Average Error: 18.4 → 0.2
Time: 8.0s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.941146194758038734:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.941146194758038734:\\
\;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}\right)\\

\end{array}
double f(double x, double y) {
        double r306868 = 1.0;
        double r306869 = x;
        double r306870 = y;
        double r306871 = r306869 - r306870;
        double r306872 = r306868 - r306870;
        double r306873 = r306871 / r306872;
        double r306874 = r306868 - r306873;
        double r306875 = log(r306874);
        double r306876 = r306868 - r306875;
        return r306876;
}
↓
double f(double x, double y) {
        double r306877 = x;
        double r306878 = y;
        double r306879 = r306877 - r306878;
        double r306880 = 1.0;
        double r306881 = r306880 - r306878;
        double r306882 = r306879 / r306881;
        double r306883 = 0.9411461947580387;
        bool r306884 = r306882 <= r306883;
        double r306885 = 1.0;
        double r306886 = r306885 / r306881;
        double r306887 = r306879 * r306886;
        double r306888 = r306880 - r306887;
        double r306889 = log(r306888);
        double r306890 = r306880 - r306889;
        double r306891 = r306877 / r306878;
        double r306892 = 2.0;
        double r306893 = pow(r306878, r306892);
        double r306894 = r306877 / r306893;
        double r306895 = r306880 * r306894;
        double r306896 = r306891 + r306895;
        double r306897 = r306880 / r306878;
        double r306898 = r306896 - r306897;
        double r306899 = log(r306898);
        double r306900 = r306880 - r306899;
        double r306901 = r306884 ? r306890 : r306900;
        return r306901;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.4
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9411461947580387

    1. Initial program 0.0

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
    2. Using strategy rm

    3. Applied div-inv0.0

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

    if 0.9411461947580387 < (/ (- x y) (- 1.0 y))

    1. Initial program 61.4

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
    2. Using strategy rm

    3. Applied div-inv60.0

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

    4. Taylor expanded around inf 0.7

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

    5. Simplified0.7

      \[\leadsto 1 - \log \color{blue}{\left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.941146194758038734:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

  (- 1 (log (- 1 (/ (- x y) (- 1 y))))))