asinacos.c   [plain text]


/*
 * Copyright (c) 2002 Apple Computer, Inc. All rights reserved.
 *
 * @APPLE_LICENSE_HEADER_START@
 * 
 * The contents of this file constitute Original Code as defined in and
 * are subject to the Apple Public Source License Version 1.1 (the
 * "License").  You may not use this file except in compliance with the
 * License.  Please obtain a copy of the License at
 * http://www.apple.com/publicsource and read it before using this file.
 * 
 * This Original Code and all software distributed under the License are
 * distributed on an "AS IS" basis, WITHOUT WARRANTY OF ANY KIND, EITHER
 * EXPRESS OR IMPLIED, AND APPLE HEREBY DISCLAIMS ALL SUCH WARRANTIES,
 * INCLUDING WITHOUT LIMITATION, ANY WARRANTIES OF MERCHANTABILITY,
 * FITNESS FOR A PARTICULAR PURPOSE OR NON-INFRINGEMENT.  Please see the
 * License for the specific language governing rights and limitations
 * under the License.
 * 
 * @APPLE_LICENSE_HEADER_END@
 */
/*******************************************************************************
*                                                                              *
*     File asinacos.c,                                                         *
*     Implementation of asin and acos for the PowerPC.                         *
*                                                                              *
*     Copyright  1991-2001 Apple Computer, Inc.  All rights reserved.         *
*                                                                              *
*     Written and ported by Robert A. Murley (ram) for Mac OS X.               *
*     Modified by A. Sazegari, September 2001.                                 *
*                                                                              *
*     A MathLib v4 file.                                                       *
*                                                                              *
*     Based on the trigonometric functions from IBM/Taligent.                  *
*                                                                              *
*     July	20 2001: replaced __setflm with FEGETENVD/FESETENVD.           *
*                        replaced DblInHex typedef with hexdouble.             *
*     September 07 2001: added #ifdef __ppc__.                                 *
*     September 09 2001: added more comments.                                  *
*     September 10 2001: added macros to detect PowerPC and correct compiler.  *
*     September 21 2001: added <CoreServices/CoreServices.h> to get to <fp.h>  *
*                        and <fenv.h>.                                         *
*     September 24 2001: used standard symbols for environment flags.          *
*     September 25 2001: added FESETENVD(0.0) at start of acos.                *
*     October   08 2001: removed <CoreServices/CoreServices.h>.                *
*                        changed compiler errors to warnings.                  *
*     November  06 2001: commented out warning about Intel architectures.      *
*                                                                              *
*     W A R N I N G:                                                           *
*     These routines require a 64-bit double precision IEEE-754 model.         *
*     They are written for PowerPC only and are expecting the compiler         *
*     to generate the correct sequence of multiply-add fused instructions.     *
*                                                                              *
*     These routines are not intended for 32-bit Intel architectures.          *
*                                                                              *
*     A version of gcc higher than 932 is required.                            *
*                                                                              *
*      GCC compiler options:                                                   *
*            optimization level 3 (-O3)                                        *
*            -fschedule-insns -finline-functions -funroll-all-loops            *
*                                                                              *
*******************************************************************************/

#include    "fp_private.h"
#include    "fenv_private.h"
#include    "math.h"

extern const unsigned short SqrtTable[];

// Floating-point constants
static const double kPiBy2Tail = 6.1232339957367660e-17;      // 0x1.1a62633145c07p-54
static const hexdouble kPiu       = HEXDOUBLE(0x400921fb, 0x54442d18);

#define      INVERSE_TRIGONOMETRIC_NAN      "34"

/****************************************************************************

FUNCTION:  double asin (double x)
    
DESCRIPTION:  returns the inverse sine of its argument in the range of
              -pi/2 to pi/2 (radians).
    
CALLS:  __FABS
    
EXCEPTIONS raised:  INEXACT, UNDERFLOW/INEXACT or INVALID.
    
ACCURACY:  Error is less than an ulp and usually less than half an ulp.
           Caller's rounding direction is honored.

