Average Error: 18.5 → 0.1
Time: 15.8s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -34474611802046.8828125:\\ \;\;\;\;1 - \log \left(\left(\sqrt[3]{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}} \cdot \sqrt[3]{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}}\right) \cdot \sqrt[3]{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}}\right)\\ \mathbf{elif}\;y \le 43744445.7007110416889190673828125:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\sqrt{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}} \cdot \sqrt{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -34474611802046.8828125:\\
\;\;\;\;1 - \log \left(\left(\sqrt[3]{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}} \cdot \sqrt[3]{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}}\right) \cdot \sqrt[3]{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}}\right)\\

\mathbf{elif}\;y \le 43744445.7007110416889190673828125:\\
\;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\

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

\end{array}
double f(double x, double y) {
        double r19219399 = 1.0;
        double r19219400 = x;
        double r19219401 = y;
        double r19219402 = r19219400 - r19219401;
        double r19219403 = r19219399 - r19219401;
        double r19219404 = r19219402 / r19219403;
        double r19219405 = r19219399 - r19219404;
        double r19219406 = log(r19219405);
        double r19219407 = r19219399 - r19219406;
        return r19219407;
}
↓
double f(double x, double y) {
        double r19219408 = y;
        double r19219409 = -34474611802046.883;
        bool r19219410 = r19219408 <= r19219409;
        double r19219411 = 1.0;
        double r19219412 = x;
        double r19219413 = r19219412 / r19219408;
        double r19219414 = r19219411 * r19219412;
        double r19219415 = r19219408 * r19219408;
        double r19219416 = r19219414 / r19219415;
        double r19219417 = r19219413 + r19219416;
        double r19219418 = r19219411 / r19219408;
        double r19219419 = r19219417 - r19219418;
        double r19219420 = cbrt(r19219419);
        double r19219421 = r19219420 * r19219420;
        double r19219422 = r19219421 * r19219420;
        double r19219423 = log(r19219422);
        double r19219424 = r19219411 - r19219423;
        double r19219425 = 43744445.70071104;
        bool r19219426 = r19219408 <= r19219425;
        double r19219427 = r19219412 - r19219408;
        double r19219428 = r19219411 - r19219408;
        double r19219429 = r19219427 / r19219428;
        double r19219430 = r19219411 - r19219429;
        double r19219431 = sqrt(r19219430);
        double r19219432 = log(r19219431);
        double r19219433 = r19219432 + r19219432;
        double r19219434 = r19219411 - r19219433;
        double r19219435 = sqrt(r19219419);
        double r19219436 = r19219435 * r19219435;
        double r19219437 = log(r19219436);
        double r19219438 = r19219411 - r19219437;
        double r19219439 = r19219426 ? r19219434 : r19219438;
        double r19219440 = r19219410 ? r19219424 : r19219439;
        return r19219440;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.5
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -34474611802046.883

    1. Initial program 53.9

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    5. Applied add-cube-cbrt0.0

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

    if -34474611802046.883 < y < 43744445.70071104

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt0.2

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

    4. Applied log-prod0.2

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

    if 43744445.70071104 < y

    1. Initial program 29.8

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    5. Applied add-sqr-sqrt0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -34474611802046.8828125:\\ \;\;\;\;1 - \log \left(\left(\sqrt[3]{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}} \cdot \sqrt[3]{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}}\right) \cdot \sqrt[3]{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}}\right)\\ \mathbf{elif}\;y \le 43744445.7007110416889190673828125:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\sqrt{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}} \cdot \sqrt{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))