Average Error: 18.9 → 0.2
Time: 16.9s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.999970095902279254573841171804815530777:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.999970095902279254573841171804815530777:\\
\;\;\;\;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 r1592812 = 1.0;
        double r1592813 = x;
        double r1592814 = y;
        double r1592815 = r1592813 - r1592814;
        double r1592816 = r1592812 - r1592814;
        double r1592817 = r1592815 / r1592816;
        double r1592818 = r1592812 - r1592817;
        double r1592819 = log(r1592818);
        double r1592820 = r1592812 - r1592819;
        return r1592820;
}
↓
double f(double x, double y) {
        double r1592821 = x;
        double r1592822 = y;
        double r1592823 = r1592821 - r1592822;
        double r1592824 = 1.0;
        double r1592825 = r1592824 - r1592822;
        double r1592826 = r1592823 / r1592825;
        double r1592827 = 0.9999700959022793;
        bool r1592828 = r1592826 <= r1592827;
        double r1592829 = 1.0;
        double r1592830 = r1592829 / r1592825;
        double r1592831 = r1592823 * r1592830;
        double r1592832 = r1592824 - r1592831;
        double r1592833 = log(r1592832);
        double r1592834 = r1592824 - r1592833;
        double r1592835 = r1592821 / r1592822;
        double r1592836 = 2.0;
        double r1592837 = pow(r1592822, r1592836);
        double r1592838 = r1592821 / r1592837;
        double r1592839 = r1592824 * r1592838;
        double r1592840 = r1592835 + r1592839;
        double r1592841 = r1592824 / r1592822;
        double r1592842 = r1592840 - r1592841;
        double r1592843 = log(r1592842);
        double r1592844 = r1592824 - r1592843;
        double r1592845 = r1592828 ? r1592834 : r1592844;
        return r1592845;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.9
Target0.1
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)) < 0.9999700959022793

    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 0.9999700959022793 < (/ (- 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 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.999970095902279254573841171804815530777:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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))))))