tt15h,tt15m,tt15l); /* 129 - 28/53 */
Mul133(&tt16h,&tt16m,&tt16l,z,t15h,t15m,t15l); /* 128 - 23/53 */
Add233(&t16h,&t16m,&t16l,tbl[i+1],tbl[i+2],tt16h,tt16m,tt16l); /* 126 - 19/53 */
Renormalize3(&polyh,&polym,&polyl,t16h,t16m,t16l); /* infty - 52/53 */
/* Reconstruction:
- Multiply by the inverted sign
- Add Pi/2 in triple-double
- Renormalize
*/
zw1h = -sign * polyh;
zw1m = -sign * polym;
zw1l = -sign * polyl;
Add33(&acoshover,&acosmover,&acoslover,PIHALFH,PIHALFM,PIHALFL,zw1h,zw1m,zw1l);
Renormalize3(acosh,acosm,acosl,acoshover,acosmover,acoslover);
}
void acos_accurate_higher(double *acosh, double *acosm, double *acosl, double z, double sign) {
double highPoly;
double tt1h, tt1l;
double t1h, t1l, t2h, t2l, t3h, t3l, t4h, t4l, t5h, t5l, t6h, t6l, t7h, t7l, t8h, t8l;
double tt10h, tt10m, tt10l, tt11h, tt11m, tt11l, tt12h, tt12m, tt12l, tt13h, tt13m, tt13l;
double tt14h, tt14m, tt14l, tt15h, tt15m, tt15l, tt16h, tt16m, tt16l, tt17h, tt17m, tt17l;
double t9h, t9l, t10h, t10m, t10l, t11h, t11m, t11l, t12h, t12m, t12l, t13h, t13m, t13l;
double t14h, t14m, t14l, t15h, t15m, t15l, t16h, t16m, t16l, t17h, t17m, t17l;
double tt18h, tt18m, tt18l, polyh, polym, polyl;
double sqrtzh, sqrtzm, sqrtzl, twoZ, pTimesSh, pTimesSm, pTimesSl;
double allh, allm, alll;
double tt13hover, tt13mover, tt13lover, tt16hover, tt16mover, tt16lover;
double polyhover, polymover, polylover;
#if EVAL_PERF
crlibm_second_step_taken++;
#endif
/* We evaluate acos(x) with x > 0 as
acos(x) = -1 * f(z) * sqrt(2*z)
with z = 1 - x and
f(z) = (asin(z) - Pi/2) / sqrt(2*z)
f(z) is approximated by p(z)
The polynomial p(z) is of degree 29
Its coefficients start at tbl[TBLIDX10]
Coefficients for degrees 29 to 18 are in double precision,
for degrees 17 to 9 in double-double precision and
finally for degrees 8 to 1 in triple-double.
The constant coefficient (-1) is not stored in the table,
the computations are nevertheless in triple-double
We evaluate the monomials in the precision in which
the correspondant coefficients are stored
The coefficients' values decrease very quickly
so even with |z| < 2^-2.18 we can compute degree 18
already in double precision
Compute than sqrt(2*z) as a triple-double
multiply in triple-double.
*/
/* Double computations */
#if defined(PROCESSOR_HAS_FMA) && !defined(AVOID_FMA)
highPoly = FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(
tbl[TBLIDX10+53] ,z,tbl[TBLIDX10+52]),z,tbl[TBLIDX10+51]),z,
tbl[TBLIDX10+50]),z,tbl[TBLIDX10+49]),z,tbl[TBLIDX10+48]),z,
tbl[TBLIDX10+47]),z,tbl[TBLIDX10+46]),z,tbl[TBLIDX10+45]),z,
tbl[TBLIDX10+44]),z,tbl[TBLIDX10+43]),z,tbl[TBLIDX10+42]);
#else
highPoly = tbl[TBLIDX10+42] + z * (tbl[TBLIDX10+43] + z * (tbl[TBLIDX10+44] + z * (
tbl[TBLIDX10+45] + z * (tbl[TBLIDX10+46] + z * (tbl[TBLIDX10+47] + z * (
