Average Error: 18.1 → 0.0
Time: 10.1s
Precision: binary64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999895614048552:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{y}{1 - y}\right) - \frac{x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{{y}^{2}} + \frac{x}{y}\right) - \left(\frac{1}{{y}^{2}} + \frac{1}{y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999895614048552:\\
\;\;\;\;1 - \log \left(\left(1 + \frac{y}{1 - y}\right) - \frac{x}{1 - y}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(\left(\frac{x}{{y}^{2}} + \frac{x}{y}\right) - \left(\frac{1}{{y}^{2}} + \frac{1}{y}\right)\right)\\

\end{array}
(FPCore (x y) :precision binary64 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.9999895614048552)
   (- 1.0 (log (- (+ 1.0 (/ y (- 1.0 y))) (/ x (- 1.0 y)))))
   (-
    1.0
    (log
     (- (+ (/ x (pow y 2.0)) (/ x y)) (+ (/ 1.0 (pow y 2.0)) (/ 1.0 y)))))))
double code(double x, double y) {
	return 1.0 - log(1.0 - ((x - y) / (1.0 - y)));
}
↓
double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9999895614048552) {
		tmp = 1.0 - log((1.0 + (y / (1.0 - y))) - (x / (1.0 - y)));
	} else {
		tmp = 1.0 - log(((x / pow(y, 2.0)) + (x / y)) - ((1.0 / pow(y, 2.0)) + (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

Original18.1
Target0.1
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.999989561404855176

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.0

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

    if 0.999989561404855176 < (/.f64 (-.f64 x y) (-.f64 1 y))

    1. Initial program 62.1

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

    2. Taylor expanded around inf 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999895614048552:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{y}{1 - y}\right) - \frac{x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{{y}^{2}} + \frac{x}{y}\right) - \left(\frac{1}{{y}^{2}} + \frac{1}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))))