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Diffstat (limited to 'libX11/src/xcms/cmsTrig.c')
| -rw-r--r-- | libX11/src/xcms/cmsTrig.c | 595 |
1 files changed, 595 insertions, 0 deletions
diff --git a/libX11/src/xcms/cmsTrig.c b/libX11/src/xcms/cmsTrig.c new file mode 100644 index 000000000..4e17e7da4 --- /dev/null +++ b/libX11/src/xcms/cmsTrig.c @@ -0,0 +1,595 @@ +/* $Xorg: cmsTrig.c,v 1.3 2000/08/17 19:45:10 cpqbld Exp $ */ + +/* + * Code and supporting documentation (c) Copyright 1990 1991 Tektronix, Inc. + * All Rights Reserved + * + * This file is a component of an X Window System-specific implementation + * of Xcms based on the TekColor Color Management System. Permission is + * hereby granted to use, copy, modify, sell, and otherwise distribute this + * software and its documentation for any purpose and without fee, provided + * that this copyright, permission, and disclaimer notice is reproduced in + * all copies of this software and in supporting documentation. TekColor + * is a trademark of Tektronix, Inc. + * + * Tektronix makes no representation about the suitability of this software + * for any purpose. It is provided "as is" and with all faults. + * + * TEKTRONIX DISCLAIMS ALL WARRANTIES APPLICABLE TO THIS SOFTWARE, + * INCLUDING THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A + * PARTICULAR PURPOSE. IN NO EVENT SHALL TEKTRONIX BE LIABLE FOR ANY + * SPECIAL, INDIRECT OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES WHATSOEVER + * RESULTING FROM LOSS OF USE, DATA, OR PROFITS, WHETHER IN AN ACTION OF + * CONTRACT, NEGLIGENCE, OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN + * CONNECTION WITH THE USE OR THE PERFORMANCE OF THIS SOFTWARE. + */ + +/* $XFree86: xc/lib/X11/cmsTrig.c,v 3.8 2001/10/28 03:32:34 tsi Exp $ */ +/* + * It should be pointed out that for simplicity's sake, the + * environment parameters are defined as floating point constants, + * rather than octal or hexadecimal initializations of allocated + * storage areas. This means that the range of allowed numbers + * may not exactly match the hardware's capabilities. For example, + * if the maximum positive double precision floating point number + * is EXACTLY 1.11...E100 and the constant "MAXDOUBLE is + * defined to be 1.11E100 then the numbers between 1.11E100 and + * 1.11...E100 are considered to be undefined. For most + * applications, this will cause no problems. + * + * An alternate method is to allocate a global static "double" variable, + * say "maxdouble", and use a union declaration and initialization + * to initialize it with the proper bits for the EXACT maximum value. + * This was not done because the only compilers available to the + * author did not fully support union initialization features. + * + */ + +#ifdef HAVE_CONFIG_H +#include <config.h> +#endif +#include "Xcmsint.h" + +/* forward/static */ +static double _XcmsModulo(double value, double base); +static double _XcmsPolynomial( + register int order, + double const *coeffs, + double x); +static double +_XcmsModuloF( + double val, + register double *dp); + +/* + * DEFINES + */ +#define XCMS_MAXERROR 0.000001 +#define XCMS_MAXITER 10000 +#define XCMS_PI 3.14159265358979323846264338327950 +#define XCMS_TWOPI 6.28318530717958620 +#define XCMS_HALFPI 1.57079632679489660 +#define XCMS_FOURTHPI 0.785398163397448280 +#define XCMS_SIXTHPI 0.523598775598298820 +#define XCMS_RADIANS(d) ((d) * XCMS_PI / 180.0) +#define XCMS_DEGREES(r) ((r) * 180.0 / XCMS_PI) +#define XCMS_X6_UNDERFLOWS (4.209340e-52) /* X**6 almost underflows */ +#define XCMS_X16_UNDERFLOWS (5.421010e-20) /* X**16 almost underflows*/ +#define XCMS_CHAR_BIT 8 +#define XCMS_LONG_MAX 0x7FFFFFFF +#define XCMS_DEXPLEN 11 +#define XCMS_NBITS(type) (XCMS_CHAR_BIT * (int)sizeof(type)) +#define XCMS_FABS(x) ((x) < 0.0 ? -(x) : (x)) + +/* XCMS_DMAXPOWTWO - largest power of two exactly representable as a double */ +#ifdef _CRAY +#define XCMS_DMAXPOWTWO ((double)(1 < 47)) +#else +#define XCMS_DMAXPOWTWO ((double)(XCMS_LONG_MAX) * \ + (1L << ((XCMS_NBITS(double)-XCMS_DEXPLEN) - XCMS_NBITS(int) + 1))) +#endif + +/* + * LOCAL VARIABLES + */ + +static double const cos_pcoeffs[] = { + 0.12905394659037374438e7, + -0.37456703915723204710e6, + 0.13432300986539084285e5, + -0.11231450823340933092e3 +}; + +static double const cos_qcoeffs[] = { + 0.12905394659037373590e7, + 0.23467773107245835052e5, + 0.20969518196726306286e3, + 1.0 +}; + +static double const sin_pcoeffs[] = { + 0.20664343336995858240e7, + -0.18160398797407332550e6, + 0.35999306949636188317e4, + -0.20107483294588615719e2 +}; + +static double const sin_qcoeffs[] = { + 0.26310659102647698963e7, + 0.39270242774649000308e5, + 0.27811919481083844087e3, + 1.0 +}; + +/* + * + * FUNCTION + * + * _XcmsCosine double precision cosine + * + * KEY WORDS + * + * cos + * machine independent routines + * trigonometric functions + * math libraries + * + * DESCRIPTION + * + * Returns double precision cosine of double precision + * floating point argument. + * + * USAGE + * + * double _XcmsCosine (x) + * double x; + * + * REFERENCES + * + * Computer Approximations, J.F. Hart et al, John Wiley & Sons, + * 1968, pp. 112-120. + * + * RESTRICTIONS + * + * The sin and cos routines are interactive in the sense that + * in the process of reducing the argument to the range -PI/4 + * to PI/4, each may call the other. Ultimately one or the + * other uses a polynomial approximation on the reduced + * argument. The sin approximation has a maximum relative error + * of 10**(-17.59) and the cos approximation has a maximum + * relative error of 10**(-16.18). + * + * These error bounds assume exact arithmetic + * in the polynomial evaluation. Additional rounding and + * truncation errors may occur as the argument is reduced + * to the range over which the polynomial approximation + * is valid, and as the polynomial is evaluated using + * finite-precision arithmetic. + * + * PROGRAMMER + * + * Fred Fish + * + * INTERNALS + * + * Computes cos(x) from: + * + * (1) Reduce argument x to range -PI to PI. + * + * (2) If x > PI/2 then call cos recursively + * using relation cos(x) = -cos(x - PI). + * + * (3) If x < -PI/2 then call cos recursively + * using relation cos(x) = -cos(x + PI). + * + * (4) If x > PI/4 then call sin using + * relation cos(x) = sin(PI/2 - x). + * + * (5) If x < -PI/4 then call cos using + * relation cos(x) = sin(PI/2 + x). + * + * (6) If x would cause underflow in approx + * evaluation arithmetic then return + * sqrt(1.0 - x**2). + * + * (7) By now x has been reduced to range + * -PI/4 to PI/4 and the approximation + * from HART pg. 119 can be used: + * + * cos(x) = ( p(y) / q(y) ) + * Where: + * + * y = x * (4/PI) + * + * p(y) = SUM [ Pj * (y**(2*j)) ] + * over j = {0,1,2,3} + * + * q(y) = SUM [ Qj * (y**(2*j)) ] + * over j = {0,1,2,3} + * + * P0 = 0.12905394659037374438571854e+7 + * P1 = -0.3745670391572320471032359e+6 + * P2 = 0.134323009865390842853673e+5 + * P3 = -0.112314508233409330923e+3 + * Q0 = 0.12905394659037373590295914e+7 + * Q1 = 0.234677731072458350524124e+5 + * Q2 = 0.2096951819672630628621e+3 + * Q3 = 1.0000... + * (coefficients from HART table #3843 pg 244) + * + * + * **** NOTE **** The range reduction relations used in + * this routine depend on the final approximation being valid + * over the negative argument range in addition to the positive + * argument range. The particular approximation chosen from + * HART satisfies this requirement, although not explicitly + * stated in the text. This may not be true of other + * approximations given in the reference. + * + */ + +double _XcmsCosine(double x) +{ + auto double y; + auto double yt2; + double retval; + + if (x < -XCMS_PI || x > XCMS_PI) { + x = _XcmsModulo (x, XCMS_TWOPI); + if (x > XCMS_PI) { + x = x - XCMS_TWOPI; + } else if (x < -XCMS_PI) { + x = x + XCMS_TWOPI; + } + } + if (x > XCMS_HALFPI) { + retval = -(_XcmsCosine (x - XCMS_PI)); + } else if (x < -XCMS_HALFPI) { + retval = -(_XcmsCosine (x + XCMS_PI)); + } else if (x > XCMS_FOURTHPI) { + retval = _XcmsSine (XCMS_HALFPI - x); + } else if (x < -XCMS_FOURTHPI) { + retval = _XcmsSine (XCMS_HALFPI + x); + } else if (x < XCMS_X6_UNDERFLOWS && x > -XCMS_X6_UNDERFLOWS) { + retval = _XcmsSquareRoot (1.0 - (x * x)); + } else { + y = x / XCMS_FOURTHPI; + yt2 = y * y; + retval = _XcmsPolynomial (3, cos_pcoeffs, yt2) / _XcmsPolynomial (3, cos_qcoeffs, yt2); + } + return (retval); +} + + +/* + * FUNCTION + * + * _XcmsModulo double precision modulo + * + * KEY WORDS + * + * _XcmsModulo + * machine independent routines + * math libraries + * + * DESCRIPTION + * + * Returns double precision modulo of two double + * precision arguments. + * + * USAGE + * + * double _XcmsModulo (value, base) + * double value; + * double base; + * + * PROGRAMMER + * + * Fred Fish + * + */ +static double _XcmsModulo(double value, double base) +{ + auto double intpart; + + value /= base; + value = _XcmsModuloF (value, &intpart); + value *= base; + return(value); +} + + +/* + * frac = (double) _XcmsModuloF(double val, double *dp) + * return fractional part of 'val' + * set *dp to integer part of 'val' + * + * Note -> only compiled for the CA or KA. For the KB/MC, + * "math.c" instantiates a copy of the inline function + * defined in "math.h". + */ +static double +_XcmsModuloF( + double val, + register double *dp) +{ + register double abs; + /* + * Don't use a register for this. The extra precision this results + * in on some systems causes problems. + */ + double ip; + + /* should check for illegal values here - nan, inf, etc */ + abs = XCMS_FABS(val); + if (abs >= XCMS_DMAXPOWTWO) { + ip = val; + } else { + ip = abs + XCMS_DMAXPOWTWO; /* dump fraction */ + ip -= XCMS_DMAXPOWTWO; /* restore w/o frac */ + if (ip > abs) /* if it rounds up */ + ip -= 1.0; /* fix it */ + ip = XCMS_FABS(ip); + } + *dp = ip; + return (val - ip); /* signed fractional part */ +} + + +/* + * FUNCTION + * + * _XcmsPolynomial double precision polynomial evaluation + * + * KEY WORDS + * + * poly + * machine independent routines + * math libraries + * + * DESCRIPTION + * + * Evaluates a polynomial and returns double precision + * result. Is passed a the order of the polynomial, + * a pointer to an array of double precision polynomial + * coefficients (in ascending order), and the independent + * variable. + * + * USAGE + * + * double _XcmsPolynomial (order, coeffs, x) + * int order; + * double *coeffs; + * double x; + * + * PROGRAMMER + * + * Fred Fish + * + * INTERNALS + * + * Evalates the polynomial using recursion and the form: + * + * P(x) = P0 + x(P1 + x(P2 +...x(Pn))) + * + */ + +static double _XcmsPolynomial( + register int order, + double const *coeffs, + double x) +{ + auto double rtn_value; + +#if 0 + auto double curr_coeff; + if (order <= 0) { + rtn_value = *coeffs; + } else { + curr_coeff = *coeffs; /* Bug in Unisoft's compiler. Does not */ + coeffs++; /* generate good code for *coeffs++ */ + rtn_value = curr_coeff + x * _XcmsPolynomial (--order, coeffs, x); + } +#else /* ++jrb -- removed tail recursion */ + coeffs += order; + rtn_value = *coeffs--; + while(order-- > 0) + rtn_value = *coeffs-- + (x * rtn_value); +#endif + + return(rtn_value); +} + + +/* + * FUNCTION + * + * _XcmsSine double precision sine + * + * KEY WORDS + * + * sin + * machine independent routines + * trigonometric functions + * math libraries + * + * DESCRIPTION + * + * Returns double precision sine of double precision + * floating point argument. + * + * USAGE + * + * double _XcmsSine (x) + * double x; + * + * REFERENCES + * + * Computer Approximations, J.F. Hart et al, John Wiley & Sons, + * 1968, pp. 112-120. + * + * RESTRICTIONS + * + * The sin and cos routines are interactive in the sense that + * in the process of reducing the argument to the range -PI/4 + * to PI/4, each may call the other. Ultimately one or the + * other uses a polynomial