/* * 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 atan2.c, * * Function atan2(y/x), * * Implementation of arc tangent of ( y / x ) for the PowerPC. * * * * Copyright c 1991-2001 Apple Computer, Inc. All rights reserved. * * * * Written by Ali Sazegari, started on November 1991, * * Modified and ported by Robert A. Murley (ram) for Mac OS X. * * * * A MathLib v4 file. * * * * December 03 1992: first rs6000 port. * * January 05 1993: added the environmental controls. * * July 07 1993: fixed the argument of __fpclassifyd. * * July 11 1993: changed feholdexcept to _feprocentry to set rounding * * to zero. removed the constant nan, fixed all edge * * cases to reflect the fpce-nceg requirements. * * September19 1994: changed all the environemntal enquiries to __setflm. * * October 06 1994: initialized CurrentEnvironment to correct an invalid * * flag problem. * * July 23 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 18 2001: added <CoreServices/CoreServices.h> to get to <fp.h> * * and <fenv.h>. * * September 25 2001: removed fenv_access pragma. * * October 05 2001: added const double pi. * * 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 "math.h" #include "fenv_private.h" #include "fp_private.h" /******************************************************************************* * Functions needed for the computation. * *******************************************************************************/ /* the following fp.h functions are used: */ /* atan, __fpclassifyd, copysign and __signbitd. */ /******************************************************************************* * This function is odd. The positive interval is computed and for * * negative values, the sign is reflected in the computation. * *******************************************************************************/ static const double kMinNormal = 0x1.0p-1022; // 2.2250738585072014e-308; static const double kMaxNormal = 1.7976931348623157e308; static const double kHalf = 0.5; extern double atanCore ( double ); extern double atanCoreInv ( double ); double atan2 ( double y, double x ) { register double result; register double FPR_env, FPR_z, FPR_half, FPR_pi, FPR_kMinNormal, FPR_kMaxNormal, FPR_absx, FPR_absy, FPR_t; FPR_z = 0.0; /******************************************************************************* * If argument is SNaN then a QNaN has to be returned and the invalid * * flag signaled. * *******************************************************************************/ if (unlikely( ( x != x ) || ( y != y ) )) return x + y; FEGETENVD( FPR_env ); FESETENVD( FPR_z ); __NOOP; // takes slot 0 following the mtfsf FPR_t = y / x; // takes slot 1 (hence fpu1) following the mtfsf FPR_pi = M_PI; FPR_kMinNormal = kMinNormal; FPR_kMaxNormal = kMaxNormal; FPR_absy = __FABS ( y ); FPR_half = kHalf; FPR_absx = __FABS ( x ); /******************************************************************************* * The next switch will decipher what sort of argument we have: * * * * atan2 ( ±0, x ) = ±0, if x > 0, * * atan2 ( ±0, +0) = ±0, * * atan2 ( ±0, x ) = ±¹, if x < 0, * * atan2 ( ±0, -0) = ±¹, * * atan2 ( y, ±0 ) = ¹/2, if y > 0, * * atan2 ( y, ±0 ) = -¹/2, if y < 0, * * atan2 ( ±y, ° ) = ±0, for finite y > 0, * * atan2 ( ±°, x ) = ±¹/2, for finite x, * * atan2 ( ±y, -°) = ±¹, for finite y > 0, * * atan2 ( ±°, ° ) = ±¹/4, * * atan2 ( ±°, -°) = ±3¹/4. * * * * note that the non obvious cases are y and x both infinite or both zero. * * for more information, see ÒBranch Cuts for Complex Elementary Functions, * * or much Much Ado About NothingÕs Sign bitÓ, by W. Kahan, Proceedings of * * the joint IMA/SIAM conference on The state of the Art in Numerical * * Analysis, 14-18 April 1986, Clarendon Press (1987). * * * * atan2(y,0) does not raise the divide-by-zero exception, nor does * * atan2(0,0) raise the invalid exception. * *******************************************************************************/ if (likely( FPR_absx <= FPR_kMaxNormal )) // slot 0 hence fpu0 { if (unlikely( x == FPR_z )) // slot 0 hence fpu 0 { if ( y > FPR_z ) result = __FMUL( FPR_half, FPR_pi ); else if ( y < FPR_z ) result = __FMUL( -FPR_half, FPR_pi ); else { if ( __signbitd ( x ) ) // x is +-0 result = copysign ( FPR_pi, y ); // y is +-0 else { FESETENVD( FPR_env ); return y; // Exact zero result. } } FESETENVD( FPR_env ); __PROG_INEXACT( FPR_pi ); return result; } } else { // Infinite x if ( x > FPR_z ) { if ( FPR_absy <= FPR_kMaxNormal ) { FESETENVD( FPR_env ); result = copysign ( FPR_z, y ); return result; } else { if ( y > FPR_z ) result = __FMUL( 0.25f, FPR_pi ); else result = __FMUL( -0.25f, FPR_pi ); } } else { if ( FPR_absy <= FPR_kMaxNormal ) result = copysign ( FPR_pi, y ); else { if ( y > FPR_z ) result = __FMUL( 0.75f, FPR_pi ); else result = __FMUL( -0.75f, FPR_pi ); } } FESETENVD( FPR_env ); __PROG_INEXACT( FPR_pi ); return result; } if (likely( FPR_absy <= FPR_kMaxNormal )) { if (unlikely( y == FPR_z )) { if ( x > FPR_z ) { FESETENVD( FPR_env ); return y; } else result = copysign ( FPR_pi, y ); // y is +-0 FESETENVD( FPR_env ); __PROG_INEXACT( FPR_pi ); return result; } } else { // Infinite y if ( y > FPR_z ) result = __FMUL( FPR_half, FPR_pi ); else result = __FMUL( -FPR_half, FPR_pi ); FESETENVD( FPR_env ); __PROG_INEXACT( FPR_pi ); return result; } /******************************************************************************* * End of the special case section. atan2 is mostly a collection of special * * case functions. Next we will carry out the main computation which at * * this point will only receive non-zero normal or denormal numbers. * *******************************************************************************/ if (FPR_absy > FPR_absx) result = atanCoreInv ( __FABS( FPR_t ) ); else result = atanCore ( __FABS( FPR_t ) ); if ( x < FPR_z ) result = FPR_pi - result; FPR_t = __FABS( result ); FESETENVD( FPR_env ); if ( FPR_t >= FPR_kMinNormal || FPR_t == FPR_z ) __PROG_INEXACT( FPR_pi ); else __PROG_UF_INEXACT( FPR_kMinNormal ); if ( y > FPR_z ) return result; else return -result; }