-#include "mp-private.h"

-

-static MPNumber pi;

-static gboolean have_pi = FALSE;

-

-static int

-mp_compare_mp_to_int(const MPNumber *x, int i)

-{

- MPNumber t;

- mp_set_from_integer(i, &t);

- return mp_compare_mp_to_mp(x, &t);

-}

-

- MPNumber t1, t2;

- break;

- mp_get_pi(&t1);

- mp_multiply(x, &t1, &t2);

- mp_divide_integer(&t2, 180, z);

- mp_get_pi(&t1);

- mp_multiply(x, &t1, &t2);

- mp_divide_integer(&t2, 200, z);

-

-void

-mp_get_pi(MPNumber *z)

-{

- if (mp_is_zero(&pi)) {

- mp_set_from_string("3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679", 10, &pi);

- have_pi = TRUE;

- }

- mp_set_from_mp(&pi, z);

-}

-

-

- MPNumber t1, t2;

- switch (unit) {

- break;

- mp_multiply_integer(x, 180, &t2);

- mp_get_pi(&t1);

- mp_divide(&t2, &t1, z);

- mp_multiply_integer(x, 200, &t2);

- mp_get_pi(&t1);

- mp_divide(&t2, &t1, z);

-/* z = sin(x) -1 >= x >= 1, do_sin = 1

- * z = cos(x) -1 >= x >= 1, do_sin = 0

- */

-static void

-mpsin1(const MPNumber *x, MPNumber *z, int do_sin)

-{

- int i, b2;

- MPNumber t1, t2;

-

- /* sin(0) = 0, cos(0) = 1 */

- if (mp_is_zero(x)) {

- if (do_sin == 0)

- mp_set_from_integer(1, z);

- else

- mp_set_from_integer(0, z);

- return;

- }

-

- mp_multiply(x, x, &t2);

- if (mp_compare_mp_to_int(&t2, 1) > 0) {

- mperr("*** ABS(X) > 1 IN CALL TO MPSIN1 ***");

- }

-

- if (do_sin == 0) {

- mp_set_from_integer(1, &t1);

- mp_set_from_integer(0, z);

- i = 1;

- } else {

- mp_set_from_mp(x, &t1);

- mp_set_from_mp(&t1, z);

- i = 2;

- }

-

- /* Taylor series */

- /* POWER SERIES LOOP. REDUCE T IF POSSIBLE */

- b2 = 2 * max(MP_BASE, 64);

- do {

- if (MP_T + t1.exponent <= 0)

- break;

-

- /* IF I*(I+1) IS NOT REPRESENTABLE AS AN INTEGER, THE FOLLOWING

- * DIVISION BY I*(I+1) HAS TO BE SPLIT UP.

- */

- mp_multiply(&t2, &t1, &t1);

- if (i > b2) {

- mp_divide_integer(&t1, -i, &t1);

- mp_divide_integer(&t1, i + 1, &t1);

- } else {

- mp_divide_integer(&t1, -i * (i + 1), &t1);

- }

- mp_add(&t1, z, z);

-

- i += 2;

- } while (t1.sign != 0);

-

- if (do_sin == 0)

- mp_add_integer(z, 1, z);

-}

-

-

-static void

-mp_sin_real(const MPNumber *x, MPAngleUnit unit, MPNumber *z)

-{

- int xs;

- MPNumber x_radians;

-

- /* sin(0) = 0 */

- if (mp_is_zero(x)) {

- mp_set_from_integer(0, z);

- return;

- }

-

- convert_to_radians(x, unit, &x_radians);

-

- xs = x_radians.sign;

- mp_abs(&x_radians, &x_radians);

-

- /* USE MPSIN1 IF ABS(X) <= 1 */

- if (mp_compare_mp_to_int(&x_radians, 1) <= 0) {

- mpsin1(&x_radians, z, 1);

- }

- /* FIND ABS(X) MODULO 2PI */

- else {

- mp_get_pi(z);

- mp_divide_integer(z, 4, z);

- mp_divide(&x_radians, z, &x_radians);

- mp_divide_integer(&x_radians, 8, &x_radians);

- mp_fractional_component(&x_radians, &x_radians);

-

- /* SUBTRACT 1/2, SAVE SIGN AND TAKE ABS */

- mp_add_fraction(&x_radians, -1, 2, &x_radians);

