Average Error: 18.5 → 0.2
Time: 4.1s
Precision: binary64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -83317507269799.2 \lor \neg \left(y \leq 720987198.4659485\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \frac{1}{y} \cdot \left(\frac{x}{y} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 + \frac{y - x}{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \leq -83317507269799.2 \lor \neg \left(y \leq 720987198.4659485\right):\\
\;\;\;\;1 - \log \left(\frac{x}{y} + \frac{1}{y} \cdot \left(\frac{x}{y} + -1\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\frac{e^{1}}{1 + \frac{y - x}{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 -83317507269799.2) (not (<= y 720987198.4659485)))
   (- 1.0 (log (+ (/ x y) (* (/ 1.0 y) (+ (/ x y) -1.0)))))
   (log (/ (exp 1.0) (+ 1.0 (/ (- y x) (- 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 <= -83317507269799.2) || !(y <= 720987198.4659485))) {
		tmp = ((double) (1.0 - ((double) log(((double) ((x / y) + ((double) ((1.0 / y) * ((double) ((x / y) + -1.0))))))))));
	} else {
		tmp = ((double) log((((double) exp(1.0)) / ((double) (1.0 + (((double) (y - x)) / ((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

Original18.5
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 < -83317507269799.2031 or 720987198.465948462 < y

    1. Initial program Error: 47.4 bits

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

    2. Taylor expanded around inf Error: 0.0 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: 0.0 bits

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

    if -83317507269799.2031 < y < 720987198.465948462

    1. Initial program Error: 0.3 bits

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

    3. Applied add-log-expError: 0.3 bits

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

    4. Applied diff-logError: 0.3 bits

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

    5. SimplifiedError: 0.3 bits

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

  4. Final simplificationError: 0.2 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -83317507269799.2 \lor \neg \left(y \leq 720987198.4659485\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \frac{1}{y} \cdot \left(\frac{x}{y} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 + \frac{y - x}{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))))))