Average Error: 18.3 → 0.2
Time: 19.8s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999998768371195501103443348256405442953:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999998768371195501103443348256405442953:\\
\;\;\;\;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 r295907 = 1.0;
        double r295908 = x;
        double r295909 = y;
        double r295910 = r295908 - r295909;
        double r295911 = r295907 - r295909;
        double r295912 = r295910 / r295911;
        double r295913 = r295907 - r295912;
        double r295914 = log(r295913);
        double r295915 = r295907 - r295914;
        return r295915;
}
↓
double f(double x, double y) {
        double r295916 = x;
        double r295917 = y;
        double r295918 = r295916 - r295917;
        double r295919 = 1.0;
        double r295920 = r295919 - r295917;
        double r295921 = r295918 / r295920;
        double r295922 = 0.9999998768371196;
        bool r295923 = r295921 <= r295922;
        double r295924 = 1.0;
        double r295925 = r295924 / r295920;
        double r295926 = r295918 * r295925;
        double r295927 = r295919 - r295926;
        double r295928 = log(r295927);
        double r295929 = r295919 - r295928;
        double r295930 = r295916 / r295917;
        double r295931 = 2.0;
        double r295932 = pow(r295917, r295931);
        double r295933 = r295916 / r295932;
        double r295934 = r295919 * r295933;
        double r295935 = r295930 + r295934;
        double r295936 = r295919 / r295917;
        double r295937 = r295935 - r295936;
        double r295938 = log(r295937);
        double r295939 = r295919 - r295938;
        double r295940 = r295923 ? r295929 : r295939;
        return r295940;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.3
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9999998768371196

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

    if 0.9999998768371196 < (/ (- x y) (- 1.0 y))

    1. Initial program 62.5

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999998768371195501103443348256405442953:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.619472414) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

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