Average Error: 29.0 → 1.6
Time: 1.0m
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.944788434501455940360221547012999398233 \cdot 10^{56}:\\ \;\;\;\;\left(x + y \cdot 3.130605476229999961645944495103321969509\right) - \frac{y}{z} \cdot 36.52704169880641416057187598198652267456\\ \mathbf{elif}\;z \le 2808458130450682678548765868032:\\ \;\;\;\;y \cdot \frac{b + \left(a + \left(t + z \cdot \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right)\right) \cdot z\right) \cdot z}{\left(\left(31.46901157490000144889563671313226222992 + z \cdot \left(z + 15.2346874069999991263557603815570473671\right)\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227} + x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{y}{z} \cdot \frac{t}{z} + y \cdot 3.130605476229999961645944495103321969509\right) - \frac{y \cdot 36.52704169880641416057187598198652267456}{z}\right) + x\\ \end{array}\]
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 -1.944788434501455940360221547012999398233 \cdot 10^{56}:\\
\;\;\;\;\left(x + y \cdot 3.130605476229999961645944495103321969509\right) - \frac{y}{z} \cdot 36.52704169880641416057187598198652267456\\

\mathbf{elif}\;z \le 2808458130450682678548765868032:\\
\;\;\;\;y \cdot \frac{b + \left(a + \left(t + z \cdot \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right)\right) \cdot z\right) \cdot z}{\left(\left(31.46901157490000144889563671313226222992 + z \cdot \left(z + 15.2346874069999991263557603815570473671\right)\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r20050809 = x;
        double r20050810 = y;
        double r20050811 = z;
        double r20050812 = 3.13060547623;
        double r20050813 = r20050811 * r20050812;
        double r20050814 = 11.1667541262;
        double r20050815 = r20050813 + r20050814;
        double r20050816 = r20050815 * r20050811;
        double r20050817 = t;
        double r20050818 = r20050816 + r20050817;
        double r20050819 = r20050818 * r20050811;
        double r20050820 = a;
        double r20050821 = r20050819 + r20050820;
        double r20050822 = r20050821 * r20050811;
        double r20050823 = b;
        double r20050824 = r20050822 + r20050823;
        double r20050825 = r20050810 * r20050824;
        double r20050826 = 15.234687407;
        double r20050827 = r20050811 + r20050826;
        double r20050828 = r20050827 * r20050811;
        double r20050829 = 31.4690115749;
        double r20050830 = r20050828 + r20050829;
        double r20050831 = r20050830 * r20050811;
        double r20050832 = 11.9400905721;
        double r20050833 = r20050831 + r20050832;
        double r20050834 = r20050833 * r20050811;
        double r20050835 = 0.607771387771;
        double r20050836 = r20050834 + r20050835;
        double r20050837 = r20050825 / r20050836;
        double r20050838 = r20050809 + r20050837;
        return r20050838;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r20050839 = z;
        double r20050840 = -1.944788434501456e+56;
        bool r20050841 = r20050839 <= r20050840;
        double r20050842 = x;
        double r20050843 = y;
        double r20050844 = 3.13060547623;
        double r20050845 = r20050843 * r20050844;
        double r20050846 = r20050842 + r20050845;
        double r20050847 = r20050843 / r20050839;
        double r20050848 = 36.527041698806414;
        double r20050849 = r20050847 * r20050848;
        double r20050850 = r20050846 - r20050849;
        double r20050851 = 2.8084581304506827e+30;
        bool r20050852 = r20050839 <= r20050851;
        double r20050853 = b;
        double r20050854 = a;
        double r20050855 = t;
        double r20050856 = 11.1667541262;
        double r20050857 = r20050844 * r20050839;
        double r20050858 = r20050856 + r20050857;
        double r20050859 = r20050839 * r20050858;
        double r20050860 = r20050855 + r20050859;
        double r20050861 = r20050860 * r20050839;
        double r20050862 = r20050854 + r20050861;
        double r20050863 = r20050862 * r20050839;
        double r20050864 = r20050853 + r20050863;
        double r20050865 = 31.4690115749;
        double r20050866 = 15.234687407;
        double r20050867 = r20050839 + r20050866;
        double r20050868 = r20050839 * r20050867;
        double r20050869 = r20050865 + r20050868;
        double r20050870 = r20050869 * r20050839;
        double r20050871 = 11.9400905721;
        double r20050872 = r20050870 + r20050871;
        double r20050873 = r20050872 * r20050839;
        double r20050874 = 0.607771387771;
        double r20050875 = r20050873 + r20050874;
        double r20050876 = r20050864 / r20050875;
        double r20050877 = r20050843 * r20050876;
        double r20050878 = r20050877 + r20050842;
        double r20050879 = r20050855 / r20050839;
        double r20050880 = r20050847 * r20050879;
        double r20050881 = r20050880 + r20050845;
        double r20050882 = r20050843 * r20050848;
        double r20050883 = r20050882 / r20050839;
        double r20050884 = r20050881 - r20050883;
        double r20050885 = r20050884 + r20050842;
        double r20050886 = r20050852 ? r20050878 : r20050885;
        double r20050887 = r20050841 ? r20050850 : r20050886;
        return r20050887;
}

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

Original29.0
Target1.0
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.944788434501456e+56

    1. Initial program 61.7

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

    2. Taylor expanded around inf 8.4

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

    3. Simplified8.5

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

    4. Taylor expanded around inf 3.5

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

    if -1.944788434501456e+56 < z < 2.8084581304506827e+30

    1. Initial program 1.9

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

    3. Applied *-un-lft-identity1.9

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227\right)}}\]

    4. Applied times-frac0.8

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}}\]

    5. Simplified0.8

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

    if 2.8084581304506827e+30 < z

    1. Initial program 59.3

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

    2. Taylor expanded around inf 8.5

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

    3. Simplified8.6

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

    5. Applied times-frac1.6

      \[\leadsto x + \left(\left(y \cdot 3.130605476229999961645944495103321969509 + \color{blue}{\frac{y}{z} \cdot \frac{t}{z}}\right) - \frac{y \cdot 36.52704169880641416057187598198652267456}{z}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.944788434501455940360221547012999398233 \cdot 10^{56}:\\ \;\;\;\;\left(x + y \cdot 3.130605476229999961645944495103321969509\right) - \frac{y}{z} \cdot 36.52704169880641416057187598198652267456\\ \mathbf{elif}\;z \le 2808458130450682678548765868032:\\ \;\;\;\;y \cdot \frac{b + \left(a + \left(t + z \cdot \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right)\right) \cdot z\right) \cdot z}{\left(\left(31.46901157490000144889563671313226222992 + z \cdot \left(z + 15.2346874069999991263557603815570473671\right)\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227} + x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{y}{z} \cdot \frac{t}{z} + y \cdot 3.130605476229999961645944495103321969509\right) - \frac{y \cdot 36.52704169880641416057187598198652267456}{z}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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))))