****************************************************************************/
static struct kBlock {
    const double k2ToM26    /* = 1.490116119384765625e-8 */ ;     // 0x1.0p-26 */ ;
    const double kPiBy2     /* = 1.570796326794896619231322 */ ;  // 0x1.921fb54442d18p0
    const double kPiBy2Tail /* = 6.1232339957367660e-17 */ ;      // 0x1.1a62633145c07p-54
    const double kMinNormal /* = 2.2250738585072014e-308 */ ;     // 0x1.0p-1022
    
    const  double a14  /* = 0.03085091303188211304259 */ ;
    const  double a13  /* =-0.02425125216179617744614 */ ;
    const  double a12  /* = 0.02273334623573271023373 */ ;
    const  double a11  /* = 0.0002983797813053653120360 */ ;
    const  double a10  /* = 0.008819738782817955152102 */ ;
    const  double a9   /* = 0.008130738583790024803650 */ ;
    const  double a8   /* = 0.009793486386035222577675 */ ;
    const  double a7   /* = 0.01154901189590083099584 */ ;
    const  double a6   /* = 0.01396501522140471126730 */ ;
    const  double a5   /* = 0.01735275722383019335276 */ ;
    const  double a4   /* = 0.02237215928757573539490 */ ;
    const  double a3   /* = 0.03038194444121688698408 */ ;
    const  double a2   /* = 0.04464285714288482921087 */ ;
    const  double a1   /* = 0.07499999999999990640378 */ ;
    const  double a0   /* = 0.1666666666666666667191 */ ;
    const  double aa11 /* = 0.0002983797813053653120360 */ ;
    const  double aa10 /* = 0.008819738782817955152102 */ ;
    const  double aa9  /* = 0.008130738583790024803650 */ ;
    const  double aa8  /* = 0.009793486386035222577675 */ ;
    const  double aa7  /* = 0.01154901189590083099584 */ ;
    const  double aa6  /* = 0.01396501522140471126730 */ ;
    const  double aa5  /* = 0.01735275722383019335276 */ ;
    const  double aa4  /* = 0.02237215928757573539490 */ ;
    const  double aa3  /* = 0.03038194444121688698408 */ ;
    const  double aa2  /* = 0.04464285714288482921087 */ ;
    const  double aa1  /* = 0.07499999999999990640378 */ ;
    const  double aa0  /* = 0.1666666666666666667191 */ ;
} kBlock =
{
/* static const double k2ToM26    = */ 0x1.0p-26,					// 1.490116119384765625e-8
/* static const double kPiBy2     = */ 1.570796326794896619231322,  // 0x1.921fb54442d18p0
/* static const double kPiBy2Tail = */ 6.1232339957367660e-17,      // 0x1.1a62633145c07p-54
/* static const double kMinNormal = */ 0x1.0p-1022,					// 2.2250738585072014e-308

/* const static double a14  = */ 0.03085091303188211304259,
/* const static double a13  = */-0.02425125216179617744614,
/* const static double a12  = */ 0.02273334623573271023373,
/* const static double a11  = */ 0.0002983797813053653120360,
/* const static double a10  = */ 0.008819738782817955152102,
/* const static double a9   = */ 0.008130738583790024803650,
/* const static double a8   = */ 0.009793486386035222577675,
/* const static double a7   = */ 0.01154901189590083099584,
/* const static double a6   = */ 0.01396501522140471126730,
/* const static double a5   = */ 0.01735275722383019335276,
/* const static double a4   = */ 0.02237215928757573539490,
/* const static double a3   = */ 0.03038194444121688698408,
/* const static double a2   = */ 0.04464285714288482921087,
/* const static double a1   = */ 0.07499999999999990640378,
/* const static double a0   = */ 0.1666666666666666667191,
/* const static double aa11 = */ 0.0002983797813053653120360,
/* const static double aa10 = */ 0.008819738782817955152102,
/* const static double aa9  = */ 0.008130738583790024803650,
/* const static double aa8  = */ 0.009793486386035222577675,
/* const static double aa7  = */ 0.01154901189590083099584,
/* const static double aa6  = */ 0.01396501522140471126730,
/* const static double aa5  = */ 0.01735275722383019335276,
/* const static double aa4  = */ 0.02237215928757573539490,
/* const static double aa3  = */ 0.03038194444121688698408,
/* const static double aa2  = */ 0.04464285714288482921087,
/* const static double aa1  = */ 0.07499999999999990640378,
/* const static double aa0  = */ 0.1666666666666666667191
};

