Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D

Average Error: 29.4 → 1.1

Time: 37.5s

Precision: 64

Math

\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.275224954987132994321982327394344911771 \cdot 10^{68}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{\frac{t}{z}}{z}, x\right)\\ \mathbf{elif}\;z \le 7597311987327426783290525665460224:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{\frac{t}{z}}{z}, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r14860705 = x;
        double r14860706 = y;
        double r14860707 = z;
        double r14860708 = 3.13060547623;
        double r14860709 = r14860707 * r14860708;
        double r14860710 = 11.1667541262;
        double r14860711 = r14860709 + r14860710;
        double r14860712 = r14860711 * r14860707;
        double r14860713 = t;
        double r14860714 = r14860712 + r14860713;
        double r14860715 = r14860714 * r14860707;
        double r14860716 = a;
        double r14860717 = r14860715 + r14860716;
        double r14860718 = r14860717 * r14860707;
        double r14860719 = b;
        double r14860720 = r14860718 + r14860719;
        double r14860721 = r14860706 * r14860720;
        double r14860722 = 15.234687407;
        double r14860723 = r14860707 + r14860722;
        double r14860724 = r14860723 * r14860707;
        double r14860725 = 31.4690115749;
        double r14860726 = r14860724 + r14860725;
        double r14860727 = r14860726 * r14860707;
        double r14860728 = 11.9400905721;
        double r14860729 = r14860727 + r14860728;
        double r14860730 = r14860729 * r14860707;
        double r14860731 = 0.607771387771;
        double r14860732 = r14860730 + r14860731;
        double r14860733 = r14860721 / r14860732;
        double r14860734 = r14860705 + r14860733;
        return r14860734;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r14860735 = z;
        double r14860736 = -1.275224954987133e+68;
        bool r14860737 = r14860735 <= r14860736;
        double r14860738 = y;
        double r14860739 = 3.13060547623;
        double r14860740 = t;
        double r14860741 = r14860740 / r14860735;
        double r14860742 = r14860741 / r14860735;
        double r14860743 = r14860739 + r14860742;
        double r14860744 = x;
        double r14860745 = fma(r14860738, r14860743, r14860744);
        double r14860746 = 7.597311987327427e+33;
        bool r14860747 = r14860735 <= r14860746;
        double r14860748 = 11.1667541262;
        double r14860749 = fma(r14860739, r14860735, r14860748);
        double r14860750 = fma(r14860749, r14860735, r14860740);
        double r14860751 = a;
        double r14860752 = fma(r14860735, r14860750, r14860751);
        double r14860753 = b;
        double r14860754 = fma(r14860752, r14860735, r14860753);
        double r14860755 = 15.234687407;
        double r14860756 = r14860755 + r14860735;
        double r14860757 = 31.4690115749;
        double r14860758 = fma(r14860735, r14860756, r14860757);
        double r14860759 = 11.9400905721;
        double r14860760 = fma(r14860735, r14860758, r14860759);
        double r14860761 = 0.607771387771;
        double r14860762 = fma(r14860735, r14860760, r14860761);
        double r14860763 = r14860754 / r14860762;
        double r14860764 = fma(r14860738, r14860763, r14860744);
        double r14860765 = r14860747 ? r14860764 : r14860745;
        double r14860766 = r14860737 ? r14860745 : r14860765;
        return r14860766;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	29.4
Target	1.0
Herbie	1.1

\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -1.275224954987133e+68 or 7.597311987327427e+33 < z

   1. Initial program 61.2

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

   2. Simplified 59.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}, x\right)}\]

   3. Taylor expanded around inf 8.9

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot y}{{z}^{2}} + 3.130605476229999961645944495103321969509 \cdot y\right)}\]

   4. Simplified 0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.130605476229999961645944495103321969509, x\right)}\]

   if -1.275224954987133e+68 < z < 7.597311987327427e+33

   1. Initial program 2.8

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

   2. Simplified 1.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}, x\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.275224954987132994321982327394344911771 \cdot 10^{68}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{\frac{t}{z}}{z}, x\right)\\ \mathbf{elif}\;z \le 7597311987327426783290525665460224:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{\frac{t}{z}}{z}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0))) (if (< z 7.066965436914287e+59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771) (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))