Average Error: 28.9 → 1.1
Time: 22.3s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -11377676592436386882634106867972293791840000:\\ \;\;\;\;\left(y \cdot 3.130605476229999961645944495103321969509 + \left(\frac{y}{\frac{z \cdot z}{t}} - \frac{36.52704169880641416057187598198652267456 \cdot y}{z}\right)\right) + x\\ \mathbf{elif}\;z \le 854771490603964610616754176:\\ \;\;\;\;x + y \cdot \frac{z \cdot \left(z \cdot \left(t + \left(\left(z \cdot z\right) \cdot 3.130605476229999961645944495103321969509 + z \cdot 11.16675412620000074070958362426608800888\right)\right) + a\right) + b}{0.6077713877710000378584709324059076607227 + \left(z \cdot \left(31.46901157490000144889563671313226222992 + \left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) + 11.94009057210000079862766142468899488449\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y \cdot \frac{\frac{t}{z}}{z} - \frac{36.52704169880641416057187598198652267456 \cdot y}{z}\right) + y \cdot 3.130605476229999961645944495103321969509\right)\\ \end{array}\]
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 -11377676592436386882634106867972293791840000:\\
\;\;\;\;\left(y \cdot 3.130605476229999961645944495103321969509 + \left(\frac{y}{\frac{z \cdot z}{t}} - \frac{36.52704169880641416057187598198652267456 \cdot y}{z}\right)\right) + x\\

\mathbf{elif}\;z \le 854771490603964610616754176:\\
\;\;\;\;x + y \cdot \frac{z \cdot \left(z \cdot \left(t + \left(\left(z \cdot z\right) \cdot 3.130605476229999961645944495103321969509 + z \cdot 11.16675412620000074070958362426608800888\right)\right) + a\right) + b}{0.6077713877710000378584709324059076607227 + \left(z \cdot \left(31.46901157490000144889563671313226222992 + \left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) + 11.94009057210000079862766142468899488449\right) \cdot z}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r341996 = x;
        double r341997 = y;
        double r341998 = z;
        double r341999 = 3.13060547623;
        double r342000 = r341998 * r341999;
        double r342001 = 11.1667541262;
        double r342002 = r342000 + r342001;
        double r342003 = r342002 * r341998;
        double r342004 = t;
        double r342005 = r342003 + r342004;
        double r342006 = r342005 * r341998;
        double r342007 = a;
        double r342008 = r342006 + r342007;
        double r342009 = r342008 * r341998;
        double r342010 = b;
        double r342011 = r342009 + r342010;
        double r342012 = r341997 * r342011;
        double r342013 = 15.234687407;
        double r342014 = r341998 + r342013;
        double r342015 = r342014 * r341998;
        double r342016 = 31.4690115749;
        double r342017 = r342015 + r342016;
        double r342018 = r342017 * r341998;
        double r342019 = 11.9400905721;
        double r342020 = r342018 + r342019;
        double r342021 = r342020 * r341998;
        double r342022 = 0.607771387771;
        double r342023 = r342021 + r342022;
        double r342024 = r342012 / r342023;
        double r342025 = r341996 + r342024;
        return r342025;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r342026 = z;
        double r342027 = -1.1377676592436387e+43;
        bool r342028 = r342026 <= r342027;
        double r342029 = y;
        double r342030 = 3.13060547623;
        double r342031 = r342029 * r342030;
        double r342032 = r342026 * r342026;
        double r342033 = t;
        double r342034 = r342032 / r342033;
        double r342035 = r342029 / r342034;
        double r342036 = 36.527041698806414;
        double r342037 = r342036 * r342029;
        double r342038 = r342037 / r342026;
        double r342039 = r342035 - r342038;
        double r342040 = r342031 + r342039;
        double r342041 = x;
        double r342042 = r342040 + r342041;
        double r342043 = 8.547714906039646e+26;
        bool r342044 = r342026 <= r342043;
        double r342045 = r342032 * r342030;
        double r342046 = 11.1667541262;
        double r342047 = r342026 * r342046;
        double r342048 = r342045 + r342047;
        double r342049 = r342033 + r342048;
        double r342050 = r342026 * r342049;
        double r342051 = a;
        double r342052 = r342050 + r342051;
        double r342053 = r342026 * r342052;
        double r342054 = b;
        double r342055 = r342053 + r342054;
        double r342056 = 0.607771387771;
        double r342057 = 31.4690115749;
        double r342058 = 15.234687407;
        double r342059 = r342026 + r342058;
        double r342060 = r342059 * r342026;
        double r342061 = r342057 + r342060;
        double r342062 = r342026 * r342061;
        double r342063 = 11.9400905721;
        double r342064 = r342062 + r342063;
        double r342065 = r342064 * r342026;
        double r342066 = r342056 + r342065;
        double r342067 = r342055 / r342066;
        double r342068 = r342029 * r342067;
        double r342069 = r342041 + r342068;
        double r342070 = r342033 / r342026;
        double r342071 = r342070 / r342026;
        double r342072 = r342029 * r342071;
        double r342073 = r342072 - r342038;
        double r342074 = r342073 + r342031;
        double r342075 = r342041 + r342074;
        double r342076 = r342044 ? r342069 : r342075;
        double r342077 = r342028 ? r342042 : r342076;
        return r342077;
}