double asin (double x)
{
    hexdouble     cD, yD ,gD , fpenv;
    double        absx, x2, x3, x4, temp1, temp2, u, v;
    double        g, y, d, y2, e;
    register int           index;
    register uint32_t ghead, yhead, rnddir;
    
    register double FPR_env, FPR_z, FPR_half, FPR_one, FPR_k2ToM26, FPR_cD, FPR_yD, FPR_kMinNormal, FPR_kPiBy2, FPR_kPiBy2Tail;
    register struct kBlock *pK;
    
    pK = &kBlock;
    
    FPR_z = 0.0;                            FPR_half = 0.5;
    FPR_k2ToM26 = pK->k2ToM26;				FPR_kMinNormal = pK->kMinNormal;
    FPR_kPiBy2 = pK->kPiBy2;				FPR_one = 1.0;
    
    if (unlikely(x != x))                           		// NaN argument
        return ( copysign ( nan ( INVERSE_TRIGONOMETRIC_NAN ), x ) );
        
    absx = __FABS( x );
    
    FEGETENVD ( FPR_env );                   		// save env, set default
    __ENSURE( FPR_z, FPR_half, FPR_k2ToM26 ); 		__ENSURE( FPR_one, FPR_kMinNormal, FPR_kPiBy2 ); // ensure register loads
    FPR_cD = __FNMSUB( FPR_half, absx, FPR_half ); 
    FESETENVD ( FPR_z );
    
    x2 = __FMUL( x, x );				// under/overflows are masked
    cD.d = FPR_cD;
    					        
    if ( absx <= FPR_half )
    {   // |x| <= 0.5
        if (unlikely( absx < FPR_k2ToM26 ))
        {   // |x| < 2^(-26)
            if ( absx == FPR_z )
            {                              		// x == 0.0 case is exact
                FESETENVD ( FPR_env );
                return ( x );
            }
            
            fpenv.d = FPR_env;
			__NOOP;
			__NOOP;
			__NOOP;
            rnddir = fpenv.i.lo & FE_ALL_RND;   		// caller's rounding direction
            FPR_yD = x;
    
            // Rounding away from zero:  return NextDouble(x,x+x)
            if (((rnddir == FE_DOWNWARD) && (x < FPR_z)) || ((rnddir == FE_UPWARD) && (x > FPR_z)))
            {
                yD.d = FPR_yD;
                __NOOP;
                __NOOP;
                __NOOP;
                yD.i.lo += 1u;
                if (yD.i.lo == 0u)
                    yD.i.hi += 1u;
                __NOOP;
                __NOOP;
                __NOOP;
                FPR_yD = yD.d;
                absx = __FABS( FPR_yD );
            }

            FESETENVD ( FPR_env );               	// restore caller environment
            __PROG_INEXACT( FPR_kPiBy2 );
            
            // Set UNDERFLOW if result is subnormal        
            if (absx < FPR_kMinNormal)             	// subnormal case
                __PROG_UF_INEXACT( FPR_kMinNormal );   	//   set UNDERFLOW
                
            return ( FPR_yD );                     	// return small result
        }   // [|x| < 2^(-26)]
        
        {
        register double r0, r1, r2, r3, r4, r5, r6, r7, r8, r9;
        register double r10, r11, r12, r13, r14;
        
        r14 = pK->a14;					r13 = pK->a13;
        temp1 = __FMADD( r14, x2, r13 );		x4 = __FMUL( x2, x2 );			
        
        x3 = __FMUL( x, x2 );				__NOOP;
        r11 = pK->a11;					r12 = pK->a12;
        
        r9 = pK->a9;					r10 = pK->a10;
        temp1 = __FMADD( temp1, x4, r11 );		__NOOP;
        
        temp1 = __FMADD( temp1, x4, r9 );		temp2 = __FMADD( r12, x4, r10 );
        r7 = pK->a7;					r8 = pK->a8;
                
