Average Error: 18.6 → 0.2
Time: 5.6s
Precision: binary64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -54595547.19675808 \lor \neg \left(y \leq 5.362376023934354 \cdot 10^{+51}\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \frac{1}{y} \cdot \left(\frac{x}{y} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 + \frac{\frac{y - x}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \leq -54595547.19675808 \lor \neg \left(y \leq 5.362376023934354 \cdot 10^{+51}\right):\\
\;\;\;\;1 - \log \left(\frac{x}{y} + \frac{1}{y} \cdot \left(\frac{x}{y} + -1\right)\right)\\

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

\end{array}
double code(double x, double y) {
	return ((double) (1.0 - ((double) log(((double) (1.0 - (((double) (x - y)) / ((double) (1.0 - y)))))))));
}
↓
double code(double x, double y) {
	double VAR;
	if (((y <= -54595547.19675808) || !(y <= 5.362376023934354e+51))) {
		VAR = ((double) (1.0 - ((double) log(((double) ((x / y) + ((double) ((1.0 / y) * ((double) ((x / y) + -1.0))))))))));
	} else {
		VAR = ((double) (1.0 - ((double) log(((double) (1.0 + ((((double) (y - x)) / ((double) (((double) cbrt(((double) (1.0 - y)))) * ((double) cbrt(((double) (1.0 - y))))))) / ((double) cbrt(((double) (1.0 - y)))))))))));
	}
	return VAR;
}

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
\[\begin{array}{l} \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -54595547.196758077 or 5.3623760239343538e51 < y

    1. Initial program 49.2

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

    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

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

    if -54595547.196758077 < y < 5.3623760239343538e51

    1. Initial program 0.2

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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -54595547.19675808 \lor \neg \left(y \leq 5.362376023934354 \cdot 10^{+51}\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \frac{1}{y} \cdot \left(\frac{x}{y} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 + \frac{\frac{y - x}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :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))))))