Average Error: 18.4 → 0.2
Time: 54.8s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -152049567.74437964 \lor \neg \left(y \le 8942476.00878707133\right):\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(1 - \log \left(1 - \frac{x - y}{1 - y}\right)\right)}^{3}}\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -152049567.74437964 \lor \neg \left(y \le 8942476.00878707133\right):\\
\;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\

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

\end{array}
double f(double x, double y) {
        double r643581 = 1.0;
        double r643582 = x;
        double r643583 = y;
        double r643584 = r643582 - r643583;
        double r643585 = r643581 - r643583;
        double r643586 = r643584 / r643585;
        double r643587 = r643581 - r643586;
        double r643588 = log(r643587);
        double r643589 = r643581 - r643588;
        return r643589;
}
↓
double f(double x, double y) {
        double r643590 = y;
        double r643591 = -152049567.74437964;
        bool r643592 = r643590 <= r643591;
        double r643593 = 8942476.008787071;
        bool r643594 = r643590 <= r643593;
        double r643595 = !r643594;
        bool r643596 = r643592 || r643595;
        double r643597 = 1.0;
        double r643598 = x;
        double r643599 = r643598 / r643590;
        double r643600 = 2.0;
        double r643601 = pow(r643590, r643600);
        double r643602 = r643598 / r643601;
        double r643603 = r643597 * r643602;
        double r643604 = r643599 + r643603;
        double r643605 = 1.0;
        double r643606 = r643605 / r643590;
        double r643607 = r643597 * r643606;
        double r643608 = r643604 - r643607;
        double r643609 = log(r643608);
        double r643610 = r643597 - r643609;
        double r643611 = r643598 - r643590;
        double r643612 = r643597 - r643590;
        double r643613 = r643611 / r643612;
        double r643614 = r643597 - r643613;
        double r643615 = log(r643614);
        double r643616 = r643597 - r643615;
        double r643617 = 3.0;
        double r643618 = pow(r643616, r643617);
        double r643619 = cbrt(r643618);
        double r643620 = r643596 ? r643610 : r643619;
        return r643620;
}

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
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -152049567.74437964 or 8942476.008787071 < y

    1. Initial program 46.8

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

    3. Applied add-cube-cbrt43.0

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

    4. Applied associate-/r*43.0

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

    5. Simplified42.9

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

    6. Taylor expanded around inf 0.2

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

    if -152049567.74437964 < y < 8942476.008787071

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.3

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

    4. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -152049567.74437964 \lor \neg \left(y \le 8942476.00878707133\right):\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(1 - \log \left(1 - \frac{x - y}{1 - y}\right)\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(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))))))