Average Error: 18.1 → 0.4
Time: 8.2s
Precision: binary64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -1.9232412763397154 \cdot 10^{+20} \lor \neg \left(y \leq 39022702.812917\right):\\ \;\;\;\;\log \left(\frac{e}{\left(\frac{x}{y \cdot y} + \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(1 - \frac{x}{1 - y}\right) + \frac{y}{1 - y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \leq -1.9232412763397154 \cdot 10^{+20} \lor \neg \left(y \leq 39022702.812917\right):\\
\;\;\;\;\log \left(\frac{e}{\left(\frac{x}{y \cdot y} + \frac{x}{y}\right) - \frac{1}{y}}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(\left(1 - \frac{x}{1 - y}\right) + \frac{y}{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 -1.9232412763397154e+20) (not (<= y 39022702.812917)))
   (log (/ E (- (+ (/ x (* y y)) (/ x y)) (/ 1.0 y))))
   (- 1.0 (log (+ (- 1.0 (/ x (- 1.0 y))) (/ 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 <= -1.9232412763397154e+20) || !(y <= 39022702.812917)) {
		tmp = log(((double) M_E) / (((x / (y * y)) + (x / y)) - (1.0 / y)));
	} else {
		tmp = 1.0 - log((1.0 - (x / (1.0 - y))) + (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

Original18.1
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -192324127633971540000 or 39022702.8129170015 < y

    1. Initial program 46.8

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

    3. Applied add-log-exp_binary64_1273446.8

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

    4. Applied diff-log_binary64_1278746.8

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

    5. Simplified46.8

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

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

    if -192324127633971540000 < y < 39022702.8129170015

    1. Initial program 0.7

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

    3. Applied div-sub_binary64_127000.7

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

    4. Applied associate--r-_binary64_126340.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9232412763397154 \cdot 10^{+20} \lor \neg \left(y \leq 39022702.812917\right):\\ \;\;\;\;\log \left(\frac{e}{\left(\frac{x}{y \cdot y} + \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(1 - \frac{x}{1 - y}\right) + \frac{y}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020308 
(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))))))