Average Error: 29.4 → 1.0
Time: 23.7s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3240660210645078190185871572992:\\ \;\;\;\;\left(\left(\frac{t}{z \cdot z} \cdot y + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\ \mathbf{elif}\;z \le 587734457247325707726001012736:\\ \;\;\;\;x + \frac{y}{\frac{z \cdot \left(11.94009057210000079862766142468899488449 + \left(31.46901157490000144889563671313226222992 + z \cdot \left(z + 15.2346874069999991263557603815570473671\right)\right) \cdot z\right) + 0.6077713877710000378584709324059076607227}{b + \left(z \cdot \left(\left(3.130605476229999961645944495103321969509 \cdot z + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) + a\right) \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{t}{z \cdot z} \cdot y + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\ \end{array}\]
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 -3240660210645078190185871572992:\\
\;\;\;\;\left(\left(\frac{t}{z \cdot z} \cdot y + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\

\mathbf{elif}\;z \le 587734457247325707726001012736:\\
\;\;\;\;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(\frac{t}{z \cdot z} \cdot y + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r22096010 = x;
        double r22096011 = y;
        double r22096012 = z;
        double r22096013 = 3.13060547623;
        double r22096014 = r22096012 * r22096013;
        double r22096015 = 11.1667541262;
        double r22096016 = r22096014 + r22096015;
        double r22096017 = r22096016 * r22096012;
        double r22096018 = t;
        double r22096019 = r22096017 + r22096018;
        double r22096020 = r22096019 * r22096012;
        double r22096021 = a;
        double r22096022 = r22096020 + r22096021;
        double r22096023 = r22096022 * r22096012;
        double r22096024 = b;
        double r22096025 = r22096023 + r22096024;
        double r22096026 = r22096011 * r22096025;
        double r22096027 = 15.234687407;
        double r22096028 = r22096012 + r22096027;
        double r22096029 = r22096028 * r22096012;
        double r22096030 = 31.4690115749;
        double r22096031 = r22096029 + r22096030;
        double r22096032 = r22096031 * r22096012;
        double r22096033 = 11.9400905721;
        double r22096034 = r22096032 + r22096033;
        double r22096035 = r22096034 * r22096012;
        double r22096036 = 0.607771387771;
        double r22096037 = r22096035 + r22096036;
        double r22096038 = r22096026 / r22096037;
        double r22096039 = r22096010 + r22096038;
        return r22096039;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r22096040 = z;
        double r22096041 = -3.240660210645078e+30;
        bool r22096042 = r22096040 <= r22096041;
        double r22096043 = t;
        double r22096044 = r22096040 * r22096040;
        double r22096045 = r22096043 / r22096044;
        double r22096046 = y;
        double r22096047 = r22096045 * r22096046;
        double r22096048 = 3.13060547623;
        double r22096049 = r22096048 * r22096046;
        double r22096050 = r22096047 + r22096049;
        double r22096051 = 36.527041698806414;
        double r22096052 = r22096040 / r22096046;
        double r22096053 = r22096051 / r22096052;
        double r22096054 = r22096050 - r22096053;
        double r22096055 = x;
        double r22096056 = r22096054 + r22096055;
        double r22096057 = 5.877344572473257e+29;
        bool r22096058 = r22096040 <= r22096057;
        double r22096059 = 11.9400905721;
        double r22096060 = 31.4690115749;
        double r22096061 = 15.234687407;
        double r22096062 = r22096040 + r22096061;
        double r22096063 = r22096040 * r22096062;
        double r22096064 = r22096060 + r22096063;
        double r22096065 = r22096064 * r22096040;
        double r22096066 = r22096059 + r22096065;
        double r22096067 = r22096040 * r22096066;
        double r22096068 = 0.607771387771;
        double r22096069 = r22096067 + r22096068;
        double r22096070 = b;
        double r22096071 = r22096048 * r22096040;
        double r22096072 = 11.1667541262;
        double r22096073 = r22096071 + r22096072;
        double r22096074 = r22096073 * r22096040;
        double r22096075 = r22096074 + r22096043;
        double r22096076 = r22096040 * r22096075;
        double r22096077 = a;
        double r22096078 = r22096076 + r22096077;
        double r22096079 = r22096078 * r22096040;
        double r22096080 = r22096070 + r22096079;
        double r22096081 = r22096069 / r22096080;
        double r22096082 = r22096046 / r22096081;
        double r22096083 = r22096055 + r22096082;
        double r22096084 = r22096058 ? r22096083 : r22096056;
        double r22096085 = r22096042 ? r22096056 : r22096084;
        return r22096085;
}

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

Original29.4
Target0.8
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.240660210645078e+30 or 5.877344572473257e+29 < z

    1. Initial program 58.8

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

    2. Taylor expanded around inf 9.2

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

    3. Simplified1.6

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

    4. Taylor expanded around 0 9.2

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

    5. Simplified1.5

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

    if -3.240660210645078e+30 < z < 5.877344572473257e+29

    1. Initial program 1.1

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

    3. Applied associate-/l*0.5

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3240660210645078190185871572992:\\ \;\;\;\;\left(\left(\frac{t}{z \cdot z} \cdot y + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\ \mathbf{elif}\;z \le 587734457247325707726001012736:\\ \;\;\;\;x + \frac{y}{\frac{z \cdot \left(11.94009057210000079862766142468899488449 + \left(31.46901157490000144889563671313226222992 + z \cdot \left(z + 15.2346874069999991263557603815570473671\right)\right) \cdot z\right) + 0.6077713877710000378584709324059076607227}{b + \left(z \cdot \left(\left(3.130605476229999961645944495103321969509 \cdot z + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) + a\right) \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{t}{z \cdot z} \cdot y + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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))))