- xs = -xs * x_radians.sign;

- if (xs == 0) {

- mp_set_from_integer(0, z);

- return;

- }

-

- x_radians.sign = 1;

- mp_multiply_integer(&x_radians, 4, &x_radians);

-

- /* IF NOT LESS THAN 1, SUBTRACT FROM 2 */

- if (x_radians.exponent > 0)

- mp_add_integer(&x_radians, -2, &x_radians);

-

- if (mp_is_zero(&x_radians)) {

- mp_set_from_integer(0, z);

- return;

- }

-

- x_radians.sign = 1;

- mp_multiply_integer(&x_radians, 2, &x_radians);

-

- /* NOW REDUCED TO FIRST QUADRANT, IF LESS THAN PI/4 USE

- * POWER SERIES, ELSE COMPUTE COS OF COMPLEMENT

- */

- if (x_radians.exponent > 0) {

- mp_add_integer(&x_radians, -2, &x_radians);

- mp_multiply(&x_radians, z, &x_radians);

- mpsin1(&x_radians, z, 0);

- } else {

- mp_multiply(&x_radians, z, &x_radians);

- mpsin1(&x_radians, z, 1);

- }

- }

-

- z->sign = xs;

-}

-

-

-static void

-mp_cos_real(const MPNumber *x, MPAngleUnit unit, MPNumber *z)

- /* cos(0) = 1 */

- if (mp_is_zero(x)) {

- mp_set_from_integer(1, z);

- return;

- }

-

- convert_to_radians(x, unit, z);

-

- /* Use power series if |x| <= 1 */

- mp_abs(z, z);

- if (mp_compare_mp_to_int(z, 1) <= 0) {

- mpsin1(z, z, 0);

- } else {

- MPNumber t;

-

- /* cos(x) = sin(π/2 - |x|) */

- mp_get_pi(&t);

- mp_divide_integer(&t, 2, &t);

- mp_subtract(&t, z, z);

- mp_sin(z, MP_RADIANS, z);

- }

- if (mp_is_complex(x)) {

- MPNumber x_real, x_im, z_real, z_im, t;

-

- mp_real_component(x, &x_real);

- mp_imaginary_component(x, &x_im);

-

- mp_sin_real(&x_real, unit, &z_real);

- mp_cosh(&x_im, &t);

- mp_multiply(&z_real, &t, &z_real);

-

- mp_cos_real(&x_real, unit, &z_im);

- mp_sinh(&x_im, &t);

- mp_multiply(&z_im, &t, &z_im);

-

- mp_set_from_complex(&z_real, &z_im, z);

- }

- mp_sin_real(x, unit, z);

- if (mp_is_complex(x)) {

- MPNumber x_real, x_im, z_real, z_im, t;

-

- mp_real_component(x, &x_real);

- mp_imaginary_component(x, &x_im);

-

- mp_cos_real(&x_real, unit, &z_real);

- mp_cosh(&x_im, &t);

- mp_multiply(&z_real, &t, &z_real);

-

- mp_sin_real(&x_real, unit, &z_im);

- mp_sinh(&x_im, &t);

- mp_multiply(&z_im, &t, &z_im);

- mp_invert_sign(&z_im, &z_im);

-

- mp_set_from_complex(&z_real, &z_im, z);

- }

- mp_cos_real(x, unit, z);

- MPNumber cos_x, sin_x;

- /* Check for undefined values */

- mp_cos(x, unit, &cos_x);

- if (mp_is_zero(&cos_x)) {

- /* tan(x) = sin(x) / cos(x) */

- mp_sin(x, unit, &sin_x);

- mp_divide(&sin_x, &cos_x, z);

- MPNumber t1, t2;

- /* asin⁻¹(0) = 0 */

- if (mp_is_zero(x)) {

-

- /* sin⁻¹(x) = tan⁻¹(x / √(1 - x²)), |x| < 1 */

- if (x->exponent <= 0) {

- mp_set_from_integer(1, &t1);

- mp_set_from_mp(&t1, &t2);

- mp_subtract(&t1, x, &t1);

- mp_add(&t2, x, &t2);

- mp_multiply(&t1, &t2, &t2);

- mp_root(&t2, -2, &t2);

- mp_multiply(x, &t2, z);

- mp_atan(z, unit, z);

