Average Error: 18.6 → 0.1
Time: 22.0s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -240248535.9542663097381591796875:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \mathbf{elif}\;y \le 4261617509.646634578704833984375:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -240248535.9542663097381591796875:\\
\;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\

\mathbf{elif}\;y \le 4261617509.646634578704833984375:\\
\;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r18480740 = 1.0;
        double r18480741 = x;
        double r18480742 = y;
        double r18480743 = r18480741 - r18480742;
        double r18480744 = r18480740 - r18480742;
        double r18480745 = r18480743 / r18480744;
        double r18480746 = r18480740 - r18480745;
        double r18480747 = log(r18480746);
        double r18480748 = r18480740 - r18480747;
        return r18480748;
}
↓
double f(double x, double y) {
        double r18480749 = y;
        double r18480750 = -240248535.9542663;
        bool r18480751 = r18480749 <= r18480750;
        double r18480752 = 1.0;
        double r18480753 = 1.0;
        double r18480754 = r18480752 / r18480749;
        double r18480755 = r18480753 + r18480754;
        double r18480756 = x;
        double r18480757 = r18480756 / r18480749;
        double r18480758 = r18480755 * r18480757;
        double r18480759 = r18480758 - r18480754;
        double r18480760 = log(r18480759);
        double r18480761 = r18480752 - r18480760;
        double r18480762 = 4261617509.6466346;
        bool r18480763 = r18480749 <= r18480762;
        double r18480764 = exp(r18480752);
        double r18480765 = r18480756 - r18480749;
        double r18480766 = r18480752 - r18480749;
        double r18480767 = r18480765 / r18480766;
        double r18480768 = r18480752 - r18480767;
        double r18480769 = r18480764 / r18480768;
        double r18480770 = log(r18480769);
        double r18480771 = r18480763 ? r18480770 : r18480761;
        double r18480772 = r18480751 ? r18480761 : r18480771;
        return r18480772;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.6
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -240248535.9542663 or 4261617509.6466346 < y

    1. Initial program 47.0

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

    if -240248535.9542663 < y < 4261617509.6466346

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied diff-log0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -240248535.9542663097381591796875:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \mathbf{elif}\;y \le 4261617509.646634578704833984375:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))