        r5 = pK->a5;					r6 = pK->a6;
        temp1 = __FMADD( temp1, x4, r7 );		temp2 = __FMADD( temp2, x4, r8 );
            
        temp1 = __FMADD( temp1, x4, r5 );		temp2 = __FMADD( temp2, x4, r6 );
        r3 = pK->a3;					r4 = pK->a4;
        
        r1 = pK->a1;					r2 = pK->a2;
        temp1 = __FMADD( temp1, x4, r3 );		temp2 = __FMADD( temp2, x4, r4 );
        
        temp1 = __FMADD( temp1, x4, r1 );		temp2 = __FMADD( temp2, x4, r2 );
        r0 = pK->a0;					__NOOP;
        
        temp2 = __FMADD( temp2, x2, temp1 );	
        temp1 = __FMADD( x2, temp2, r0 );
        }
        
        FESETENVD ( FPR_env );                   	// restore caller's mode
        __PROG_INEXACT( FPR_kPiBy2 );
        
        return (__FMADD( x3, temp1, x ));
    }   // [|x| <= 0.5]
    
/****************************************************************************
    For 0.5 < |x| < 1.0, a polynomial approximation is used to estimate
    the power series resulting from the identity:
       ArcSin(x) = pi/2 - 2*ArcSin(Sqrt(0.5(1.0 - x))).
    The square root is evaluated in-line and concurrently with the
    evaluation of the ArcSin.
 ***************************************************************************/
    
    if (absx < FPR_one)
    {   // |x| < 1.0
        register double FPR_gD, FPR_two;
        register double r0, r1, r2, r3, r4, r5, r6, r7, r8, r9;
        register double r10, r11, r12, r13, r14;
        register uint32_t GPR_p, GPR_q, GPR_r, GPR_s, GPR_t;
        
        x2 = FPR_cD;	/* fmr */			x4 = __FMUL( FPR_cD, FPR_cD); 	// Was x2*x2;
        GPR_t = cD.i.hi;				r0 = pK->aa0;

        GPR_s = GPR_t + 0x3ff00000u;			__NOOP;
        r14 = pK->a14;					r13 = pK->a13;

        r12 = pK->a12;					r11 = pK->a11;
        gD.i.lo = 0u;        

        index = (GPR_t >> 13) & 0xffu;
        GPR_p = SqrtTable[index]; // 3 instrs issue
        
        temp1 = __FMADD( r13, x4, r11 ); 		temp2 = __FMADD( r14, x4, r12 );
        r10 = pK->a10;					r9 = pK->a9;
        
        GPR_r = ((0xff00u & GPR_p) << 4);		
        ghead = (GPR_s >> 1) & 0x7ff00000u;
        gD.i.hi = ghead + GPR_r;
        
        r8 = pK->a8;					r7 = pK->aa7;
        yhead = 0x7fc00000u - ghead;

        yD.i.lo = 0u;					GPR_q = ((0xffu & GPR_p) << 12);
        yD.i.hi = yhead + GPR_q;			
        
        r6 = pK->aa6;					r5 = pK->aa5;
        temp1 = __FMADD( temp1, x4, r9 ); 		temp2 = __FMADD( temp2, x4, r10 );
                
        temp1 = __FMADD( temp1, x4, r7 ); 		temp2 = __FMADD( temp2, x4, r8 );
        r4 = pK->aa4;					r3 = pK->aa3;
                                        			
        r2 = pK->aa2;					r1 = pK->aa1;        
        temp1 = __FMADD( temp1, x4, r5 );		temp2 = __FMADD( temp2, x4, r6 );
        
        temp1 = __FMADD( temp1, x4, r3 ); 		temp2 = __FMADD( temp2, x4, r4 );
        FPR_two = 2.0;
        