tbl[TBLIDX10+48] + z * (tbl[TBLIDX10+49] + z * (tbl[TBLIDX10+50] + z * (
tbl[TBLIDX10+51] + z * (tbl[TBLIDX10+52] + z * tbl[TBLIDX10+53]))))))))));
#endif
/* Double-double computations */
Mul12(&tt1h,&tt1l,z,highPoly);
Add22(&t1h,&t1l,tbl[TBLIDX10+40],tbl[TBLIDX10+41],tt1h,tt1l);
MulAdd212(&t2h,&t2l,tbl[TBLIDX10+38],tbl[TBLIDX10+39],z,t1h,t1l);
MulAdd212(&t3h,&t3l,tbl[TBLIDX10+36],tbl[TBLIDX10+37],z,t2h,t2l);
MulAdd212(&t4h,&t4l,tbl[TBLIDX10+34],tbl[TBLIDX10+35],z,t3h,t3l);
MulAdd212(&t5h,&t5l,tbl[TBLIDX10+32],tbl[TBLIDX10+33],z,t4h,t4l);
MulAdd212(&t6h,&t6l,tbl[TBLIDX10+30],tbl[TBLIDX10+31],z,t5h,t5l);
MulAdd212(&t7h,&t7l,tbl[TBLIDX10+28],tbl[TBLIDX10+29],z,t6h,t6l);
MulAdd212(&t8h,&t8l,tbl[TBLIDX10+26],tbl[TBLIDX10+27],z,t7h,t7l);
MulAdd212(&t9h,&t9l,tbl[TBLIDX10+24],tbl[TBLIDX10+25],z,t8h,t8l);
/* Triple-double computations */
Mul123(&tt10h,&tt10m,&tt10l,z,t9h,t9l); /* 154 - 47/53 */
Add33(&t10h,&t10m,&t10l,tbl[TBLIDX10+21],tbl[TBLIDX10+22],tbl[TBLIDX10+23],tt10h,tt10m,tt10l); /* 144 - 42/53 */
Mul133(&tt11h,&tt11m,&tt11l,z,t10h,t10m,t10l); /* 142 - 37/53 */
Add33(&t11h,&t11m,&t11l,tbl[TBLIDX10+18],tbl[TBLIDX10+19],tbl[TBLIDX10+20],tt11h,tt11m,tt11l); /* 134 - 32/53 */
Mul133(&tt12h,&tt12m,&tt12l,z,t11h,t11m,t11l); /* 132 - 27/53 */
Add33(&t12h,&t12m,&t12l,tbl[TBLIDX10+15],tbl[TBLIDX10+16],tbl[TBLIDX10+17],tt12h,tt12m,tt12l); /* 124 - 22/53 */
Mul133(&tt13hover,&tt13mover,&tt13lover,z,t12h,t12m,t12l); /* 122 - 17/53 */
Renormalize3(&tt13h,&tt13m,&tt13l,tt13hover,tt13mover,tt13lover); /* infty - 52/53 */
Add33(&t13h,&t13m,&t13l,tbl[TBLIDX10+12],tbl[TBLIDX10+13],tbl[TBLIDX10+14],tt13h,tt13m,tt13l); /* 149 - 47/53 */
Mul133(&tt14h,&tt14m,&tt14l,z,t13h,t13m,t13l); /* 147 - 42/53 */
Add33(&t14h,&t14m,&t14l,tbl[TBLIDX10+9],tbl[TBLIDX10+10],tbl[TBLIDX10+11],tt14h,tt14m,tt14l); /* 139 - 37/53 */
Mul133(&tt15h,&tt15m,&tt15l,z,t14h,t14m,t14l); /* 137 - 32/53 */
Add33(&t15h,&t15m,&t15l,tbl[TBLIDX10+6],tbl[TBLIDX10+7],tbl[TBLIDX10+8],tt15h,tt15m,tt15l); /* 129 - 27/53 */
Mul133(&tt16hover,&tt16mover,&tt16lover,z,t15h,t15m,t15l); /* 127 - 22/53 */
Renormalize3(&tt16h,&tt16m,&tt16l,tt16hover,tt16mover,tt16lover); /* infty - 52/53 */
Add33(&t16h,&t16m,&t16l,tbl[TBLIDX10+3],tbl[TBLIDX10+4],tbl[TBLIDX10+5],tt16h,tt16m,tt16l); /* 149 - 47/53 */
Mul133(&tt17h,&tt17m,&tt17l,z,t16h,t16m,t16l); /* 147 - 42/53 */
Add33(&t17h,&t17m,&t17l,tbl[TBLIDX10+0],tbl[TBLIDX10+1],tbl[TBLIDX10+2],tt17h,tt17m,tt17l); /* 139 - 37/53 */
Mul133(&tt18h,&tt18m,&tt18l,z,t17h,t17m,t17l); /* 137 - 32/53 */
Add133(&polyhover,&polymover,&polylover,-1,tt18h,tt18m,tt18l); /* 136 - 30/53 */
Renormalize3(&polyh,&polym,&polyl,polyhover,polymover,polylover); /* infty - 52/53 */
/* Compute sqrt(2*z) as a triple-double */