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

Original28.9
Target0.8
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.1377676592436387e+43

    1. Initial program 60.4

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

    2. Simplified58.2

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

    3. Taylor expanded around inf 8.3

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

    4. Simplified1.1

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

    5. Taylor expanded around 0 8.4

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

    6. Simplified1.1

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

    if -1.1377676592436387e+43 < z < 8.547714906039646e+26

    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. Simplified0.5

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

    3. Taylor expanded around 0 0.5

      \[\leadsto x + \frac{z \cdot \left(a + z \cdot \left(t + \color{blue}{\left(11.16675412620000074070958362426608800888 \cdot z + 3.130605476229999961645944495103321969509 \cdot {z}^{2}\right)}\right)\right) + b}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.2346874069999991263557603815570473671\right) + 31.46901157490000144889563671313226222992\right) + 11.94009057210000079862766142468899488449\right) + 0.6077713877710000378584709324059076607227} \cdot y\]

    4. Simplified0.5

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

    if 8.547714906039646e+26 < z

    1. Initial program 58.1

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

    2. Simplified54.8

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

    3. Taylor expanded around inf 9.6

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

    4. Simplified2.2

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

    6. Applied div-inv2.2

      \[\leadsto x + \left(y \cdot 3.130605476229999961645944495103321969509 + \left(\color{blue}{\left(y \cdot \frac{1}{z}\right)} \cdot \frac{t}{z} - \frac{y \cdot 36.52704169880641416057187598198652267456}{z}\right)\right)\]

    7. Applied associate-*l*2.2

      \[\leadsto x + \left(y \cdot 3.130605476229999961645944495103321969509 + \left(\color{blue}{y \cdot \left(\frac{1}{z} \cdot \frac{t}{z}\right)} - \frac{y \cdot 36.52704169880641416057187598198652267456}{z}\right)\right)\]

    8. Simplified2.2

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -11377676592436386882634106867972293791840000:\\ \;\;\;\;\left(y \cdot 3.130605476229999961645944495103321969509 + \left(\frac{y}{\frac{z \cdot z}{t}} - \frac{36.52704169880641416057187598198652267456 \cdot y}{z}\right)\right) + x\\ \mathbf{elif}\;z \le 854771490603964610616754176:\\ \;\;\;\;x + y \cdot \frac{z \cdot \left(z \cdot \left(t + \left(\left(z \cdot z\right) \cdot 3.130605476229999961645944495103321969509 + z \cdot 11.16675412620000074070958362426608800888\right)\right) + a\right) + b}{0.6077713877710000378584709324059076607227 + \left(z \cdot \left(31.46901157490000144889563671313226222992 + \left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) + 11.94009057210000079862766142468899488449\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y \cdot \frac{\frac{t}{z}}{z} - \frac{36.52704169880641416057187598198652267456 \cdot y}{z}\right) + y \cdot 3.130605476229999961645944495103321969509\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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))))