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-rw-r--r--libX11/src/xcms/cmsTrig.c1186
1 files changed, 593 insertions, 593 deletions
diff --git a/libX11/src/xcms/cmsTrig.c b/libX11/src/xcms/cmsTrig.c
index 26ae05f75..5a01a56c8 100644
--- a/libX11/src/xcms/cmsTrig.c
+++ b/libX11/src/xcms/cmsTrig.c
@@ -1,593 +1,593 @@
-
-/*
- * 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.
- */
-
-/*
- * 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));
-}
+
+/*
+ * 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.
+ */
+
+/*
+ * 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));
+}
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