twoZ = 2 * z;
Sqrt13(&sqrtzh,&sqrtzm,&sqrtzl,twoZ); /* 146 - 52/53 */
/* Multiply p(z) by sqrt(2*z) */
Mul33(&pTimesSh,&pTimesSm,&pTimesSl,polyh,polym,polyl,sqrtzh,sqrtzm,sqrtzl); /* 139 - 48/53 */
/* Reconstruction:
If the sign of x in acos(x) was positive:
- Multiply pTimesSh + pTimesSm + pTimesSl approx f(x) * sqrt(2 * z) by -1
- Renormalize
- Return
Otherwise:
- Add Pi in triple-double to pTimesSh + pTimesSm + pTimesSl approx f(x) * sqrt(2 * z)
- Renormalize
- Return
*/
if (sign > 0) {
allh = -1.0 * pTimesSh;
allm = -1.0 * pTimesSm;
alll = -1.0 * pTimesSl; /* 139 - 48/53 */
} else {
Add33(&allh,&allm,&alll,PIH,PIM,PIL,pTimesSh,pTimesSm,pTimesSl); /* 130 - 43/53 */
}
Renormalize3(acosh,acosm,acosl,allh,allm,alll); /* infty - 52/53 */
}
double acos_rn(double x) {
db_number xdb;
double sign, z, acosh, acosm, acosl;
int i;
double xSqh, xSql;
double tt1h, tt1l;
double tt6h, tt6l;
double t1h, t1l, t2h, t2l, t3h, t3l, t4h, t4l, t5h, t5l, t6h, t6l;
double t7h, t7l, t8h, t8l, polyh, polyl, twoZ, sqrtzh, sqrtzl;
double pTimesSh, pTimesSl, highPoly, xCubeh, xCubel;
double tmp1, tmp2, tmp3, tmp4, tmp5;
double zw1h, zw1l;
/*
#if CRLIBM_REQUIRES_ROUNDING_MODE_CHANGE
SAVE_STATE_AND_SET_RNDOUBLE
#endif
*/
/* Transform the argument into integer */
xdb.d = x;
/* Special case handling */
/* Exact algebraic case x = 1, acos(1) = 0 */
if (x == 1.0) return 0.0;
/* Strip off the sign of argument x */
if (xdb.i[HI] & 0x80000000) sign = -1; else sign = 1;
xdb.i[HI] &= 0x7fffffff;
/* acos is defined on -1 <= x <= 1, elsewhere it is NaN */
if ((xdb.i[HI] > 0x3ff00000) || ((xdb.i[HI] == 0x3ff00000) && (xdb.i[LO] != 0x00000000))) {
return (x-x)/0.0; /* return NaN */
}
/* If |x| < 2^(-120) we have
round(acos(x)) = round(pi/2)
So we can decide the rounding without any computation
*/
if (xdb.i[HI] < 0x38700000) {
return PIHALFDOUBLERN;
}
/* Recast x */
x = xdb.d;
/* Find correspondant interval and compute index to the table
We start by filtering the two special cases around 0 and 1
*/
if (xdb.i[HI] < BOUND1) {
/* Special interval 0..BOUND1
The polynomial has no even monomials
We must prove extra accuracy in the interval 0..sin(2^(-18))
*/
/* Quick phase starts */
/* Compute square of x for both quick and accurate phases */
Mul12(&xSqh,&xSql,x,x);
tmp4 = tbl[3];
tmp5 = tbl[4];
t4h = tmp4;
t4l = tmp5;
if (xdb.i[HI] > EXTRABOUND) {
/* Double precision evaluation */
#if defined(PROCESSOR_HAS_FMA) && !defined(AVOID_FMA)
highPoly = FMA(FMA(FMA(FMA(tbl[23],xSqh,tbl[21]),xSqh,tbl[19]),xSqh,tbl[17]),xSqh,tbl[15]);
#else
highPoly = tbl[15] + xSqh * (tbl[17] + xSqh * (tbl[19] + xSqh * (tbl[21] + xSqh * tbl[23])));
#endif
/* Double-double precision evaluation */
Mul12(&tt1h,&tt1l,xSqh,highPoly);
