/* * Copyright (c) 2002 Apple Computer, Inc. All rights reserved. * * @APPLE_LICENSE_HEADER_START@ * * Copyright (c) 1999-2003 Apple Computer, Inc. All Rights Reserved. * * This file contains Original Code and/or Modifications of Original Code * as defined in and that are subject to the Apple Public Source License * Version 2.0 (the 'License'). You may not use this file except in * compliance with the License. Please obtain a copy of the License at * http://www.opensource.apple.com/apsl/ and read it before using this * file. * * The 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, QUIET ENJOYMENT OR NON-INFRINGEMENT. * Please see the License for the specific language governing rights and * limitations under the License. * * @APPLE_LICENSE_HEADER_END@ */ /******************************************************************************* * * * File power.c, * * Function pow ( x, y ), * * Implementation of x^y for the PowerPC. * * * * W A R N I N G: For double precision IEEE machines with MAF * * instructions. * * * * Copyright © 1991-1997 Apple Computer, Inc. All rights reserved. * * * * Written by Ali Sazegari. * * * * June 2 1997 first port of the ibm/taligent power. * * * *******************************************************************************/ #ifdef __APPLE_CC__ #if __APPLE_CC__ > 930 #include "math.h" #include "fenv.h" #include "fp_private.h" #include "fenv_private.h" #define POWER_NAN "37" #define SET_INVALID 0x01000000 /******************************************************************************* * Floating-point constants. * *******************************************************************************/ static const double ln2 = 6.931471805599452862e-1; static const double ln2Tail = 2.319046813846299558e-17; static const double kRint = 6.7553994410557440000e+15; // 0x43380000, 0x00000000 static const double maxExp = 7.0978271289338397000e+02; // 0x40862e42, 0xfefa39ef static const double min2Exp = -7.4513321910194122000e+02; // 0xc0874910, 0xd52d3052 static const double denormal = 2.9387358770557188000e-39; // 0x37f00000, 0x00000000 static const double twoTo128 = 3.402823669209384635e+38; // 0x1.0p+128 static const double twoTo52 = 4503599627370496.0; // 0x1p52 static const double oneOverLn2 = 1.4426950408889633000e+00; // 0x3ff71547, 0x652b82fe static const hexdouble huge = HEXDOUBLE(0x7ff00000, 0x00000000); /******************************************************************************* * Approximation coefficients for high precision exp and log. * *******************************************************************************/ static const double c2 = -0.5000000000000000042725; static const double c3 = 0.3333333333296328456505; static const double c4 = -0.2499999999949632195759; static const double c5 = 0.2000014541571685319141; static const double c6 = -0.1666681511199968257150; static const double d2 = -0.5000000000000000000000; // bfe0000000000000 static const double d3 = 0.3333333333333360968212; // 3FD5555555555587 static const double d4 = -0.2500000000000056849516; // BFD0000000000066 static const double d5 = 0.1999999996377879912068; // 3FC9999998D278CB static const double d6 = -0.1666666662009609743117; // BFC5555554554F15 static const double d7 = 0.1428690115662276259345; // 3FC249882224F72F static const double d8 = -0.1250122079043555957999; // BFC0006668466517 static const double cc4 = 0.5000000000000000000; static const double cc3 = 0.16666666666663809522820; static const double cc2 = 0.04166666666666111110939; static const double cc1 = 0.008333338095239329810170; static const double cc0 = 0.001388889583333492938381; /******************************************************************************* * Log and exp table entries definition. * *******************************************************************************/ extern unsigned long logTable[]; extern unsigned long expTable[]; struct logTableEntry { double X; double G; hexdouble F; }; struct expTableEntry { double x; double f; }; static double PowerInner ( double, double ); static double _NearbyInt ( double x ); double pow ( double base, double exponent ) { register long int isExpOddInt; double absBase, result; hexdouble u, v, OldEnvironment; register unsigned long int expExp; v.d = exponent; u.d = base; if ( ( ( v.i.hi & 0x000fffff ) | v.i.lo ) == 0 ) { // exponent is power of 2 (or +-Inf) expExp = v.i.hi & 0xfff00000ul; if ( expExp == 0x40000000 ) return base*base; // if exponent == 2.0 else if ( exponent == 0.0 ) return 1.0; // if exponent == 0.0 else if ( expExp == 0x3ff00000 ) return base; // if exponent == 1.0 else if ( expExp == 0xbff00000 ) return 1.0/base; // if exponent == -1.0 } if ( ( v.i.hi & 0x7ff00000ul ) < 0x7ff00000ul ) { // exponent is finite if ( ( u.i.hi & 0x7ff00000ul ) < 0x7ff00000ul ) { // base is finite if ( base > 0.0 ) return PowerInner( base, exponent ); // base is positive else if ( base < 0.0 ) { if ( _NearbyInt ( exponent ) != exponent ) // exponent is non-integer { fegetenvd( OldEnvironment.d ); OldEnvironment.i.lo |= SET_INVALID; result = nan ( POWER_NAN ); fesetenvd( OldEnvironment.d ); return result; } result = PowerInner( -base, exponent ); if ( _NearbyInt ( 0.5 * exponent ) != 0.5 * exponent ) // exponent is odd result = - result; return ( result ); } else { // base is 0.0: isExpOddInt = ( ( _NearbyInt( exponent ) == exponent ) ? ( _NearbyInt( 0.5 * exponent ) != 0.5 * exponent ) : 0 ); if ( exponent > 0.0 ) return ( ( isExpOddInt ) ? base : 0.0 ); else // exponent < 0.0 return ( ( isExpOddInt ) ? 1.0/base : 1.0/__fabs( base ) ); } } else if (base != base) return base; else { // base == +-Inf if ( base > 0 ) return ( exponent > 0 ) ? huge.d : +0.0; else { isExpOddInt = ( ( _NearbyInt ( exponent ) == exponent ) ? ( _NearbyInt ( 0.5 * exponent ) != 0.5 * exponent ) : 0 ); return ( exponent > 0 ) ? ( isExpOddInt ? -huge.d : +huge.d ) : ( isExpOddInt ? -0.0 : +0.0 ); } } } else if ( exponent != exponent ) return base + exponent; else { // exponent is +-Inf if ( base != base ) return base; absBase = __fabs( base ); if ( ( exponent > 0 && absBase > 1 ) || ( exponent < 0 && absBase < 1 ) ) return huge.d; else if ( ( exponent > 0 && absBase < 1 ) || ( exponent < 0 && absBase > 1 ) ) return +0.0; else // now: Abs( base ) == 1.0 return 1.0; } } /******************************************************************************* * Private function _NearbyInt. * *******************************************************************************/ static double _NearbyInt ( double x ) { hexdouble xInHex; double OldEnvironment; xInHex.d = x; if ( ( xInHex.i.hi & 0x7ffffffful ) < 0x43300000ul ) { // |x| < 2.0^52 fegetenvd( OldEnvironment ); // save environment, set default fesetenvd( 0.0 ); if ( xInHex.i.hi & 0x80000000ul ) // x non-positive x = ( ( x - twoTo52 ) + twoTo52 ); else x = ( ( x + twoTo52 ) - twoTo52 ); fesetenvd( OldEnvironment ); return x; } else // |x| >= 2.0^52 return ( x ); } /******************************************************************************* * * * Function PowerInner computes the base to the exponent power. This * * routine is called internally by the power function. It assumes that * * base is strictly positive, exponent is normal or denormal and the * * rounding direction is round-to-nearest. * * * *******************************************************************************/ static double PowerInner ( double base, double exponent ) { hexdouble u, v, OldEnvironment, scale, exp; register long int i, n; register unsigned long int head; register double z, zLow, high, low, zSquared, temp1, temp2, temp3, result, tail, d, x, xLow, y, yLow, power; struct logTableEntry *logTablePointer = ( struct logTableEntry * ) logTable; struct expTableEntry *expTablePointer = ( struct expTableEntry * ) expTable + 177; u.d = base; head = u.i.hi; if ( head >= 0x000ffffful ) // normal case d = ( u.i.hi >> 20 ) - 1023.0; else { // denormal case u.d = base * twoTo128; // scale to normal range d = ( u.i.hi >> 20 ) - 1151.0; } i = u.i.hi & 0x000fffff; u.i.hi = i | 0x3ff00000; // Handle exact integral powers and roots of two specially if ( ( ( u.i.hi & 0x000fffff ) | u.i.lo ) == 0 ) // base is power of 2 { z = d * exponent; if ( z == _NearbyInt( z ) ) { // Clip z so conversion to int is safe if (z > 2097) n = 2098; else if (z < -2098) n = -2099; else n = z; return scalbn( 1.0, n ); } } if ( u.i.hi & 0x00080000 ) { // 1.5 <= x < 2.0 n = 1; u.d *= 0.5; i = ( i >> 13 ) & 0x3f; // table lookup index } else { // 1.0 <= x < 1.5 n = 0; i = ( i >> 12 ) + 64; // table lookup index } fegetenvd( OldEnvironment.d ); // save environment, set default fesetenvd( 0.0 ); z = ( u.d - logTablePointer[i].X ) * logTablePointer[i].G; zLow = ( ( u.d - logTablePointer[i].X ) - z * logTablePointer[i].X ) * logTablePointer[i].G; zSquared = z * z; if ( n == 0 ) { // n = 0 v.d = base; if ( ( v.i.hi == 0x3ff00000 ) && ( v.i.lo == 0x0 ) ) { //base = 1.0 fesetenvd( OldEnvironment.d ); return 1.0; } temp1 = d8 * zSquared + d6; temp2 = d7 * zSquared + d5; temp1 = temp1 * zSquared + d4; temp2 = temp2 * zSquared + d3; temp1 = temp1 * z + temp2; temp2 = temp1 * z + d2; temp3 = z + logTablePointer[i].F.d; low = logTablePointer[i].F.d - temp3 + z + ( zLow - z * zLow ); result = ( temp2 * zSquared + low ) + temp3; tail = temp3 - result + ( temp2 * zSquared + low ); } else if ( logTablePointer[i].F.i.hi != 0 ) { // n = 1 low = ln2Tail + zLow; high = z + logTablePointer[i].F.d; zLow = logTablePointer[i].F.d - high + z; temp1 = ln2 + low; low = ( ln2 - temp1 ) + low; temp3 = c6 * zSquared + c4; temp2 = c5 * zSquared + c3; temp3 = temp3 * zSquared; temp2 = ( temp2 * z + temp3 ) + c2; temp3 = high + temp1; temp1 = ( temp1 - temp3 ) + high; result = ( ( ( temp2 * zSquared + low ) + temp1 ) + zLow ) + temp3; tail = temp3 - result + zLow + temp1 + ( temp2 * zSquared + low ); } else { // n = 1. low = ln2Tail + zLow; temp3 = ln2 + low; low = ( ln2 - temp3 ) + low; temp1 = d8 * zSquared + d6; temp2 = d7 * zSquared + d5; temp1 = temp1 * zSquared + d4; temp2 = temp2 * zSquared + d3; temp1 = temp1 * z + temp2; temp2 = temp1 * z + d2; result = ( ( temp2 * zSquared + low ) + z ) + temp3; tail = temp3 - result + z + ( temp2 * zSquared + low ); } temp1 = d * ln2; temp2 = __fmsub(d, ln2, temp1); // d * ln2 - temp1; z = temp1 + result; zLow = temp1 - z + result + __fmadd(d, ln2Tail, tail + temp2); // ( d * ln2Tail + tail + temp2 ); temp3 = exponent * zLow; x = __fmadd(exponent, z, temp3); // exponent * z + temp3; xLow = __fmsub(exponent, z, x) + temp3; // ( exponent * z - x ) + temp3; if ( base == _NearbyInt(base) && exponent > 0 && exponent == _NearbyInt(exponent)) { /* NOTHING: +ve integral base and +ve integral exponent should deliver exact result */ } else OldEnvironment.i.lo |= FE_INEXACT; // set inexact flag u.d = x; if ( ( u.i.hi & 0x7ff00000ul ) < 0x40800000ul ) { // abs( x ) < 512 scale.d = 0.0; exp.d = x * oneOverLn2 + kRint; scale.i.hi = ( exp.i.lo + 1023 ) << 20; exp.d -= kRint; y = x - ln2 * exp.d; yLow = ln2Tail * exp.d; u.d = 512.0 * y + kRint; i = u.i.lo; d = y - expTablePointer[i].x; z = d - yLow; zLow = d - z - yLow + xLow; zSquared = z * z; temp1 = cc0 * zSquared + cc2; temp2 = cc1 * zSquared + cc3; temp1 = temp1 * zSquared + cc4; temp2 = temp1 + temp2 * z; temp1 = temp2 * zSquared + z; result = scale.d * expTablePointer[i].f; #if 1 temp2 = __prod( result , ( temp1 + ( zLow + zLow * temp1 ) )); // thwart gcc MAF #else temp2 = result * ( temp1 + ( zLow + zLow * temp1 ) ); #endif result += temp2; fesetenvd( OldEnvironment.d ); return result; } if ( ( x <= maxExp ) && ( x > min2Exp ) ) { scale.d = 0.0; exp.d = x * oneOverLn2 + kRint; if ( x >= 512.0 ) { power = 2.0; scale.i.hi = ( exp.i.lo + 1023 - 1 ) << 20; } else { power = denormal; scale.i.hi = ( exp.i.lo + 1023 + 128 ) << 20; } exp.d -= kRint; y = x - ln2 * exp.d; yLow = ln2Tail * exp.d; u.d = 512.0 * y + kRint; i = u.i.lo; d = y - expTablePointer[i].x; z = d - yLow; zLow = d - z - yLow + xLow; zSquared = z * z; temp1 = cc0 * zSquared + cc2; temp2 = cc1 * zSquared + cc3; temp1 = temp1 * zSquared + cc4; temp2 = temp1 + temp2 * z; temp1 = temp2 * zSquared + z; result = scale.d * expTablePointer[i].f; temp2 = __prod( result , ( temp1 + ( zLow + zLow * temp1 ) ) ); // __prod thwarts gcc MAF u.d = ( result + temp2 ) * power; } else if ( x > maxExp ) u.d = huge.d; else u.d = +0.0; i = u.i.hi & 0x7ff00000; if ( i == 0x7ff00000 ) OldEnvironment.i.lo |= (FE_OVERFLOW | FE_INEXACT); // Inf: set overflow/inexact flag else if ( i == 0x0 ) OldEnvironment.i.lo |= (FE_UNDERFLOW | FE_INEXACT); // zero: set underflow/inexact flag fesetenvd( OldEnvironment.d ); return u.d; } #ifdef notdef float powf ( float x, float y ) { return (float)pow ( x, y ); } #endif #else /* __APPLE_CC__ version */ #error Version gcc-932 or higher required. Compilation terminated. #endif /* __APPLE_CC__ version */ #endif /* __APPLE_CC__ */