Average Error: 18.4 → 0.1
Time: 16.9s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.999963631452672641053425195423187687993:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{1 \cdot x}{y \cdot y} + \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.999963631452672641053425195423187687993:\\
\;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r7717882 = 1.0;
        double r7717883 = x;
        double r7717884 = y;
        double r7717885 = r7717883 - r7717884;
        double r7717886 = r7717882 - r7717884;
        double r7717887 = r7717885 / r7717886;
        double r7717888 = r7717882 - r7717887;
        double r7717889 = log(r7717888);
        double r7717890 = r7717882 - r7717889;
        return r7717890;
}
↓
double f(double x, double y) {
        double r7717891 = x;
        double r7717892 = y;
        double r7717893 = r7717891 - r7717892;
        double r7717894 = 1.0;
        double r7717895 = r7717894 - r7717892;
        double r7717896 = r7717893 / r7717895;
        double r7717897 = 0.9999636314526726;
        bool r7717898 = r7717896 <= r7717897;
        double r7717899 = exp(r7717894);
        double r7717900 = r7717894 - r7717896;
        double r7717901 = r7717899 / r7717900;
        double r7717902 = log(r7717901);
        double r7717903 = r7717894 * r7717891;
        double r7717904 = r7717892 * r7717892;
        double r7717905 = r7717903 / r7717904;
        double r7717906 = r7717891 / r7717892;
        double r7717907 = r7717905 + r7717906;
        double r7717908 = r7717894 / r7717892;
        double r7717909 = r7717907 - r7717908;
        double r7717910 = log(r7717909);
        double r7717911 = r7717894 - r7717910;
        double r7717912 = r7717898 ? r7717902 : r7717911;
        return r7717912;
}

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
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9999636314526726

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied diff-log0.1

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

    if 0.9999636314526726 < (/ (- x y) (- 1.0 y))

    1. Initial program 62.4

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.999963631452672641053425195423187687993:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{1 \cdot x}{y \cdot y} + \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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))))))