Average Error: 29.5 → 1.4
Time: 24.9s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -982813980482942554198071844188130490974200:\\ \;\;\;\;\left(\left(y \cdot 3.130605476229999961645944495103321969509 - \frac{y}{z} \cdot 36.52704169880641416057187598198652267456\right) + \frac{t}{\frac{z \cdot z}{y}}\right) + x\\ \mathbf{elif}\;z \le 58050431039188128680116224:\\ \;\;\;\;x + \frac{y}{\frac{z \cdot \left(11.94009057210000079862766142468899488449 + \left(31.46901157490000144889563671313226222992 + z \cdot \left(z + 15.2346874069999991263557603815570473671\right)\right) \cdot z\right) + 0.6077713877710000378584709324059076607227}{b + \left(z \cdot \left(\left(3.130605476229999961645944495103321969509 \cdot z + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) + a\right) \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot 3.130605476229999961645944495103321969509 - \frac{y}{z} \cdot 36.52704169880641416057187598198652267456\right) + \frac{t}{\frac{z \cdot z}{y}}\right) + x\\ \end{array}\]
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 -982813980482942554198071844188130490974200:\\
\;\;\;\;\left(\left(y \cdot 3.130605476229999961645944495103321969509 - \frac{y}{z} \cdot 36.52704169880641416057187598198652267456\right) + \frac{t}{\frac{z \cdot z}{y}}\right) + x\\

\mathbf{elif}\;z \le 58050431039188128680116224:\\
\;\;\;\;x + \frac{y}{\frac{z \cdot \left(11.94009057210000079862766142468899488449 + \left(31.46901157490000144889563671313226222992 + z \cdot \left(z + 15.2346874069999991263557603815570473671\right)\right) \cdot z\right) + 0.6077713877710000378584709324059076607227}{b + \left(z \cdot \left(\left(3.130605476229999961645944495103321969509 \cdot z + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) + a\right) \cdot z}}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(y \cdot 3.130605476229999961645944495103321969509 - \frac{y}{z} \cdot 36.52704169880641416057187598198652267456\right) + \frac{t}{\frac{z \cdot z}{y}}\right) + x\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r19233861 = x;
        double r19233862 = y;
        double r19233863 = z;
        double r19233864 = 3.13060547623;
        double r19233865 = r19233863 * r19233864;
        double r19233866 = 11.1667541262;
        double r19233867 = r19233865 + r19233866;
        double r19233868 = r19233867 * r19233863;
        double r19233869 = t;
        double r19233870 = r19233868 + r19233869;
        double r19233871 = r19233870 * r19233863;
        double r19233872 = a;
        double r19233873 = r19233871 + r19233872;
        double r19233874 = r19233873 * r19233863;
        double r19233875 = b;
        double r19233876 = r19233874 + r19233875;
        double r19233877 = r19233862 * r19233876;
        double r19233878 = 15.234687407;
        double r19233879 = r19233863 + r19233878;
        double r19233880 = r19233879 * r19233863;
        double r19233881 = 31.4690115749;
        double r19233882 = r19233880 + r19233881;
        double r19233883 = r19233882 * r19233863;
        double r19233884 = 11.9400905721;
        double r19233885 = r19233883 + r19233884;
        double r19233886 = r19233885 * r19233863;
        double r19233887 = 0.607771387771;
        double r19233888 = r19233886 + r19233887;
        double r19233889 = r19233877 / r19233888;
        double r19233890 = r19233861 + r19233889;
        return r19233890;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r19233891 = z;
        double r19233892 = -9.828139804829426e+41;
        bool r19233893 = r19233891 <= r19233892;
        double r19233894 = y;
        double r19233895 = 3.13060547623;
        double r19233896 = r19233894 * r19233895;
        double r19233897 = r19233894 / r19233891;
        double r19233898 = 36.527041698806414;
        double r19233899 = r19233897 * r19233898;
        double r19233900 = r19233896 - r19233899;
        double r19233901 = t;
        double r19233902 = r19233891 * r19233891;
        double r19233903 = r19233902 / r19233894;
        double r19233904 = r19233901 / r19233903;
        double r19233905 = r19233900 + r19233904;
        double r19233906 = x;
        double r19233907 = r19233905 + r19233906;
        double r19233908 = 5.805043103918813e+25;
        bool r19233909 = r19233891 <= r19233908;
        double r19233910 = 11.9400905721;
        double r19233911 = 31.4690115749;
        double r19233912 = 15.234687407;
        double r19233913 = r19233891 + r19233912;
        double r19233914 = r19233891 * r19233913;
        double r19233915 = r19233911 + r19233914;
        double r19233916 = r19233915 * r19233891;
        double r19233917 = r19233910 + r19233916;
        double r19233918 = r19233891 * r19233917;
        double r19233919 = 0.607771387771;
        double r19233920 = r19233918 + r19233919;
        double r19233921 = b;
        double r19233922 = r19233895 * r19233891;
        double r19233923 = 11.1667541262;
        double r19233924 = r19233922 + r19233923;
        double r19233925 = r19233924 * r19233891;
        double r19233926 = r19233925 + r19233901;
        double r19233927 = r19233891 * r19233926;
        double r19233928 = a;
        double r19233929 = r19233927 + r19233928;
        double r19233930 = r19233929 * r19233891;
        double r19233931 = r19233921 + r19233930;
        double r19233932 = r19233920 / r19233931;
        double r19233933 = r19233894 / r19233932;
        double r19233934 = r19233906 + r19233933;
        double r19233935 = r19233909 ? r19233934 : r19233907;
        double r19233936 = r19233893 ? r19233907 : r19233935;
        return r19233936;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target1.1
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -9.828139804829426e+41 or 5.805043103918813e+25 < z

    1. Initial program 59.4

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

    3. Applied associate-/l*56.8

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

    4. Taylor expanded around inf 8.9

      \[\leadsto x + \color{blue}{\left(\left(\frac{t \cdot y}{{z}^{2}} + 3.130605476229999961645944495103321969509 \cdot y\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)}\]

    5. Simplified2.2

      \[\leadsto x + \color{blue}{\left(\left(y \cdot 3.130605476229999961645944495103321969509 - \frac{y}{z} \cdot 36.52704169880641416057187598198652267456\right) + \frac{t}{\frac{z \cdot z}{y}}\right)}\]

    if -9.828139804829426e+41 < z < 5.805043103918813e+25

    1. Initial program 1.3

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

    3. Applied associate-/l*0.7

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -982813980482942554198071844188130490974200:\\ \;\;\;\;\left(\left(y \cdot 3.130605476229999961645944495103321969509 - \frac{y}{z} \cdot 36.52704169880641416057187598198652267456\right) + \frac{t}{\frac{z \cdot z}{y}}\right) + x\\ \mathbf{elif}\;z \le 58050431039188128680116224:\\ \;\;\;\;x + \frac{y}{\frac{z \cdot \left(11.94009057210000079862766142468899488449 + \left(31.46901157490000144889563671313226222992 + z \cdot \left(z + 15.2346874069999991263557603815570473671\right)\right) \cdot z\right) + 0.6077713877710000378584709324059076607227}{b + \left(z \cdot \left(\left(3.130605476229999961645944495103321969509 \cdot z + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) + a\right) \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot 3.130605476229999961645944495103321969509 - \frac{y}{z} \cdot 36.52704169880641416057187598198652267456\right) + \frac{t}{\frac{z \cdot z}{y}}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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))))