Average Error: 18.2 → 0.2
Time: 18.6s
Precision: 64
\[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.8581617207853955:\\ \;\;\;\;1.0 - \left(\log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right) + \log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} + \left(\frac{1.0}{y} \cdot \frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]
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 0.8581617207853955:\\
\;\;\;\;1.0 - \left(\log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right) + \log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)\right)\\

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

\end{array}
double f(double x, double y) {
        double r22264854 = 1.0;
        double r22264855 = x;
        double r22264856 = y;
        double r22264857 = r22264855 - r22264856;
        double r22264858 = r22264854 - r22264856;
        double r22264859 = r22264857 / r22264858;
        double r22264860 = r22264854 - r22264859;
        double r22264861 = log(r22264860);
        double r22264862 = r22264854 - r22264861;
        return r22264862;
}
↓
double f(double x, double y) {
        double r22264863 = x;
        double r22264864 = y;
        double r22264865 = r22264863 - r22264864;
        double r22264866 = 1.0;
        double r22264867 = r22264866 - r22264864;
        double r22264868 = r22264865 / r22264867;
        double r22264869 = 0.8581617207853955;
        bool r22264870 = r22264868 <= r22264869;
        double r22264871 = r22264866 - r22264868;
        double r22264872 = sqrt(r22264871);
        double r22264873 = log(r22264872);
        double r22264874 = r22264873 + r22264873;
        double r22264875 = r22264866 - r22264874;
        double r22264876 = r22264863 / r22264864;
        double r22264877 = r22264866 / r22264864;
        double r22264878 = r22264877 * r22264876;
        double r22264879 = r22264878 - r22264877;
        double r22264880 = r22264876 + r22264879;
        double r22264881 = log(r22264880);
        double r22264882 = r22264866 - r22264881;
        double r22264883 = r22264870 ? r22264875 : r22264882;
        return r22264883;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.2
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)) < 0.8581617207853955

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied log-prod0.0

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

    if 0.8581617207853955 < (/ (- x y) (- 1.0 y))

    1. Initial program 59.3

      \[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.7

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.8581617207853955:\\ \;\;\;\;1.0 - \left(\log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right) + \log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} + \left(\frac{1.0}{y} \cdot \frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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))))))