Average Error: 18.8 → 0.2
Time: 50.1s
Precision: 64
\[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.9998974867815034:\\ \;\;\;\;1.0 - \log \left(1.0 - \frac{\frac{x - y}{\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}}}{\sqrt[3]{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\left(\frac{1.0}{y} \cdot \frac{x}{y} - \frac{1.0}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]
1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.9998974867815034:\\
\;\;\;\;1.0 - \log \left(1.0 - \frac{\frac{x - y}{\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}}}{\sqrt[3]{1.0 - y}}\right)\\

\mathbf{else}:\\
\;\;\;\;1.0 - \log \left(\left(\frac{1.0}{y} \cdot \frac{x}{y} - \frac{1.0}{y}\right) + \frac{x}{y}\right)\\

\end{array}
double f(double x, double y) {
        double r17115849 = 1.0;
        double r17115850 = x;
        double r17115851 = y;
        double r17115852 = r17115850 - r17115851;
        double r17115853 = r17115849 - r17115851;
        double r17115854 = r17115852 / r17115853;
        double r17115855 = r17115849 - r17115854;
        double r17115856 = log(r17115855);
        double r17115857 = r17115849 - r17115856;
        return r17115857;
}
↓
double f(double x, double y) {
        double r17115858 = x;
        double r17115859 = y;
        double r17115860 = r17115858 - r17115859;
        double r17115861 = 1.0;
        double r17115862 = r17115861 - r17115859;
        double r17115863 = r17115860 / r17115862;
        double r17115864 = 0.9998974867815034;
        bool r17115865 = r17115863 <= r17115864;
        double r17115866 = cbrt(r17115862);
        double r17115867 = r17115866 * r17115866;
        double r17115868 = r17115860 / r17115867;
        double r17115869 = r17115868 / r17115866;
        double r17115870 = r17115861 - r17115869;
        double r17115871 = log(r17115870);
        double r17115872 = r17115861 - r17115871;
        double r17115873 = r17115861 / r17115859;
        double r17115874 = r17115858 / r17115859;
        double r17115875 = r17115873 * r17115874;
        double r17115876 = r17115875 - r17115873;
        double r17115877 = r17115876 + r17115874;
        double r17115878 = log(r17115877);
        double r17115879 = r17115861 - r17115878;
        double r17115880 = r17115865 ? r17115872 : r17115879;
        return r17115880;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.8
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.61947241:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9998974867815034

    1. Initial program 0.1

      \[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt0.1

      \[\leadsto 1.0 - \log \left(1.0 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}\right) \cdot \sqrt[3]{1.0 - y}}}\right)\]

    4. Applied associate-/r*0.1

      \[\leadsto 1.0 - \log \left(1.0 - \color{blue}{\frac{\frac{x - y}{\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}}}{\sqrt[3]{1.0 - y}}}\right)\]

    if 0.9998974867815034 < (/ (- x y) (- 1.0 y))

    1. Initial program 61.9

      \[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt56.8

      \[\leadsto 1.0 - \log \left(1.0 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}\right) \cdot \sqrt[3]{1.0 - y}}}\right)\]

    4. Applied associate-/r*56.8

      \[\leadsto 1.0 - \log \left(1.0 - \color{blue}{\frac{\frac{x - y}{\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}}}{\sqrt[3]{1.0 - y}}}\right)\]

    5. Taylor expanded around inf 0.4

      \[\leadsto 1.0 - \log \color{blue}{\left(\left(\frac{x}{y} + 1.0 \cdot \frac{x}{{y}^{2}}\right) - 1.0 \cdot \frac{1}{y}\right)}\]

    6. Simplified0.4

      \[\leadsto 1.0 - \log \color{blue}{\left(\left(\frac{1.0}{y} \cdot \frac{x}{y} - \frac{1.0}{y}\right) + \frac{x}{y}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.9998974867815034:\\ \;\;\;\;1.0 - \log \left(1.0 - \frac{\frac{x - y}{\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}}}{\sqrt[3]{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\left(\frac{1.0}{y} \cdot \frac{x}{y} - \frac{1.0}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))