Average Error: 17.8 → 0.1
Time: 7.9s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1630045945.92276692 \lor \neg \left(y \le 76447033.3308169693\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\left(\frac{x}{{y}^{2}} + \frac{x}{y}\right) - \frac{1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -1630045945.92276692 \lor \neg \left(y \le 76447033.3308169693\right):\\
\;\;\;\;1 - \log \left(1 \cdot \left(\left(\frac{x}{{y}^{2}} + \frac{x}{y}\right) - \frac{1}{y}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\

\end{array}
double f(double x, double y) {
        double r381617 = 1.0;
        double r381618 = x;
        double r381619 = y;
        double r381620 = r381618 - r381619;
        double r381621 = r381617 - r381619;
        double r381622 = r381620 / r381621;
        double r381623 = r381617 - r381622;
        double r381624 = log(r381623);
        double r381625 = r381617 - r381624;
        return r381625;
}
↓
double f(double x, double y) {
        double r381626 = y;
        double r381627 = -1630045945.922767;
        bool r381628 = r381626 <= r381627;
        double r381629 = 76447033.33081697;
        bool r381630 = r381626 <= r381629;
        double r381631 = !r381630;
        bool r381632 = r381628 || r381631;
        double r381633 = 1.0;
        double r381634 = x;
        double r381635 = 2.0;
        double r381636 = pow(r381626, r381635);
        double r381637 = r381634 / r381636;
        double r381638 = r381634 / r381626;
        double r381639 = r381637 + r381638;
        double r381640 = 1.0;
        double r381641 = r381640 / r381626;
        double r381642 = r381639 - r381641;
        double r381643 = r381633 * r381642;
        double r381644 = log(r381643);
        double r381645 = r381633 - r381644;
        double r381646 = r381633 - r381626;
        double r381647 = cbrt(r381646);
        double r381648 = r381647 * r381647;
        double r381649 = r381640 / r381648;
        double r381650 = r381634 - r381626;
        double r381651 = r381650 / r381647;
        double r381652 = r381649 * r381651;
        double r381653 = r381633 - r381652;
        double r381654 = log(r381653);
        double r381655 = r381633 - r381654;
        double r381656 = r381632 ? r381645 : r381655;
        return r381656;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original17.8
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 y < -1630045945.922767 or 76447033.33081697 < y

    1. Initial program 46.1

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

    3. Applied add-cube-cbrt42.3

      \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}\right)\]

    4. Applied *-un-lft-identity42.3

      \[\leadsto 1 - \log \left(1 - \frac{\color{blue}{1 \cdot \left(x - y\right)}}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}\right)\]

    5. Applied times-frac42.2

      \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}}\right)\]
    6. Using strategy rm

    7. Applied flip3--49.3

      \[\leadsto 1 - \log \color{blue}{\left(\frac{{1}^{3} - {\left(\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)}^{3}}{1 \cdot 1 + \left(\left(\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right) \cdot \left(\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right) + 1 \cdot \left(\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\right)}\right)}\]

    8. Simplified49.3

      \[\leadsto 1 - \log \left(\frac{{1}^{3} - {\left(\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)}^{3}}{\color{blue}{\left(\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right) \cdot \left(\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}} + 1\right) + 1 \cdot 1}}\right)\]

    9. Taylor expanded around inf 0.1

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

    10. Simplified0.1

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

    if -1630045945.922767 < y < 76447033.33081697

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

      \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}\right)\]

    4. Applied *-un-lft-identity0.1

      \[\leadsto 1 - \log \left(1 - \frac{\color{blue}{1 \cdot \left(x - y\right)}}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}\right)\]

    5. Applied times-frac0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1630045945.92276692 \lor \neg \left(y \le 76447033.3308169693\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\left(\frac{x}{{y}^{2}} + \frac{x}{y}\right) - \frac{1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))))))