/* * 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 ashachath.c, * * Function ArcSinh(x), ArcCosh(x), ArcTanh(x); * * Implementation of arc sine, arc cosine and arc tangent hyperbolic for * * the PowerPC. * * * * Copyright © 1991 Apple Computer, Inc. All rights reserved. * * * * Written by Ali Sazegari, started on December 1991, * * Modified and ported by Robert A. Murley (ram) for Mac OS X. * * * * A MathLib v4 file. * * * * January 28 1992: added velvelÕs algorithms for ArcSinh and ArcCosh. * * * * December 03 1992: first rs6000 port. * * January 05 1993: added the environmental controls. * * July 07 1993: changed the comment for square root of epsilon, * * made atanh symmetric about zero using copysign, * * changed the exception switch argument for asinh from * * x to PositiveX, switched to the use of string * * oriented nan. * * July 29 1993: corrected the string nan. * * July 18 1994: changed the logic to avoid checking for NaNs, * * introduced an automatic xMinusOne in acosh. Replaced * * fenv functions with __setflm. * * August 08 2001: replaced __setflm with fegetenvd/fesetenvd. * * replaced DblInHex typedef with hexdouble. * * Sept 19 2001: added macros to detect PowerPC and correct compiler. * * Sept 19 2001: added <CoreServices/CoreServices.h> to get to <fp.h> * * and <fenv.h>, removed denormal comments. * * 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 * * * *******************************************************************************/ #ifdef __APPLE_CC__ #if __APPLE_CC__ > 930 #include "math.h" #include "fenv_private.h" #include "fp_private.h" #pragma fenv_access on /******************************************************************************* * Functions needed for the computation. * *******************************************************************************/ /* the following fp.h functions are used: */ /* __fpclassifyd, log1p, log, sqrt, copysign and __fabs; */ #define INVERSE_HYPERBOLIC_NAN "40" static hexdouble SqrtNegEps = HEXDOUBLE(0x3e400000, 0x00000000); /* = 7.4505805969238281e-09 */ static hexdouble Log2 = HEXDOUBLE(0x3FE62E42, 0xFEFA39EF); /* = 6.9314718055994530942E-1 */ static hexdouble FiveFourth = HEXDOUBLE(0x3FF40000, 0x00000000); /* = 1.250000000000000E+00 */ /******************************************************************************* * A R C T A N H * *******************************************************************************/ double atanh ( double x ) { register double PositiveX; hexdouble OldEnvironment, NewEnvironment; fegetenvd ( OldEnvironment.d ); fesetenvd ( 0.0 ); PositiveX = __fabs ( x ); /******************************************************************************* * * * The expression for computing ArcTanh(x) is: * * * * ArcTanh(x) = 1/2log[(1+x)/(1-x)], |x| < 1. * * * * We use the more accurate log(1+x) for the evaluation, then the ArcTanh * * representation is: * * * * ArcTanh(x) = 1/2log1p(2x/(1-x)), |x| < 1, * * * * and for small enough x ( |x| < SqrtNegEps, where 1 - SqrtNegEps^2 = 1 ) * * the first term of the taylor series expansion of log[(1+x)/(1-x)] is * * 2x/(1-x) ~= 2x. x is returned for ArcTanh(x). * * * * the value of SqrtNegEps is: * * * * SqrtNegEps = 0x3e40000000000000, then 1-SqrtNegEps^2 = 1. * * * * The monotonicity of the function has been preserved across double * * valued intervals. * * * * -inf -1 -SqrtNegEps 0 +SqrtNegEps +1 +inf * * ---------+--------------------+-----+-----+--------------------+------- * * < N a N >|-inf <= ArcTanh(x) >|< x 0 x >|< ArcTanh(x) => +inf|< N a N >* * * *******************************************************************************/ if ( PositiveX <= 1.0 ) { if ( PositiveX >= SqrtNegEps.d ) { PositiveX = 0.5 * log1p ( 2.0 * PositiveX / ( 1 - PositiveX ) ); x = copysign ( PositiveX, x ); } } /******************************************************************************* * If argument is SNaN then a QNaN has to be returned and the invalid * * flag signaled for SNaN. Otherwise, the argument is less than 1.0 and * * a hyperbolic nan is returned. * *******************************************************************************/ else { if ( x != x ) x *= 2.0; else { x = nan ( INVERSE_HYPERBOLIC_NAN ); OldEnvironment.i.lo |= SET_INVALID; } } switch ( __fpclassifyd ( x ) ) { case FP_SUBNORMAL: OldEnvironment.i.lo |= FE_UNDERFLOW; /* FALL THROUGH */ case FP_NORMAL: OldEnvironment.i.lo |= FE_INEXACT; /* FALL THROUGH */ default: break; } fegetenvd ( NewEnvironment.d ); OldEnvironment.i.lo |= NewEnvironment.i.lo; fesetenvd ( OldEnvironment.d ); return x; } #ifdef notdef float atanhf( float x ) { return (float)atanh( x ); } #endif /******************************************************************************* * A R