static void special (void) { mpfr_t x, y; mpfr_init (x); mpfr_init (y); mpfr_set_inf (y, -1); mpfr_set_inf (x, 1); mpfr_digamma (y, x, MPFR_RNDN); if (mpfr_inf_p (y) == 0 || mpfr_sgn (y) < 0) { printf ("error for Psi(+Inf)\n"); printf ("expected +Inf\n"); printf ("got "); mpfr_dump (y); exit (1); } mpfr_clear (x); mpfr_clear (y); }
/* Use the reflection formula Digamma(1-x) = Digamma(x) + Pi * cot(Pi*x), i.e., Digamma(x) = Digamma(1-x) - Pi * cot(Pi*x). Assume x < 1/2. */ static int mpfr_digamma_reflection (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) { mpfr_prec_t p = MPFR_PREC(y) + 10, q; mpfr_t t, u, v; mpfr_exp_t e1, expv; int inex; MPFR_ZIV_DECL (loop); /* we want that 1-x is exact with precision q: if 0 < x < 1/2, then q = PREC(x)-EXP(x) is ok, otherwise if -1 <= x < 0, q = PREC(x)-EXP(x) is ok, otherwise for x < -1, PREC(x) is ok if EXP(x) <= PREC(x), otherwise we need EXP(x) */ if (MPFR_EXP(x) < 0) q = MPFR_PREC(x) + 1 - MPFR_EXP(x); else if (MPFR_EXP(x) <= MPFR_PREC(x)) q = MPFR_PREC(x) + 1; else q = MPFR_EXP(x); mpfr_init2 (u, q); MPFR_DBGRES(inex = mpfr_ui_sub (u, 1, x, MPFR_RNDN)); MPFR_ASSERTN(inex == 0); /* if x is half an integer, cot(Pi*x) = 0, thus Digamma(x) = Digamma(1-x) */ mpfr_mul_2exp (u, u, 1, MPFR_RNDN); inex = mpfr_integer_p (u); mpfr_div_2exp (u, u, 1, MPFR_RNDN); if (inex) { inex = mpfr_digamma (y, u, rnd_mode); goto end; } mpfr_init2 (t, p); mpfr_init2 (v, p); MPFR_ZIV_INIT (loop, p); for (;;) { mpfr_const_pi (v, MPFR_RNDN); /* v = Pi*(1+theta) for |theta|<=2^(-p) */ mpfr_mul (t, v, x, MPFR_RNDN); /* (1+theta)^2 */ e1 = MPFR_EXP(t) - (mpfr_exp_t) p + 1; /* bound for t: err(t) <= 2^e1 */ mpfr_cot (t, t, MPFR_RNDN); /* cot(t * (1+h)) = cot(t) - theta * (1 + cot(t)^2) with |theta|<=t*h */ if (MPFR_EXP(t) > 0) e1 = e1 + 2 * MPFR_EXP(t) + 1; else e1 = e1 + 1; /* now theta * (1 + cot(t)^2) <= 2^e1 */ e1 += (mpfr_exp_t) p - MPFR_EXP(t); /* error is now 2^e1 ulps */ mpfr_mul (t, t, v, MPFR_RNDN); e1 ++; mpfr_digamma (v, u, MPFR_RNDN); /* error <= 1/2 ulp */ expv = MPFR_EXP(v); mpfr_sub (v, v, t, MPFR_RNDN); if (MPFR_EXP(v) < MPFR_EXP(t)) e1 += MPFR_EXP(t) - MPFR_EXP(v); /* scale error for t wrt new v */ /* now take into account the 1/2 ulp error for v */ if (expv - MPFR_EXP(v) - 1 > e1) e1 = expv - MPFR_EXP(v) - 1; else e1 ++; e1 ++; /* rounding error for mpfr_sub */ if (MPFR_CAN_ROUND (v, p - e1, MPFR_PREC(y), rnd_mode)) break; MPFR_ZIV_NEXT (loop, p); mpfr_set_prec (t, p); mpfr_set_prec (v, p); } MPFR_ZIV_FREE (loop); inex = mpfr_set (y, v, rnd_mode); mpfr_clear (t); mpfr_clear (v); end: mpfr_clear (u); return inex; }