        FPR_gD = gD.d;					y = yD.d;
        temp1 = __FMADD( temp1, x4, r1 );		temp2 = __FMADD( temp2, x4, r2 );
        
        d = __FNMSUB( FPR_gD, FPR_gD, FPR_cD );		FPR_kPiBy2Tail = pK->kPiBy2Tail;
        g = __FMADD( y, d, FPR_gD );			y2 = y + y;                 		// 16-bit g
        e = __FNMSUB( y, g, FPR_half );			d = __FNMSUB( g, g, FPR_cD );
        y = __FMADD( e, y2, y);               		temp1 = __FMADD( x2, temp2, temp1 );	// 16-bit y
        g = __FMADD( y, d, g );                		y2 = y + y;				// 32-bit g
            
        e = __FNMSUB( y, g, FPR_half );			d = __FNMSUB( g, g, FPR_cD );
        y = __FMADD( e, y2, y );               		temp1 = __FMADD( x2, temp1, r0 );	// 32-bit y
        g = __FMADD( y, d, g);                							// 64-bit Sqrt
        d = __FNMSUB( g, g, FPR_cD );			u = __FNMSUB( FPR_two, g, FPR_kPiBy2 );	// pi/2 - 2.0*Sqrt (head)
        d = __FMUL( d, y );         			v = __FNMSUB( FPR_two, g, (FPR_kPiBy2 - u) );// tail of 0.5*Sqrt (32 bits)
        g = __FMUL( g, x2 );				d = __FNMSUB( FPR_half, FPR_kPiBy2Tail, d );
        
        temp2 = __FNMSUB( FPR_half, v, d );
        temp2 = __FMADD( g, temp1, temp2 );
    
        FESETENVD ( FPR_env );        // restore caller's mode
        __PROG_INEXACT( FPR_kPiBy2 );
        
        if (x > FPR_z)
        {                           // take care of sign of result
            return __FNMSUB( FPR_two, temp2, u );
        }
        else
            return __FMSUB( FPR_two, temp2, u );
    }   // [|x| < 1.0]
    
    
    // If |x| == 1.0, return properly signed pi/2.
    if (absx == FPR_one)
    {
        FPR_kPiBy2Tail = pK->kPiBy2Tail;
        
        FESETENVD (FPR_env);            // restore caller's mode
        if (x > FPR_z)
            return __FMADD( x, FPR_kPiBy2Tail, FPR_kPiBy2 );
        else
            return __FMSUB( x, FPR_kPiBy2Tail, FPR_kPiBy2 );
    }
    
    // |x| > 1.0 signals INVALID and returns a NaN
    fpenv.d = FPR_env;
	__NOOP;
	__NOOP;
	__NOOP;
    fpenv.i.lo |= SET_INVALID;
    FESETENVD_GRP (fpenv.d);
    return ( nan ( INVERSE_TRIGONOMETRIC_NAN ) );
}   // ArcSin(x)

/****************************************************************************

FUNCTION:  double ArcCos(double x)
    
DESCRIPTION:  returns the inverse cosine of its argument in the range of
              0 to pi (radians).
    
CALLS:  __FABS
    
EXCEPTIONS raised:  INEXACT or INVALID.
    
ACCURACY:  Error is less than an ulp and usually less than half an ulp.
           Caller's rounding direction is honored.

****************************************************************************/

double acos(double x)
{
    hexdouble     cD, yD, gD, fpenv;
    double        absx, x2, x3, x4, temp1, temp2, s, u, v, w;
    double        g, y, d, y2, e;
    int           index;
    uint32_t	  ghead, yhead;

    register double FPR_env, FPR_z, FPR_half, FPR_one, FPR_k2ToM26, FPR_cD, FPR_kMinNormal, FPR_kPiBy2, FPR_kPiBy2Tail;
    register struct kBlock *pK;
    
    pK = &kBlock;
    
    FPR_z = 0.0;					FPR_half = 0.5;
    FPR_k2ToM26 = pK->k2ToM26;				FPR_kMinNormal = pK->kMinNormal;
    FPR_kPiBy2 = pK->kPiBy2;				FPR_one = 1.0;

    if (unlikely(x != x))                           // NaN
        return ( copysign ( nan ( INVERSE_TRIGONOMETRIC_NAN ), x ) );
        
    absx = __FABS(x);
    