- return;

- }

-

- /* sin⁻¹(1) = π/2, sin⁻¹(-1) = -π/2 */

- mp_set_from_integer(x->sign, &t2);

- if (mp_is_equal(x, &t2)) {

- mp_get_pi(z);

- mp_divide_integer(z, 2 * t2.sign, z);

- return;

- }

-

- /* Translators: Error displayed when inverse sine value is undefined */

- mperr(_("Inverse sine is undefined for values outside [-1, 1]"));

- mp_set_from_integer(0, z);

- MPNumber t1, t2;

- MPNumber MPn1, pi, MPy;

-

- mp_get_pi(&pi);

- mp_set_from_integer(1, &t1);

- mp_set_from_integer(-1, &MPn1);

- if (mp_is_greater_than(x, &t1) || mp_is_less_than(x, &MPn1)) {

- /* Translators: Error displayed when inverse cosine value is undefined */

- } else if (mp_is_zero(x)) {

- mp_divide_integer(&pi, 2, z);

- } else if (mp_is_equal(x, &t1)) {

- mp_set_from_integer(0, z);

- } else if (mp_is_equal(x, &MPn1)) {

- mp_set_from_mp(&pi, z);

- } else {

- /* cos⁻¹(x) = tan⁻¹(√(1 - x²) / x) */

- mp_multiply(x, x, &t2);

- mp_subtract(&t1, &t2, &t2);

- mp_sqrt(&t2, &t2);

- mp_divide(&t2, x, &t2);

- mp_atan(&t2, MP_RADIANS, &MPy);

- if (x->sign > 0) {

- mp_set_from_mp(&MPy, z);

- } else {

- mp_add(&MPy, &pi, z);

- }

-

- convert_from_radians(z, unit, z);

- int i, q;

- float rx = 0.0;

- MPNumber t1, t2;

- if (mp_is_zero(x)) {

-

- mp_set_from_mp(x, &t2);

- if (abs(x->exponent) <= 2)

- rx = mp_cast_to_float(x);

-

- /* REDUCE ARGUMENT IF NECESSARY BEFORE USING SERIES */

- q = 1;

- while (t2.exponent >= 0)

- {

- if (t2.exponent == 0 && 2 * (t2.fraction[0] + 1) <= MP_BASE)

- break;

-

- q *= 2;

-

- /* t = t / (√(t² + 1) + 1) */

- mp_multiply(&t2, &t2, z);

- mp_add_integer(z, 1, z);

- mp_sqrt(z, z);

- mp_add_integer(z, 1, z);

- mp_divide(&t2, z, &t2);

- }

-

- /* USE POWER SERIES NOW ARGUMENT IN (-0.5, 0.5) */

- mp_set_from_mp(&t2, z);

- mp_multiply(&t2, &t2, &t1);

-

- /* SERIES LOOP. REDUCE T IF POSSIBLE. */

- for (i = 1; ; i += 2) {

- if (MP_T + 2 + t2.exponent <= 1)

- break;

-

- mp_multiply(&t2, &t1, &t2);

- mp_multiply_fraction(&t2, -i, i + 2, &t2);

-

- mp_add(z, &t2, z);

- if (mp_is_zero(&t2))

- break;

- }

-

- /* CORRECT FOR ARGUMENT REDUCTION */

- mp_multiply_integer(z, q, z);

-

- /* CHECK THAT RELATIVE ERROR LESS THAN 0.01 UNLESS EXPONENT

- * OF X IS LARGE (WHEN ATAN MIGHT NOT WORK)

- */

- if (abs(x->exponent) <= 2) {

- float ry = mp_cast_to_float(z);

- /* THE FOLLOWING MESSAGE MAY INDICATE THAT B**(T-1) IS TOO SMALL. */

- if (fabs(ry - atan(rx)) >= fabs(ry) * 0.01)

- mperr("*** ERROR OCCURRED IN MP_ATAN, RESULT INCORRECT ***");

- }

-

- convert_from_radians(z, unit, z);

- MPNumber abs_x;

-

- /* sinh(0) = 0 */

- if (mp_is_zero(x)) {

- mp_set_from_integer(0, z);

- return;

- }

-

- /* WORK WITH ABS(X) */

- mp_abs(x, &abs_x);

