Average Error: 17.7 → 0.3
Time: 5.5s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.40788806408471906:\\ \;\;\;\;\log \left(e^{1 - \log \left(1 - \frac{x - y}{1 - y}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.40788806408471906:\\
\;\;\;\;\log \left(e^{1 - \log \left(1 - \frac{x - y}{1 - y}\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)}\right)\\

\end{array}
double code(double x, double y) {
	return (1.0 - log((1.0 - ((x - y) / (1.0 - y)))));
}
↓
double code(double x, double y) {
	double temp;
	if ((((x - y) / (1.0 - y)) <= 0.40788806408471906)) {
		temp = log(exp((1.0 - log((1.0 - ((x - y) / (1.0 - y)))))));
	} else {
		temp = log(exp((1.0 - log(fma(1.0, ((x / pow(y, 2.0)) - (1.0 / y)), (x / y))))));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original17.7
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.40788806408471906

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    4. Applied diff-log0.0

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

    6. Applied add-exp-log0.0

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

    7. Applied div-exp0.0

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

    if 0.40788806408471906 < (/ (- x y) (- 1.0 y))

    1. Initial program 61.3

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

    3. Applied add-log-exp61.3

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

    4. Applied diff-log61.3

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

    6. Applied add-exp-log61.3

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

    7. Applied div-exp61.3

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

    8. Taylor expanded around inf 0.8

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

    9. Simplified0.8

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.40788806408471906:\\ \;\;\;\;\log \left(e^{1 - \log \left(1 - \frac{x - y}{1 - y}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

  (- 1 (log (- 1 (/ (- x y) (- 1 y))))))