Add22(&t1h,&t1l,tbl[12],tbl[13],tt1h,tt1l);
MulAdd212(&t2h,&t2l,tbl[9],tbl[10],xSqh,t1h,t1l);
MulAdd212(&t3h,&t3l,tbl[6],tbl[7],xSqh,t2h,t2l);
MulAdd22(&t4h,&t4l,tmp4,tmp5,xSqh,xSql,t3h,t3l);
}
MulAdd22(&t5h,&t5l,tbl[0],tbl[1],xSqh,xSql,t4h,t4l);
Mul122(&xCubeh,&xCubel,x,xSqh,xSql);
Mul22(&tt6h,&tt6l,xCubeh,xCubel,t5h,t5l);
Add12(tmp1,tmp2,x,tt6h);
tmp3 = tmp2 + tt6l;
Add12(polyh,polyl,tmp1,tmp3);
/* Reconstruction:
- Multiply by the inverted sign
- Add Pi/2 in double-double precision
*/
zw1h = -sign * polyh;
zw1l = -sign * polyl;
Add22(&acosh,&acosm,PIHALFH,PIHALFM,zw1h,zw1l);
/* Rounding test
The RN rounding constant is at tbl[34]
*/
if(acosh == (acosh + (acosm * tbl[34])))
return acosh;
/* Launch accurate phase */
acos_accurate_lower(&acosh,&acosm,&acosl,x,xSqh,xSql,sign);
ReturnRoundToNearest3(acosh,acosm,acosl);
}
if (xdb.i[HI] >= BOUND9) {
/* Special interval BOUND9..1
We use an asymptotic development of arcsin in sqrt(1 - x)
*/
/* Argument reduction for quick and accurate phase
z = 1 - x
The operation is exact as per Sterbenz' lemma
*/
z = 1 - x;
/* Quick phase starts */
/* Double precision evaluation */
#if defined(PROCESSOR_HAS_FMA) && !defined(AVOID_FMA)
highPoly = FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(
tbl[TBLIDX10+42] ,z,tbl[TBLIDX10+40]),z,tbl[TBLIDX10+38]),z,
tbl[TBLIDX10+36]),z,tbl[TBLIDX10+34]),z,tbl[TBLIDX10+32]),z,
tbl[TBLIDX10+30]),z,tbl[TBLIDX10+28]),z,tbl[TBLIDX10+26]),z,
tbl[TBLIDX10+24]);
#else
highPoly = tbl[TBLIDX10+24] + z * (tbl[TBLIDX10+26] + z * (tbl[TBLIDX10+28] + z * (
tbl[TBLIDX10+30] + z * (tbl[TBLIDX10+32] + z * (tbl[TBLIDX10+34] + z * (
tbl[TBLIDX10+36] + z * (tbl[TBLIDX10+38] + z * (tbl[TBLIDX10+40] + z *
tbl[TBLIDX10+42]))))))));
#endif
/* Double-double precision evaluation */
Mul12(&tt1h,&tt1l,z,highPoly);
Add22(&t1h,&t1l,tbl[TBLIDX10+21],tbl[TBLIDX10+22],tt1h,tt1l);
MulAdd212(&t2h,&t2l,tbl[TBLIDX10+18],tbl[TBLIDX10+19],z,t1h,t1l);
MulAdd212(&t3h,&t3l,tbl[TBLIDX10+15],tbl[TBLIDX10+16],z,t2h,t2l);
MulAdd212(&t4h,&t4l,tbl[TBLIDX10+12],tbl[TBLIDX10+13],z,t3h,t3l);
MulAdd212(&t5h,&t5l,tbl[TBLIDX10+9],tbl[TBLIDX10+10],z,t4h,t4l);
MulAdd212(&t6h,&t6l,tbl[TBLIDX10+6],tbl[TBLIDX10+7],z,t5h,t5l);
MulAdd212(&t7h,&t7l,tbl[TBLIDX10+3],tbl[TBLIDX10+4],z,t6h,t6l);
MulAdd212(&t8h,&t8l,tbl[TBLIDX10+0],tbl[TBLIDX10+1],z,t7h,t7l);
MulAdd212(&polyh,&polyl,-1,0,z,t8h,t8l);
/* Compute sqrt(2*z) as a double-double */
twoZ = 2 * z;
sqrt12(&sqrtzh,&sqrtzl,twoZ);
/* Multiply p(z) by sqrt(2*z) and add Pi/2 */
Mul22(&pTimesSh,&pTimesSl,polyh,polyl,sqrtzh,sqrtzl);
/* Reconstruction:
If the sign of x in acos(x) was positive:
- Multiply pTimesSh + pTimesSl approx f(x) * sqrt(2 * z) by -1
- Return
Otherwise:
- Add Pi in triple-double to pTimesSh + pTimesSl approx f(x) * sqrt(2 * z)
