Average Error: 18.6 → 0.2
Time: 5.1s
Precision: binary64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -67922377.24733216 \lor \neg \left(y \leq 53148174.24022449\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \frac{1}{y} \cdot \left(\frac{x}{y} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 + \frac{\frac{y - x}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \leq -67922377.24733216 \lor \neg \left(y \leq 53148174.24022449\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}
(FPCore (x y) :precision binary64 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))
↓
(FPCore (x y)
 :precision binary64
 (if (or (<= y -67922377.24733216) (not (<= y 53148174.24022449)))
   (- 1.0 (log (+ (/ x y) (* (/ 1.0 y) (+ (/ x y) -1.0)))))
   (-
    1.0
    (log
     (+
      1.0
      (/
       (/ (- y x) (* (cbrt (- 1.0 y)) (cbrt (- 1.0 y))))
       (cbrt (- 1.0 y))))))))
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 tmp;
	if (((y <= -67922377.24733216) || !(y <= 53148174.24022449))) {
		tmp = ((double) (1.0 - ((double) log(((double) ((x / y) + ((double) ((1.0 / y) * ((double) ((x / y) + -1.0))))))))));
	} else {
		tmp = ((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 tmp;
}

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 < -67922377.247332156 or 53148174.2402244881 < y

    1. Initial program Error: 48.0 bits

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

    2. Taylor expanded around inf Error: 0.2 bits

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

    3. SimplifiedError: 0.2 bits

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

    if -67922377.247332156 < y < 53148174.2402244881

    1. Initial program Error: 0.1 bits

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

    3. Applied add-cube-cbrtError: 0.1 bits

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

    4. Applied associate-/r*Error: 0.1 bits

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

  4. Final simplificationError: 0.2 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -67922377.24733216 \lor \neg \left(y \leq 53148174.24022449\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \frac{1}{y} \cdot \left(\frac{x}{y} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 + \frac{\frac{y - x}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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))))))