Average Error: 18.6 → 0.2
Time: 10.1s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99999997937035734:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.99999997937035734:\\
\;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\

\end{array}
double f(double x, double y) {
        double r487410 = 1.0;
        double r487411 = x;
        double r487412 = y;
        double r487413 = r487411 - r487412;
        double r487414 = r487410 - r487412;
        double r487415 = r487413 / r487414;
        double r487416 = r487410 - r487415;
        double r487417 = log(r487416);
        double r487418 = r487410 - r487417;
        return r487418;
}
↓
double f(double x, double y) {
        double r487419 = x;
        double r487420 = y;
        double r487421 = r487419 - r487420;
        double r487422 = 1.0;
        double r487423 = r487422 - r487420;
        double r487424 = r487421 / r487423;
        double r487425 = 0.9999999793703573;
        bool r487426 = r487424 <= r487425;
        double r487427 = cbrt(r487423);
        double r487428 = r487427 * r487427;
        double r487429 = r487421 / r487428;
        double r487430 = r487429 / r487427;
        double r487431 = r487422 - r487430;
        double r487432 = log(r487431);
        double r487433 = r487422 - r487432;
        double r487434 = r487419 / r487420;
        double r487435 = 2.0;
        double r487436 = pow(r487420, r487435);
        double r487437 = r487419 / r487436;
        double r487438 = r487422 * r487437;
        double r487439 = r487434 + r487438;
        double r487440 = 1.0;
        double r487441 = r487440 / r487420;
        double r487442 = r487422 * r487441;
        double r487443 = r487439 - r487442;
        double r487444 = log(r487443);
        double r487445 = r487422 - r487444;
        double r487446 = r487426 ? r487433 : r487445;
        return r487446;
}

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

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9999999793703573

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.2

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

    4. Applied associate-/r*0.2

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

    if 0.9999999793703573 < (/ (- x y) (- 1.0 y))

    1. Initial program 62.9

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

    3. Applied flip3--62.9

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

    4. Simplified62.9

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

    5. 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. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99999997937035734:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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))))))