- Return
*/
if (sign > 0) {
acosh = -1.0 * pTimesSh;
acosm = -1.0 * pTimesSl;
} else {
Add22(&acosh,&acosm,PIH,PIM,pTimesSh,pTimesSl);
}
/* Rounding test
The RN rounding constant is at tbl[TBLIDX10+54]
*/
if(acosh == (acosh + (acosm * tbl[TBLIDX10+54])))
return acosh;
/* Launch accurate phase */
acos_accurate_higher(&acosh,&acosm,&acosl,z,sign);
ReturnRoundToNearest3(acosh,acosm,acosl);
}
/* General 8 main intervals
We can already suppose that BOUND1 <= x <= BOUND9
*/
if (xdb.i[HI] < BOUND5) {
if (xdb.i[HI] < BOUND3) {
if (xdb.i[HI] < BOUND2) i = TBLIDX2; else i = TBLIDX3;
} else {
if (xdb.i[HI] < BOUND4) i = TBLIDX4; else i = TBLIDX5;
}
} else {
if (xdb.i[HI] < BOUND7) {
if (xdb.i[HI] < BOUND6) i = TBLIDX6; else i = TBLIDX7;
} else {
if (xdb.i[HI] < BOUND8) i = TBLIDX8; else i = TBLIDX9;
}
}
/* Argument reduction
i points to the interval midpoint value in the table
*/
z = x - tbl[i];
/* Quick phase starts */
/* Double precision evaluation */
#if defined(PROCESSOR_HAS_FMA) && !defined(AVOID_FMA)
highPoly = FMA(FMA(FMA(FMA(FMA(FMA(FMA(
tbl[i+35] ,z,tbl[i+33]),z,tbl[i+31]),z,tbl[i+29]),z,
tbl[i+27]),z,tbl[i+25]),z,tbl[i+23]),z,tbl[i+21]);
#else
highPoly = tbl[i+21] + z * (tbl[i+23] + z * (tbl[i+25] + z * (
tbl[i+27] + z * (tbl[i+29] + z * (tbl[i+31] + z * (
tbl[i+33] + z * tbl[i+35]))))));
#endif
/* Double-double precision evaluation */
Mul12(&tt1h,&tt1l,z,highPoly);
Add22(&t1h,&t1l,tbl[i+18],tbl[i+19],tt1h,tt1l);
MulAdd212(&t2h,&t2l,tbl[i+15],tbl[i+16],z,t1h,t1l);
MulAdd212(&t3h,&t3l,tbl[i+12],tbl[i+13],z,t2h,t2l);
MulAdd212(&t4h,&t4l,tbl[i+9],tbl[i+10],z,t3h,t3l);
MulAdd212(&t5h,&t5l,tbl[i+6],tbl[i+7],z,t4h,t4l);
MulAdd212(&t6h,&t6l,tbl[i+3],tbl[i+4],z,t5h,t5l);
MulAdd212(&polyh,&polyl,tbl[i+1],tbl[i+2],z,t6h,t6l);
/* Reconstruction:
- Multiply by the inverted sign
- Add Pi/2 in double-double precision
*/
zw1h = -sign * polyh;
zw1l = -sign * polyl;
Add22(&acosh,&acosm,PIHALFH,PIHALFM,zw1h,zw1l);
/* Rounding test
The RN rounding constant is at tbl[i+59]
*/
if(acosh == (acosh + (acosm * tbl[i+59])))
return acosh;
/* Launch accurate phase */
acos_accurate_middle(&acosh,&acosm,&acosl,z,i,sign);
ReturnRoundToNearest3(acosh,acosm,acosl);
}
double acos_ru(double x) {
db_number xdb;
double sign, z, acosh, acosm, acosl;
int i;
double xSqh, xSql;
double tt1h, tt1l;
double tt6h, tt6l;
double t1h, t1l, t2h, t2l, t3h, t3l, t4h, t4l, t5h, t5l, t6h, t6l;
double t7h, t7l, t8h, t8l, polyh, polyl, twoZ, sqrtzh, sqrtzl;
double pTimesSh, pTimesSl, highPoly, xCubeh, xCubel;
double tmp1, tmp2, tmp3, tmp4, tmp5;
double zw1h, zw1l;
/* Transform the argument into integer */
xdb.d = x;
/* Special case handling */
/* Exact algebraic case x = 1, acos(1) = 0 */
if (x == 1.0) return 0.0;
/* Strip off the sign of argument x */
