Average Error: 18.2 → 0.1
Time: 12.5s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99999804576706985:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.99999804576706985:\\
\;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\

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

\end{array}
double f(double x, double y) {
        double r428919 = 1.0;
        double r428920 = x;
        double r428921 = y;
        double r428922 = r428920 - r428921;
        double r428923 = r428919 - r428921;
        double r428924 = r428922 / r428923;
        double r428925 = r428919 - r428924;
        double r428926 = log(r428925);
        double r428927 = r428919 - r428926;
        return r428927;
}
↓
double f(double x, double y) {
        double r428928 = x;
        double r428929 = y;
        double r428930 = r428928 - r428929;
        double r428931 = 1.0;
        double r428932 = r428931 - r428929;
        double r428933 = r428930 / r428932;
        double r428934 = 0.9999980457670699;
        bool r428935 = r428933 <= r428934;
        double r428936 = 1.0;
        double r428937 = r428936 / r428932;
        double r428938 = r428930 * r428937;
        double r428939 = r428931 - r428938;
        double r428940 = log(r428939);
        double r428941 = r428931 - r428940;
        double r428942 = r428931 / r428929;
        double r428943 = r428942 + r428936;
        double r428944 = r428928 / r428929;
        double r428945 = r428943 * r428944;
        double r428946 = r428945 - r428942;
        double r428947 = log(r428946);
        double r428948 = r428931 - r428947;
        double r428949 = r428935 ? r428941 : r428948;
        return r428949;
}

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

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9999980457670699

    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 0.9999980457670699 < (/ (- x y) (- 1.0 y))

    1. Initial program 62.3

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

    3. Applied div-inv60.8

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

    4. Taylor expanded around inf 0.3

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

    5. Simplified0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99999804576706985:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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))))))