    FEGETENVD ( FPR_env );                   		// save env, set default
    __ENSURE( FPR_z, FPR_half, FPR_k2ToM26 ); 		__ENSURE( FPR_one, FPR_kMinNormal, FPR_kPiBy2 ); // ensure register loads
    FPR_cD = __FNMSUB( FPR_half, absx, FPR_half ); 
    FESETENVD ( FPR_z );
    
    x2 = __FMUL( x, x );				// under/overflows are masked
    cD.d = FPR_cD;

    if ( absx <= FPR_half )
    {   						// |x| <= 0.5
        register double r0, r1, r2, r3, r4, r5, r6, r7, r8, r9;
        register double r10, r11, r12, r13, r14;
        
        r14 = pK->a14;					r13 = pK->a13;
        temp1 = __FMADD( r14, x2, r13 );		x4 = __FMUL( x2, x2 );			
        
        x3 = __FMUL( x, x2 );				u = FPR_kPiBy2 - x; // pi/2 - x (high-order term)
        r11 = pK->a11;					r12 = pK->a12;
        
        r9 = pK->a9;					r10 = pK->a10;
        temp1 = __FMADD( temp1, x4, r11 );		__NOOP;
        
        temp1 = __FMADD( temp1, x4, r9 );		temp2 = __FMADD( r12, x4, r10 );
        r7 = pK->a7;					r8 = pK->a8;
                
        r5 = pK->a5;					r6 = pK->a6;
        temp1 = __FMADD( temp1, x4, r7 );		temp2 = __FMADD( temp2, x4, r8 );
        
        temp1 = __FMADD( temp1, x4, r5 );		temp2 = __FMADD( temp2, x4, r6 );
        r3 = pK->a3;					r4 = pK->a4;
        
        r1 = pK->a1;					r2 = pK->a2;
        temp1 = __FMADD( temp1, x4, r3 );		temp2 = __FMADD( temp2, x4, r4 );
        
        temp1 = __FMADD( temp1, x4, r1 );		temp2 = __FMADD( temp2, x4, r2 );
        r0 = pK->a0;					FPR_kPiBy2Tail = pK->kPiBy2Tail;
        
        // finish up ArcCos(x) = pi/2 - ArcSin(x) 
        
        temp1 = __FMADD( x2, temp2, temp1 );
        v = FPR_kPiBy2 - u - x;                 	// pi/2 - x (tail)
        temp1 = __FMADD( x2, temp1, pK->a0 );
        w = __FNMSUB( x3, temp1, (v + FPR_kPiBy2Tail) );// combine low order terms
        
        FESETENVD ( FPR_env );                   	// restore caller's mode
        __PROG_INEXACT( FPR_kPiBy2 );
        
        return ( u + w );
    }   // [|x| <= 0.5]
    
/****************************************************************************
    For 0.5 < |x| < 1.0, a polynomial approximation is used to estimate
    the power series resulting from the identity:
      ArcSin(x) = pi/2 - 2*ArcSin(Sqrt(0.5(1.0 - x))).
    The square root is evaluated in-line and concurrently with the
    evaluation of the ArcSin.
 ****************************************************************************/
    if ( absx < FPR_one ) 
    {   // 0.5 < |x| < 1.0
        register double FPR_gD, FPR_two;
        register double r0, r1, r2, r3, r4, r5, r6, r7, r8, r9;
        register double r10, r11, r12, r13, r14;
        register uint32_t GPR_p, GPR_q, GPR_r, GPR_s, GPR_t;
        
        x2 = FPR_cD;	/* fmr */			x4 = __FMUL( x2, x2);
        GPR_t = cD.i.hi;				r0 = pK->aa0;

        GPR_s = GPR_t + 0x3ff00000u;			__NOOP;
        r14 = pK->a14;					r13 = pK->a13;

        r12 = pK->a12;					r11 = pK->aa11;
        gD.i.lo = 0u;        

        index = (GPR_t >> 13) & 0xffu;
        GPR_p = SqrtTable[index]; // 3 instrs issue
        
        temp1 = __FMADD( r13, x4, r11 ); 		temp2 = __FMADD( r14, x4, r12 );
        r10 = pK->aa10;					r9 = pK->aa9;
        
