Average Error: 29.8 → 1.8
Time: 18.8s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.076810882413635365417964376133599293418 \cdot 10^{56} \lor \neg \left(z \le 128535262688296208\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{{z}^{2}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\\ \end{array}\]
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 -3.076810882413635365417964376133599293418 \cdot 10^{56} \lor \neg \left(z \le 128535262688296208\right):\\
\;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{{z}^{2}}, x\right)\\

\mathbf{else}:\\
\;\;\;\;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}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r251831 = x;
        double r251832 = y;
        double r251833 = z;
        double r251834 = 3.13060547623;
        double r251835 = r251833 * r251834;
        double r251836 = 11.1667541262;
        double r251837 = r251835 + r251836;
        double r251838 = r251837 * r251833;
        double r251839 = t;
        double r251840 = r251838 + r251839;
        double r251841 = r251840 * r251833;
        double r251842 = a;
        double r251843 = r251841 + r251842;
        double r251844 = r251843 * r251833;
        double r251845 = b;
        double r251846 = r251844 + r251845;
        double r251847 = r251832 * r251846;
        double r251848 = 15.234687407;
        double r251849 = r251833 + r251848;
        double r251850 = r251849 * r251833;
        double r251851 = 31.4690115749;
        double r251852 = r251850 + r251851;
        double r251853 = r251852 * r251833;
        double r251854 = 11.9400905721;
        double r251855 = r251853 + r251854;
        double r251856 = r251855 * r251833;
        double r251857 = 0.607771387771;
        double r251858 = r251856 + r251857;
        double r251859 = r251847 / r251858;
        double r251860 = r251831 + r251859;
        return r251860;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r251861 = z;
        double r251862 = -3.0768108824136354e+56;
        bool r251863 = r251861 <= r251862;
        double r251864 = 1.2853526268829621e+17;
        bool r251865 = r251861 <= r251864;
        double r251866 = !r251865;
        bool r251867 = r251863 || r251866;
        double r251868 = y;
        double r251869 = 3.13060547623;
        double r251870 = t;
        double r251871 = 2.0;
        double r251872 = pow(r251861, r251871);
        double r251873 = r251870 / r251872;
        double r251874 = r251869 + r251873;
        double r251875 = x;
        double r251876 = fma(r251868, r251874, r251875);
        double r251877 = r251861 * r251869;
        double r251878 = 11.1667541262;
        double r251879 = r251877 + r251878;
        double r251880 = r251879 * r251861;
        double r251881 = r251880 + r251870;
        double r251882 = r251881 * r251861;
        double r251883 = a;
        double r251884 = r251882 + r251883;
        double r251885 = r251884 * r251861;
        double r251886 = b;
        double r251887 = r251885 + r251886;
        double r251888 = r251868 * r251887;
        double r251889 = 15.234687407;
        double r251890 = r251861 + r251889;
        double r251891 = r251890 * r251861;
        double r251892 = 31.4690115749;
        double r251893 = r251891 + r251892;
        double r251894 = r251893 * r251861;
        double r251895 = 11.9400905721;
        double r251896 = r251894 + r251895;
        double r251897 = r251896 * r251861;
        double r251898 = 0.607771387771;
        double r251899 = r251897 + r251898;
        double r251900 = r251888 / r251899;
        double r251901 = r251875 + r251900;
        double r251902 = r251867 ? r251876 : r251901;
        return r251902;
}

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

Original29.8
Target1.0
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.0768108824136354e+56 or 1.2853526268829621e+17 < z

    1. Initial program 59.6

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

    2. Simplified57.8

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

    3. Taylor expanded around inf 8.6

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

    4. Simplified1.8

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

    if -3.0768108824136354e+56 < z < 1.2853526268829621e+17

    1. Initial program 1.8

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.076810882413635365417964376133599293418 \cdot 10^{56} \lor \neg \left(z \le 128535262688296208\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{{z}^{2}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1))) (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)))))

  (+ 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))))