Average Error: 18.2 → 0.1
Time: 13.0s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99999804576706985:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \end{array}\]
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 r652995 = 1.0;
        double r652996 = x;
        double r652997 = y;
        double r652998 = r652996 - r652997;
        double r652999 = r652995 - r652997;
        double r653000 = r652998 / r652999;
        double r653001 = r652995 - r653000;
        double r653002 = log(r653001);
        double r653003 = r652995 - r653002;
        return r653003;
}
↓
double f(double x, double y) {
        double r653004 = x;
        double r653005 = y;
        double r653006 = r653004 - r653005;
        double r653007 = 1.0;
        double r653008 = r653007 - r653005;
        double r653009 = r653006 / r653008;
        double r653010 = 0.9999980457670699;
        bool r653011 = r653009 <= r653010;
        double r653012 = 1.0;
        double r653013 = r653012 / r653008;
        double r653014 = r653006 * r653013;
        double r653015 = r653007 - r653014;
        double r653016 = log(r653015);
        double r653017 = r653007 - r653016;
        double r653018 = r653007 / r653005;
        double r653019 = r653018 + r653012;
        double r653020 = r653004 / r653005;
        double r653021 = r653019 * r653020;
        double r653022 = r653021 - r653018;
        double r653023 = log(r653022);
        double r653024 = r653007 - r653023;
        double r653025 = r653011 ? r653017 : r653024;
        return r653025;
}

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. 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

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