Average Error: 18.4 → 0.1
Time: 4.6s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -73256728.0171860307 \lor \neg \left(y \le 73789365.95162582\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\left(\sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}} \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}\right) \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}}\right) + \log \left(\sqrt{1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -73256728.0171860307 \lor \neg \left(y \le 73789365.95162582\right):\\
\;\;\;\;1 - \log \left(1 \cdot \left(\left(\sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}} \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}\right) \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}\right) + \frac{x}{y}\right)\\

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

\end{array}
double f(double x, double y) {
        double r334596 = 1.0;
        double r334597 = x;
        double r334598 = y;
        double r334599 = r334597 - r334598;
        double r334600 = r334596 - r334598;
        double r334601 = r334599 / r334600;
        double r334602 = r334596 - r334601;
        double r334603 = log(r334602);
        double r334604 = r334596 - r334603;
        return r334604;
}
↓
double f(double x, double y) {
        double r334605 = y;
        double r334606 = -73256728.01718603;
        bool r334607 = r334605 <= r334606;
        double r334608 = 73789365.95162582;
        bool r334609 = r334605 <= r334608;
        double r334610 = !r334609;
        bool r334611 = r334607 || r334610;
        double r334612 = 1.0;
        double r334613 = x;
        double r334614 = 2.0;
        double r334615 = pow(r334605, r334614);
        double r334616 = r334613 / r334615;
        double r334617 = 1.0;
        double r334618 = r334617 / r334605;
        double r334619 = r334616 - r334618;
        double r334620 = cbrt(r334619);
        double r334621 = r334620 * r334620;
        double r334622 = r334621 * r334620;
        double r334623 = r334612 * r334622;
        double r334624 = r334613 / r334605;
        double r334625 = r334623 + r334624;
        double r334626 = log(r334625);
        double r334627 = r334612 - r334626;
        double r334628 = r334613 - r334605;
        double r334629 = cbrt(r334628);
        double r334630 = r334629 * r334629;
        double r334631 = r334612 - r334605;
        double r334632 = r334629 / r334631;
        double r334633 = r334630 * r334632;
        double r334634 = r334612 - r334633;
        double r334635 = sqrt(r334634);
        double r334636 = log(r334635);
        double r334637 = r334636 + r334636;
        double r334638 = r334612 - r334637;
        double r334639 = r334611 ? r334627 : r334638;
        return r334639;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.4
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 < -73256728.01718603 or 73789365.95162582 < y

    1. Initial program 46.8

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

    5. Applied add-cube-cbrt0.2

      \[\leadsto 1 - \log \left(1 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}} \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}\right) \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}\right)} + \frac{x}{y}\right)\]

    if -73256728.01718603 < y < 73789365.95162582

    1. Initial program 0.1

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

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

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied times-frac0.1

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

    6. Simplified0.1

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

    8. Applied add-sqr-sqrt0.1

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

    9. Applied log-prod0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -73256728.0171860307 \lor \neg \left(y \le 73789365.95162582\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\left(\sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}} \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}\right) \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}}\right) + \log \left(\sqrt{1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))))))