-

- /* If |x| < 1 USE MPEXP TO AVOID CANCELLATION, otherwise IF TOO LARGE MP_EPOWY GIVES ERROR MESSAGE */

- if (abs_x.exponent <= 0) {

- MPNumber exp_x, a, b;

-

- /* ((e^|x| + 1) * (e^|x| - 1)) / e^|x| */

- // FIXME: Solves to e^|x| - e^-|x|, why not lower branch always? */

- mp_epowy(&abs_x, &exp_x);

- mp_add_integer(&exp_x, 1, &a);

- mp_add_integer(&exp_x, -1, &b);

- mp_multiply(&a, &b, z);

- mp_divide(z, &exp_x, z);

- }

- else {

- MPNumber exp_x;

-

- /* e^|x| - e^-|x| */

- mp_epowy(&abs_x, &exp_x);

- mp_reciprocal(&exp_x, z);

- mp_subtract(&exp_x, z, z);

- }

-

- /* DIVIDE BY TWO AND RESTORE SIGN */

- mp_divide_integer(z, 2, z);

- mp_multiply_integer(z, x->sign, z);

- MPNumber t;

-

- /* cosh(0) = 1 */

- if (mp_is_zero(x)) {

- mp_set_from_integer(1, z);

- return;

- }

-

- /* cosh(x) = (e^x + e^-x) / 2 */

- mp_abs(x, &t);

- mp_epowy(&t, &t);

- mp_reciprocal(&t, z);

- mp_add(&t, z, z);

- mp_divide_integer(z, 2, z);

- float r__1;

- MPNumber t;

-

- /* tanh(0) = 0 */

- if (mp_is_zero(x)) {

- mp_set_from_integer(0, z);

- return;

- }

-

- mp_abs(x, &t);

-

- /* SEE IF ABS(X) SO LARGE THAT RESULT IS +-1 */

- r__1 = (float) MP_T * 0.5 * log((float) MP_BASE);

- mp_set_from_float(r__1, z);

- if (mp_compare_mp_to_mp(&t, z) > 0) {

- mp_set_from_integer(x->sign, z);

- return;

- }

-

- /* If |x| >= 1/2 use ?, otherwise use ? to avoid cancellation */

- /* |tanh(x)| = (e^|2x| - 1) / (e^|2x| + 1) */

- mp_multiply_integer(&t, 2, &t);

- if (t.exponent > 0) {

- mp_epowy(&t, &t);

- mp_add_integer(&t, -1, z);

- mp_add_integer(&t, 1, &t);

- mp_divide(z, &t, z);

- } else {

- mp_epowy(&t, &t);

- mp_add_integer(&t, 1, z);

- mp_add_integer(&t, -1, &t);

- mp_divide(&t, z, z);

- }

-

- /* Restore sign */

- z->sign = x->sign * z->sign;

- MPNumber t;

-

- /* sinh⁻¹(x) = ln(x + √(x² + 1)) */

- mp_multiply(x, x, &t);

- mp_add_integer(&t, 1, &t);

- mp_sqrt(&t, &t);

- mp_add(x, &t, &t);

- mp_ln(&t, z);

- MPNumber t;

- if (mp_is_less_than(x, &t)) {

- /* cosh⁻¹(x) = ln(x + √(x² - 1)) */

- mp_multiply(x, x, &t);

- mp_add_integer(&t, -1, &t);

- mp_sqrt(&t, &t);

- mp_add(x, &t, &t);

- mp_ln(&t, z);

- MPNumber one, minus_one, n, d;

-

- /* Check -1 <= x <= 1 */

- mp_set_from_integer(1, &one);

- mp_set_from_integer(-1, &minus_one);

- if (mp_is_greater_equal(x, &one) || mp_is_less_equal(x, &minus_one)) {

- /* Translators: Error displayed when inverse hyperbolic tangent value is undefined */

- mperr(_("Inverse hyperbolic tangent is undefined for values outside [-1, 1]"));

-

- /* atanh(x) = 0.5 * ln((1 + x) / (1 - x)) */

- mp_add_integer(x, 1, &n);

- mp_set_from_mp(x, &d);

- mp_invert_sign(&d, &d);

- mp_add_integer(&d, 1, &d);

- mp_divide(&n, &d, z);

- mp_ln(z, z);

- mp_divide_integer(z, 2, z);