if (xdb.i[HI] & 0x80000000) sign = -1; else sign = 1;
xdb.i[HI] &= 0x7fffffff;
/* acos is defined on -1 <= x <= 1, elsewhere it is NaN */
if ((xdb.i[HI] > 0x3ff00000) || ((xdb.i[HI] == 0x3ff00000) && (xdb.i[LO] != 0x00000000))) {
return (x-x)/0.0; /* return NaN */
}
/* If |x| < 2^(-120) we have
round(acos(x)) = round(pi/2)
So we can decide the rounding without any computation
*/
if (xdb.i[HI] < 0x38700000) {
return PIHALFDOUBLERU;
}
/* Recast x */
x = xdb.d;
/* Find correspondant interval and compute index to the table
We start by filtering the two special cases around 0 and 1
*/
if (xdb.i[HI] < BOUND1) {
/* Special interval 0..BOUND1
The polynomial has no even monomials
We must prove extra accuracy in the interval 0..sin(2^(-18))
*/
/* Quick phase starts */
/* Compute square of x for both quick and accurate phases */
Mul12(&xSqh,&xSql,x,x);
tmp4 = tbl[3];
tmp5 = tbl[4];
t4h = tmp4;
t4l = tmp5;
if (xdb.i[HI] > EXTRABOUND) {
/* Double precision evaluation */
#if defined(PROCESSOR_HAS_FMA) && !defined(AVOID_FMA)
highPoly = FMA(FMA(FMA(FMA(tbl[23],xSqh,tbl[21]),xSqh,tbl[19]),xSqh,tbl[17]),xSqh,tbl[15]);
#else
highPoly = tbl[15] + xSqh * (tbl[17] + xSqh * (tbl[19] + xSqh * (tbl[21] + xSqh * tbl[23])));
#endif
/* Double-double precision evaluation */
Mul12(&tt1h,&tt1l,xSqh,highPoly);
Add22(&t1h,&t1l,tbl[12],tbl[13],tt1h,tt1l);
MulAdd212(&t2h,&t2l,tbl[9],tbl[10],xSqh,t1h,t1l);
MulAdd212(&t3h,&t3l,tbl[6],tbl[7],xSqh,t2h,t2l);
MulAdd22(&t4h,&t4l,tmp4,tmp5,xSqh,xSql,t3h,t3l);
}
MulAdd22(&t5h,&t5l,tbl[0],tbl[1],xSqh,xSql,t4h,t4l);
Mul122(&xCubeh,&xCubel,x,xSqh,xSql);
Mul22(&tt6h,&tt6l,xCubeh,xCubel,t5h,t5l);
Add12(tmp1,tmp2,x,tt6h);
tmp3 = tmp2 + tt6l;
Add12(polyh,polyl,tmp1,tmp3);
/* Reconstruction:
- Multiply by the inverted sign
- Add Pi/2 in double-double precision
*/
zw1h = -sign * polyh;
zw1l = -sign * polyl;
Add22(&acosh,&acosm,PIHALFH,PIHALFM,zw1h,zw1l);
/* Rounding test
The RU rounding constant is at tbl[35]
*/
TEST_AND_RETURN_RU(acosh, acosm, tbl[35]);
/* Launch accurate phase */
acos_accurate_lower(&acosh,&acosm,&acosl,x,xSqh,xSql,sign);
ReturnRoundUpwards3(acosh,acosm,acosl);
}
if (xdb.i[HI] >= BOUND9) {
/* Special interval BOUND9..1
We use an asymptotic development of arcsin in sqrt(1 - x)
*/
/* Argument reduction for quick and accurate phase
z = 1 - x
The operation is exact as per Sterbenz' lemma
*/
z = 1 - x;
/* Quick phase starts */
/* Double precision evaluation */
#if defined(PROCESSOR_HAS_FMA) && !defined(AVOID_FMA)
highPoly = FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(FMA(
tbl[TBLIDX10+42] ,z,tbl[TBLIDX10+40]),z,tbl[TBLIDX10+38]),z,
tbl[TBLIDX10+36]),z,tbl[TBLIDX10+34]),z,tbl[TBLIDX10+32]),z,
tbl[TBLIDX10+30]),z,tbl[TBLIDX10+28]),z,tbl[TBLIDX10+26]),z,