C S I N H * *******************************************************************************/ double asinh ( double x ) { register double PositiveX, InvPositiveX; hexdouble OldEnvironment, NewEnvironment; fegetenvd ( OldEnvironment.d ); fesetenvd ( 0.0 ); PositiveX = __fabs ( x ); /******************************************************************************* * * * The expression for computing ArcSinh(x) is: * * * * ArcSinh(x) = log(x+sqrt(x^2+1)). * * * * SqrtNegEps = 0x3e40000000000000, then 1-SqrtNegEps^2 = 1. * * * * Asymtotic behaviors, ( exact with respect to machine arithmetic ) * * are filtered out and computed seperately to avoid spurious over and * * underflows. * * * *-inf -1/sqrtNegEps -4/3 -sqrtNegEps 0 +sqrtNegEps +4/3 +1/sqrtNegEps +inf* *------------+--------+--------+-------+-------+--------+----------+-----------* *<-log(2|x|)>|< ArcSinh(x) >|< (x) >|< ArcSinh(x) >|< log(2x) >* * * * The constant Log2 can be obtained using the 68040 instruction * * * * FMOVECR.X #$30, FP0 ; -=> FP0 = log(2) = 0x3FE62E42FEFA39EF (in double)* * ; = 6.9314718055994530940e-01 * * * *******************************************************************************/ if ( PositiveX < SqrtNegEps.d ) ; else if ( PositiveX <= 4.0 / 3.0 ) { InvPositiveX = 1.0 / PositiveX; PositiveX = log1p ( PositiveX + PositiveX / ( InvPositiveX + sqrt ( 1.0 + InvPositiveX * InvPositiveX ) ) ); } else if ( PositiveX <= 1.0 / SqrtNegEps.d ) PositiveX = log ( 2.0 * PositiveX + 1.0 / ( PositiveX + sqrt ( 1.0 + PositiveX * PositiveX ) ) ); else if ( x != x ) x *= 2.0; else PositiveX = Log2.d + log ( PositiveX ); switch ( __fpclassifyd ( PositiveX ) ) { case FP_SUBNORMAL: OldEnvironment.i.lo |= FE_UNDERFLOW; /* FALL THROUGH */ case FP_NORMAL: OldEnvironment.i.lo |= FE_INEXACT; /* FALL THROUGH */ default: break; } fegetenvd ( NewEnvironment.d ); OldEnvironment.i.lo |= NewEnvironment.i.lo; fesetenvd ( OldEnvironment.d ); return ( copysign ( PositiveX, x ) ); } #ifdef notdef float asinhf( float x ) { return (float)asinh( x ); } #endif /******************************************************************************* * A R C C O S H * *******************************************************************************/ double acosh ( double x ) { register double xMinusOne; hexdouble OldEnvironment; fegetenvd ( OldEnvironment.d ); fesetenvd ( 0.0 ); xMinusOne = x - 1.0; /******************************************************************************* * * * The expression for computing ArcCosh(x) is: * * * * ArcCosh(x) = log(x+sqrt(x^2-1)), for x ³ 1. * * * * SqrtNegEps = 0x3e40000000000000, then 1-SqrtNegEps^2 = 1. * * * * (1) if x is in [1, 5/4] then we would like to use the more accurate * * log1p. Make the change of variable x => x-1 to handle operations on a * * lower binade. * * * * (2) If x is in a regular range (5/4,1/sqrteps], then multiply * * x+sqrt(x^2-1) by sqrt(x^2-1)/sqrt(x^2-1) and simplify to get * * 2x-1/(x+sqrt(x^2-1). This operation will increase the accuracy of the * * computation. * * * * (3) For large x, such that x^2 - 1 = x^2, ArcCosh(x) = log(2x). * * This asymtotical behavior ( exact with respect to machine arithmetic, * * is filtered out and computed seperately to avoid spurious overflows. * * * * -inf +1 +5/4 +1/sqrtNegEps +inf * * ---------+---------------------+---------------------+--------------- * * < N a N >|< (1) ArcCosh(x) >|< (2) ArcCosh(x) >|< (3) log(2x) > * * * *******************************************************************************/ if ( x >= 1.0 ) { if ( x <= FiveFourth.d ) x = log1p ( xMinusOne + sqrt ( 2.0 * xMinusOne + xMinusOne * xMinusOne ) ); else if ( ( FiveFourth.d < x ) && ( x <= 1.0 / SqrtNegEps.d ) ) x = log ( 2 * x - 1.0 / ( x + sqrt ( x * x - 1.0 ) ) ); else x = Log2.d + log ( x ); } /******************************************************************************* * If argument is SNaN then a QNaN has to be returned and the invalid * * flag signaled for SNaN. Otherwise, the argument is less than 1.0 and * * a hyperbolic nan is returned. * *******************************************************************************/ else { if ( x != x ) /* check for argument = NaN confirmed. */ x *= 2.0; else { OldEnvironment.i.lo |= SET_INVALID; x = nan ( INVERSE_HYPERBOLIC_NAN ); } } switch ( __fpclassifyd ( x ) ) { case FP_SUBNORMAL: OldEnvironment.i.lo |= FE_UNDERFLOW; /* FALL THROUGH */ case FP_NORMAL: OldEnvironment.i.lo |= FE_INEXACT; /* FALL THROUGH */ default: break; } fesetenvd ( OldEnvironment.d ); return x; } #ifdef notdef float acoshf( float x ) { return (float)acosh( x ); } #endif #else /* __APPLE_CC__ version */ #warning A higher version than gcc-932 is required. #endif /* __APPLE_CC__ version */ #endif /* __APPLE_CC__ */