Average Error: 18.4 → 0.1
Time: 12.3s
Precision: binary64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -156594998.93521395 \lor \neg \left(y \leq 137886941383.1554\right):\\ \;\;\;\;1 - \log \left(\frac{-1}{y} + \left(\frac{x}{y \cdot y} + \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \leq -156594998.93521395 \lor \neg \left(y \leq 137886941383.1554\right):\\
\;\;\;\;1 - \log \left(\frac{-1}{y} + \left(\frac{x}{y \cdot y} + \frac{x}{y}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{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 (or (<= y -156594998.93521395) (not (<= y 137886941383.1554)))
   (- 1.0 (log (+ (/ -1.0 y) (+ (/ x (* y y)) (/ x y)))))
   (-
    1.0
    (+
     (log (sqrt (- 1.0 (/ (- x y) (- 1.0 y)))))
     (log (sqrt (- 1.0 (/ (- x y) (- 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 ((y <= -156594998.93521395) || !(y <= 137886941383.1554)) {
		tmp = 1.0 - log((-1.0 / y) + ((x / (y * y)) + (x / y)));
	} else {
		tmp = 1.0 - (log(sqrt(1.0 - ((x - y) / (1.0 - y)))) + log(sqrt(1.0 - ((x - y) / (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

Original18.4
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 < -156594998.935213953 or 137886941383.1554 < y

    1. Initial program 47.8

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

    if -156594998.935213953 < y < 137886941383.1554

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt_binary64_123760.1

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

    4. Applied log-prod_binary64_124400.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -156594998.93521395 \lor \neg \left(y \leq 137886941383.1554\right):\\ \;\;\;\;1 - \log \left(\frac{-1}{y} + \left(\frac{x}{y \cdot y} + \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(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))))))