tbl[TBLIDX10+24]);
#else
highPoly = tbl[TBLIDX10+24] + z * (tbl[TBLIDX10+26] + z * (tbl[TBLIDX10+28] + z * (
tbl[TBLIDX10+30] + z * (tbl[TBLIDX10+32] + z * (tbl[TBLIDX10+34] + z * (
tbl[TBLIDX10+36] + z * (tbl[TBLIDX10+38] + z * (tbl[TBLIDX10+40] + z *
tbl[TBLIDX10+42]))))))));
#endif
/* Double-double precision evaluation */
Mul12(&tt1h,&tt1l,z,highPoly);
Add22(&t1h,&t1l,tbl[TBLIDX10+21],tbl[TBLIDX10+22],tt1h,tt1l);
MulAdd212(&t2h,&t2l,tbl[TBLIDX10+18],tbl[TBLIDX10+19],z,t1h,t1l);
MulAdd212(&t3h,&t3l,tbl[TBLIDX10+15],tbl[TBLIDX10+16],z,t2h,t2l);
MulAdd212(&t4h,&t4l,tbl[TBLIDX10+12],tbl[TBLIDX10+13],z,t3h,t3l);
MulAdd212(&t5h,&t5l,tbl[TBLIDX10+9],tbl[TBLIDX10+10],z,t4h,t4l);
MulAdd212(&t6h,&t6l,tbl[TBLIDX10+6],tbl[TBLIDX10+7],z,t5h,t5l);
MulAdd212(&t7h,&t7l,tbl[TBLIDX10+3],tbl[TBLIDX10+4],z,t6h,t6l);
MulAdd212(&t8h,&t8l,tbl[TBLIDX10+0],tbl[TBLIDX10+1],z,t7h,t7l);
MulAdd212(&polyh,&polyl,-1,0,z,t8h,t8l);
/* Compute sqrt(2*z) as a double-double */
twoZ = 2 * z;
sqrt12(&sqrtzh,&sqrtzl,twoZ);
/* Multiply p(z) by sqrt(2*z) and add Pi/2 */
Mul22(&pTimesSh,&pTimesSl,polyh,polyl,sqrtzh,sqrtzl);
/* Reconstruction:
If the sign of x in acos(x) was positive:
- Multiply pTimesSh + pTimesSl approx f(x) * sqrt(2 * z) by -1
- Return
Otherwise:
- Add Pi in triple-double to pTimesSh + pTimesSl approx f(x) * sqrt(2 * z)
- Return
*/
if (sign > 0) {
acosh = -1.0 * pTimesSh;
acosm = -1.0 * pTimesSl;
} else {
Add22(&acosh,&acosm,PIH,PIM,pTimesSh,pTimesSl);
}
/* Rounding test
The RU rounding constant is at tbl[TBLIDX10+55]
*/
TEST_AND_RETURN_RU(acosh, acosm, tbl[TBLIDX10+55]);
/* Launch accurate phase */
acos_accurate_higher(&acosh,&acosm,&acosl,z,sign);
ReturnRoundUpwards3(acosh,acosm,acosl);
}
/* General 8 main intervals
We can already suppose that BOUND1 <= x <= BOUND9
*/
if (xdb.i[HI] < BOUND5) {
if (xdb.i[HI] < BOUND3) {
if (xdb.i[HI] < BOUND2) i = TBLIDX2; else i = TBLIDX3;
} else {
if (xdb.i[HI] < BOUND4) i = TBLIDX4; else i = TBLIDX5;
}
} else {
if (xdb.i[HI] < BOUND7) {
if (xdb.i[HI] < BOUND6) i = TBLIDX6; else i = TBLIDX7;
} else {
if (xdb.i[HI] < BOUND8) i = TBLIDX8; else i = TBLIDX9;
}
}
/* Argument reduction
i points to the interval midpoint value in the table
*/
z = x - tbl[i];
/* Quick phase starts */
/* Double precision evaluation */
#if defined(PROCESSOR_HAS_FMA) && !defined(AVOID_FMA)
highPoly = FMA(FMA(FMA(FMA(FMA(FMA(FMA(
tbl[i+35] ,z,tbl[i+33]),z,tbl[i+31]),z,tbl[i+29]),z,
tbl[i+27]),z,tbl[i+25]),z,tbl[i+23]),z,tbl[i+21]);
#else
highPoly = tbl[i+21] + z * (tbl[i+23] + z * (tbl[i+25] + z * (
tbl[i+27] + z * (tbl[i+29] + z * (tbl[i+31] + z * (
tbl[i+33] + z * tbl[i+35]))))));
#endif
/* Double-double precision evaluation */
Mul12(&tt1h,&tt1l,z,highPoly);
Add22(&t1h,&t1l,tbl[i+18],tbl[i+19],tt1h,tt1l);
