Average Error: 18.6 → 0.1
Time: 6.8s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999999132201147222787085411255247890949:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999999132201147222787085411255247890949:\\
\;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\

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

\end{array}
double f(double x, double y) {
        double r350909 = 1.0;
        double r350910 = x;
        double r350911 = y;
        double r350912 = r350910 - r350911;
        double r350913 = r350909 - r350911;
        double r350914 = r350912 / r350913;
        double r350915 = r350909 - r350914;
        double r350916 = log(r350915);
        double r350917 = r350909 - r350916;
        return r350917;
}
↓
double f(double x, double y) {
        double r350918 = x;
        double r350919 = y;
        double r350920 = r350918 - r350919;
        double r350921 = 1.0;
        double r350922 = r350921 - r350919;
        double r350923 = r350920 / r350922;
        double r350924 = 0.9999999132201147;
        bool r350925 = r350923 <= r350924;
        double r350926 = r350921 - r350923;
        double r350927 = sqrt(r350926);
        double r350928 = log(r350927);
        double r350929 = r350928 + r350928;
        double r350930 = r350921 - r350929;
        double r350931 = 2.0;
        double r350932 = pow(r350919, r350931);
        double r350933 = r350918 / r350932;
        double r350934 = 1.0;
        double r350935 = r350934 / r350919;
        double r350936 = r350933 - r350935;
        double r350937 = r350921 * r350936;
        double r350938 = r350918 / r350919;
        double r350939 = r350937 + r350938;
        double r350940 = log(r350939);
        double r350941 = r350921 - r350940;
        double r350942 = r350925 ? r350930 : r350941;
        return r350942;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.6
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)) < 0.9999999132201147

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied log-prod0.1

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

    if 0.9999999132201147 < (/ (- x y) (- 1.0 y))

    1. Initial program 62.9

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999999132201147222787085411255247890949:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(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))))))