        GPR_r = ((0xff00u & GPR_p) << 4);		
        ghead = (GPR_s >> 1) & 0x7ff00000u;
        gD.i.hi = ghead + GPR_r;
        
        r8 = pK->aa8;					r7 = pK->aa7;
        yhead = 0x7fc00000u - ghead;

        yD.i.lo = 0u;					GPR_q = ((0xffu & GPR_p) << 12);
        yD.i.hi = yhead + GPR_q;			
        
        r6 = pK->aa6;					r5 = pK->aa5;
        temp1 = __FMADD( temp1, x4, r9 ); 		temp2 = __FMADD( temp2, x4, r10 );
                
        temp1 = __FMADD( temp1, x4, r7 ); 		temp2 = __FMADD( temp2, x4, r8 );
        r4 = pK->aa4;					r3 = pK->aa3;
                                        			
        r2 = pK->aa2;					r1 = pK->aa1;        
        temp1 = __FMADD( temp1, x4, r5 );		temp2 = __FMADD( temp2, x4, r6 );
        
        temp1 = __FMADD( temp1, x4, r3 ); 		temp2 = __FMADD( temp2, x4, r4 );
        FPR_two = 2.0;
        
        FPR_gD = gD.d;					y = yD.d;
        temp1 = __FMADD( temp1, x4, r1 );		temp2 = __FMADD( temp2, x4, r2 );
        
        d = __FNMSUB( FPR_gD, FPR_gD, x2 );
        g = __FMADD( y, d, FPR_gD );			y2 = y + y;                 	// 16-bit g
        e = __FNMSUB( y, g, FPR_half );			d = __FNMSUB( g, g, x2 );
        y = __FMADD( e, y2, y);                       	temp1 = __FMADD( x2, temp2, temp1 );// 16-bit y
        g = __FMADD( y, d, g );                		y2 = y + y;			// 32-bit g
        e = __FNMSUB( y, g, FPR_half );			d = __FNMSUB( g, g, x2 );
        y = __FMADD( e, y2, y );                       					// 32-bit y
        g = __FMADD( y, d, g);                						// 64-bit Sqrt
        d = __FNMSUB( g, g, x2 );                   					// tail of 0.5*Sqrt (32 bits)
        y2 = g + g;                         						// 2*Sqrt
        
        if (x > FPR_z)
        {   // positive x
            s = __FMADD( __FMUL( g, x2 ), __FMADD( x2, temp1, pK->aa0), d );// low order terms
            
            FESETENVD ( FPR_env );                   			// restore caller's mode
            __PROG_INEXACT( FPR_kPiBy2 );
            
            return __FMADD( FPR_two, s, y2);                		// combine terms
        }   // [positive x]
        else
        {   // negative x
            register double q,r,ss,t;
            
            u = __FMSUB( FPR_two, FPR_kPiBy2, y2 );            		// high-order term
            r = d - kPiBy2Tail;
            ss = __FMADD( x2, temp1, pK->aa0 );
            t = __FMUL( g, x2 );
            v = __FMSUB( FPR_two, FPR_kPiBy2, u ) - y2;       		// tail correction
            q = __FNMSUB( FPR_half, v, r );
            s = __FMADD( ss, t, q );					// collect low-order terms
            
            FESETENVD ( FPR_env );                   			// restore caller's mode
            __PROG_INEXACT( FPR_kPiBy2 );
            
            return __FNMSUB( FPR_two, s, u);                		// combine terms
        }   // [negative x]
    }   // [0.5 < |x| < 1.0]


    // |x| == 1.0:  return exact +0.0 or inexact pi
    if ( x == FPR_one )
    {
        FESETENVD ( FPR_env );            				// restore caller's mode
        return ( FPR_z );
    }
    
    if ( x == -1.0 )
    {
        FESETENVD ( FPR_env );                   			// restore caller's mode
        __PROG_INEXACT( FPR_kPiBy2 );
        return (kPiu.d + 2.0*kPiBy2Tail);
    }
    
    // |x| > 1.0 signals INVALID and returns a NaN
    fpenv.d = FPR_env;
	__NOOP;
	__NOOP;
	__NOOP;
    fpenv.i.lo |= SET_INVALID;
    FESETENVD_GRP ( fpenv.d );
    return ( nan ( INVERSE_TRIGONOMETRIC_NAN ) );
}   // ArcCos(x)