MulAdd212(&t2h,&t2l,tbl[i+15],tbl[i+16],z,t1h,t1l);
MulAdd212(&t3h,&t3l,tbl[i+12],tbl[i+13],z,t2h,t2l);
MulAdd212(&t4h,&t4l,tbl[i+9],tbl[i+10],z,t3h,t3l);
MulAdd212(&t5h,&t5l,tbl[i+6],tbl[i+7],z,t4h,t4l);
MulAdd212(&t6h,&t6l,tbl[i+3],tbl[i+4],z,t5h,t5l);
MulAdd212(&polyh,&polyl,tbl[i+1],tbl[i+2],z,t6h,t6l);
/* Reconstruction:
- Multiply by the inverted sign
- Add Pi/2 in double-double precision
*/
zw1h = -sign * polyh;
zw1l = -sign * polyl;
Add22(&acosh,&acosm,PIHALFH,PIHALFM,zw1h,zw1l);
/* Rounding test
The RU rounding constant is at tbl[i+60]
*/
TEST_AND_RETURN_RU(acosh, acosm, tbl[i+60]);
/* Launch accurate phase */
acos_accurate_middle(&acosh,&acosm,&acosl,z,i,sign);
ReturnRoundUpwards3(acosh,acosm,acosl);
}
double acos_rd(double x) {
db_number xdb;
double sign, z, acosh, acosm, acosl;
int i;
double xSqh, xSql;
double tt1h, tt1l;
double tt6h, tt6l;
double t1h, t1l, t2h, t2l, t3h, t3l, t4h, t4l, t5h, t5l, t6h, t6l;
double t7h, t7l, t8h, t8l, polyh, polyl, twoZ, sqrtzh, sqrtzl;
double pTimesSh, pTimesSl, highPoly, xCubeh, xCubel;
double tmp1, tmp2, tmp3, tmp4, tmp5;
double zw1h, zw1l;
/* Transform the argument into integer */
xdb.d = x;
/* Special case handling */
/* Exact algebraic case x = 1, acos(1) = 0 */
if (x == 1.0) return 0.0;
/* Strip off the sign of argument x */
if (xdb.i[HI] & 0x80000000) sign = -1; else sign = 1;
xdb.i[HI] &= 0x7fffffff;
/* acos is defined on -1 <= x <= 1, elsewhere it is NaN */
if ((xdb.i[HI] > 0x3ff00000) || ((xdb.i[HI] == 0x3ff00000) && (xdb.i[LO] != 0x00000000))) {
return (x-x)/0.0; /* return NaN */
}
/* If |x| < 2^(-120) we have
round(acos(x)) = round(pi/2)
So we can decide the rounding without any computation
*/
if (xdb.i[HI] < 0x38700000) {
return PIHALFDOUBLERD;
}
/* Recast x */
x = xdb.d;
/* Find correspondant interval and compute index to the table
We start by filtering the two special cases around 0 and 1
*/
if (xdb.i[HI] < BOUND1) {
/* Special interval 0..BOUND1
The polynomial has no even monomials
We must prove extra accuracy in the interval 0..sin(2^(-18))
*/
/* Quick phase starts */
/* Compute square of x for both quick and accurate phases */
Mul12(&xSqh,&xSql,x,x);
tmp4 = tbl[3];
tmp5 = tbl[4];
t4h = tmp4;
t4l = tmp5;
if (xdb.i[HI] > EXTRABOUND) {
/* Double precision evaluation */
#if defined(PROCESSOR_HAS_FMA) && !defined(AVOID_FMA)
highPoly = FMA(FMA(FMA(FMA(tbl[23],xSqh,tbl[21]),xSqh,tbl[19]),xSqh,tbl[17]),xSqh,tbl[15]);
#else
highPoly = tbl[15] + xSqh * (tbl[17] + xSqh * (tbl[19] + xSqh * (tbl[21] + xSqh * tbl[23])));
#endif
/* Double-double precision evaluation */
Mul12(&tt1h,&tt1l,xSqh,highPoly);
Add22(&t1h,&t1l,tbl[12],tbl[13],tt1h,tt1l);
MulAdd212(&t2h,&t2l,tbl[9],tbl[10],xSqh,t1h,t1l);
MulAdd212(&t3h,&t3l,tbl[6],tbl[7],xSqh,t2h,t2l);