Average Error: 18.3 → 0.1
Time: 4.2s
Precision: binary64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -93755730.34262806 \lor \neg \left(y \leq 85028056.792489\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\frac{x}{y} + 1 \cdot \frac{\frac{x}{y} + -1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 + \left(y + 1\right) \cdot \frac{y - x}{1 \cdot 1 - y \cdot y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \leq -93755730.34262806 \lor \neg \left(y \leq 85028056.792489\right):\\
\;\;\;\;\log \left(\frac{e^{1}}{\frac{x}{y} + 1 \cdot \frac{\frac{x}{y} + -1}{y}}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(1 + \left(y + 1\right) \cdot \frac{y - x}{1 \cdot 1 - y \cdot 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 <= -93755730.34262806) || !(y <= 85028056.792489))) {
		VAR = ((double) log((((double) exp(1.0)) / ((double) ((x / y) + ((double) (1.0 * (((double) ((x / y) + -1.0)) / y))))))));
	} else {
		VAR = ((double) (1.0 - ((double) log(((double) (1.0 + ((double) (((double) (y + 1.0)) * (((double) (y - x)) / ((double) (((double) (1.0 * 1.0)) - ((double) (y * 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

Original18.3
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -93755730.342628062 or 85028056.7924890071 < y

    1. Initial program Error: 47.0 bits

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

    2. Taylor expanded around inf Error: 0.1 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: 0.1 bits

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

    5. Applied add-log-expError: 0.1 bits

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

    6. Applied diff-logError: 0.2 bits

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

    7. SimplifiedError: 0.2 bits

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

    if -93755730.342628062 < y < 85028056.7924890071

    1. Initial program Error: 0.1 bits

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

    3. Applied flip--Error: 0.1 bits

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

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

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

  4. Final simplificationError: 0.1 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -93755730.34262806 \lor \neg \left(y \leq 85028056.792489\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\frac{x}{y} + 1 \cdot \frac{\frac{x}{y} + -1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 + \left(y + 1\right) \cdot \frac{y - x}{1 \cdot 1 - y \cdot y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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))))))