Average Error: 18.7 → 0.1
Time: 13.8s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -339938991.337108433 \lor \neg \left(y \le 19835852.986995392\right):\\ \;\;\;\;\log \left(\frac{\sqrt{e^{1}}}{\left(\frac{\sqrt{e^{-1}} \cdot x}{y} + \left(1 \cdot \sqrt{e^{-1}}\right) \cdot \frac{x}{{y}^{2}}\right) - \frac{1 \cdot \sqrt{e^{-1}}}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\sqrt{e^{1}}}{\frac{1 - \frac{x - y}{1 - y}}{\sqrt{e^{1}}}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -339938991.337108433 \lor \neg \left(y \le 19835852.986995392\right):\\
\;\;\;\;\log \left(\frac{\sqrt{e^{1}}}{\left(\frac{\sqrt{e^{-1}} \cdot x}{y} + \left(1 \cdot \sqrt{e^{-1}}\right) \cdot \frac{x}{{y}^{2}}\right) - \frac{1 \cdot \sqrt{e^{-1}}}{y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r408581 = 1.0;
        double r408582 = x;
        double r408583 = y;
        double r408584 = r408582 - r408583;
        double r408585 = r408581 - r408583;
        double r408586 = r408584 / r408585;
        double r408587 = r408581 - r408586;
        double r408588 = log(r408587);
        double r408589 = r408581 - r408588;
        return r408589;
}
↓
double f(double x, double y) {
        double r408590 = y;
        double r408591 = -339938991.33710843;
        bool r408592 = r408590 <= r408591;
        double r408593 = 19835852.98699539;
        bool r408594 = r408590 <= r408593;
        double r408595 = !r408594;
        bool r408596 = r408592 || r408595;
        double r408597 = 1.0;
        double r408598 = exp(r408597);
        double r408599 = sqrt(r408598);
        double r408600 = -r408597;
        double r408601 = exp(r408600);
        double r408602 = sqrt(r408601);
        double r408603 = x;
        double r408604 = r408602 * r408603;
        double r408605 = r408604 / r408590;
        double r408606 = r408597 * r408602;
        double r408607 = 2.0;
        double r408608 = pow(r408590, r408607);
        double r408609 = r408603 / r408608;
        double r408610 = r408606 * r408609;
        double r408611 = r408605 + r408610;
        double r408612 = r408606 / r408590;
        double r408613 = r408611 - r408612;
        double r408614 = r408599 / r408613;
        double r408615 = log(r408614);
        double r408616 = r408603 - r408590;
        double r408617 = r408597 - r408590;
        double r408618 = r408616 / r408617;
        double r408619 = r408597 - r408618;
        double r408620 = r408619 / r408599;
        double r408621 = r408599 / r408620;
        double r408622 = log(r408621);
        double r408623 = r408596 ? r408615 : r408622;
        return r408623;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.7
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 < -339938991.33710843 or 19835852.98699539 < y

    1. Initial program 47.2

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

    3. Applied add-log-exp47.2

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

    4. Applied diff-log47.2

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

    6. Applied add-sqr-sqrt47.2

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

    7. Applied associate-/l*47.2

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

    8. Taylor expanded around inf 0.1

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

    9. Simplified0.1

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

    if -339938991.33710843 < y < 19835852.98699539

    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

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

    6. Applied add-sqr-sqrt0.1

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

    7. Applied associate-/l*0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -339938991.337108433 \lor \neg \left(y \le 19835852.986995392\right):\\ \;\;\;\;\log \left(\frac{\sqrt{e^{1}}}{\left(\frac{\sqrt{e^{-1}} \cdot x}{y} + \left(1 \cdot \sqrt{e^{-1}}\right) \cdot \frac{x}{{y}^{2}}\right) - \frac{1 \cdot \sqrt{e^{-1}}}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\sqrt{e^{1}}}{\frac{1 - \frac{x - y}{1 - y}}{\sqrt{e^{1}}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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))))))