approximation on the reduced + * argument. The sin approximation has a maximum relative error + * of 10**(-17.59) and the cos approximation has a maximum + * relative error of 10**(-16.18). + * + * These error bounds assume exact arithmetic + * in the polynomial evaluation. Additional rounding and + * truncation errors may occur as the argument is reduced + * to the range over which the polynomial approximation + * is valid, and as the polynomial is evaluated using + * finite-precision arithmetic. + * + * PROGRAMMER + * + * Fred Fish + * + * INTERNALS + * + * Computes sin(x) from: + * + * (1) Reduce argument x to range -PI to PI. + * + * (2) If x > PI/2 then call sin recursively + * using relation sin(x) = -sin(x - PI). + * + * (3) If x < -PI/2 then call sin recursively + * using relation sin(x) = -sin(x + PI). + * + * (4) If x > PI/4 then call cos using + * relation sin(x) = cos(PI/2 - x). + * + * (5) If x < -PI/4 then call cos using + * relation sin(x) = -cos(PI/2 + x). + * + * (6) If x is small enough that polynomial + * evaluation would cause underflow + * then return x, since sin(x) + * approaches x as x approaches zero. + * + * (7) By now x has been reduced to range + * -PI/4 to PI/4 and the approximation + * from HART pg. 118 can be used: + * + * sin(x) = y * ( p(y) / q(y) ) + * Where: + * + * y = x * (4/PI) + * + * p(y) = SUM [ Pj * (y**(2*j)) ] + * over j = {0,1,2,3} + * + * q(y) = SUM [ Qj * (y**(2*j)) ] + * over j = {0,1,2,3} + * + * P0 = 0.206643433369958582409167054e+7 + * P1 = -0.18160398797407332550219213e+6 + * P2 = 0.359993069496361883172836e+4 + * P3 = -0.2010748329458861571949e+2 + * Q0 = 0.263106591026476989637710307e+7 + * Q1 = 0.3927024277464900030883986e+5 + * Q2 = 0.27811919481083844087953e+3 + * Q3 = 1.0000... + * (coefficients from HART table #3063 pg 234) + * + * + * **** NOTE **** The range reduction relations used in + * this routine depend on the final approximation being valid + * over the negative argument range in addition to the positive + * argument range. The particular approximation chosen from + * HART satisfies this requirement, although not explicitly + * stated in the text. This may not be true of other + * approximations given in the reference. + * + */ + +double +_XcmsSine (double x) +{ + double y; + double yt2; + double retval; + + if (x < -XCMS_PI || x > XCMS_PI) { + x = _XcmsModulo (x, XCMS_TWOPI); + if (x > XCMS_PI) { + x = x - XCMS_TWOPI; + } else if (x < -XCMS_PI) { + x = x + XCMS_TWOPI; + } + } + if (x > XCMS_HALFPI) { + retval = -(_XcmsSine (x - XCMS_PI)); + } else if (x < -XCMS_HALFPI) { + retval = -(_XcmsSine (x + XCMS_PI)); + } else if (x > XCMS_FOURTHPI) { + retval = _XcmsCosine (XCMS_HALFPI - x); + } else if (x < -XCMS_FOURTHPI) { + retval = -(_XcmsCosine (XCMS_HALFPI + x)); + } else if (x < XCMS_X6_UNDERFLOWS && x > -XCMS_X6_UNDERFLOWS) { + retval = x; + } else { + y = x / XCMS_FOURTHPI; + yt2 = y * y; + retval = y * (_XcmsPolynomial (3, sin_pcoeffs, yt2) / _XcmsPolynomial(3, sin_qcoeffs, yt2)); + } + return(retval); +} + + +/* + * NAME + * _XcmsArcTangent + * + * SYNOPSIS + */ +double +_XcmsArcTangent(double x) +/* + * DESCRIPTION + * Computes the arctangent. + * This is an implementation of the Gauss algorithm as + * described in: + * Forman S. Acton, Numerical Methods That Work, + * New York, NY, Harper & Row, 1970. + * + * RETURNS + * Returns the arctangent + */ +{ + double ai, a1 = 0.0, bi, b1 = 0.0, l, d; + double maxerror; + int i; + + if (x == 0.0) { + return (0.0); + } + if (x < 1.0) { + maxerror = x * XCMS_MAXERROR; + } else { + maxerror = XCMS_MAXERROR; + } + ai = _XcmsSquareRoot( 1.0 / (1.0 + (x * x)) ); + bi = 1.0; + for (i = 0; i < XCMS_MAXITER; i++) { + a1 = (ai + bi) / 2.0; + b1 = _XcmsSquareRoot((a1 * bi)); + if (a1 == b1) + break; + d = XCMS_FABS(a1 - b1); + if (d < maxerror) + break; + ai = a1; + bi = b1; + } + + l = ((a1 > b1) ? b1 : a1); + + a1 = _XcmsSquareRoot(1 + (x * x)); + return (x / (a1 * l)); +} |