Average Error: 18.2 → 0.3
Time: 18.0s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 - \frac{x - y}{1 - y} \le 4.669598041573408409021794795989990234375 \cdot 10^{-13}:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;1 - \frac{x - y}{1 - y} \le 4.669598041573408409021794795989990234375 \cdot 10^{-13}:\\
\;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\

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

\end{array}
double f(double x, double y) {
        double r221656 = 1.0;
        double r221657 = x;
        double r221658 = y;
        double r221659 = r221657 - r221658;
        double r221660 = r221656 - r221658;
        double r221661 = r221659 / r221660;
        double r221662 = r221656 - r221661;
        double r221663 = log(r221662);
        double r221664 = r221656 - r221663;
        return r221664;
}
↓
double f(double x, double y) {
        double r221665 = 1.0;
        double r221666 = x;
        double r221667 = y;
        double r221668 = r221666 - r221667;
        double r221669 = r221665 - r221667;
        double r221670 = r221668 / r221669;
        double r221671 = r221665 - r221670;
        double r221672 = 4.669598041573408e-13;
        bool r221673 = r221671 <= r221672;
        double r221674 = 1.0;
        double r221675 = r221665 / r221667;
        double r221676 = r221674 + r221675;
        double r221677 = r221666 / r221667;
        double r221678 = r221676 * r221677;
        double r221679 = r221678 - r221675;
        double r221680 = log(r221679);
        double r221681 = r221665 - r221680;
        double r221682 = sqrt(r221671);
        double r221683 = log(r221682);
        double r221684 = r221683 + r221683;
        double r221685 = r221665 - r221684;
        double r221686 = r221673 ? r221681 : r221685;
        return r221686;
}

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

Derivation

  1. Split input into 2 regimes

  2. if (- 1.0 (/ (- x y) (- 1.0 y))) < 4.669598041573408e-13

    1. Initial program 63.6

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if 4.669598041573408e-13 < (- 1.0 (/ (- x y) (- 1.0 y)))

    1. Initial program 0.3

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

    3. Applied add-sqr-sqrt0.4

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

    4. Applied log-prod0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{x - y}{1 - y} \le 4.669598041573408409021794795989990234375 \cdot 10^{-13}:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))))))