Average Error: 18.7 → 0.1
Time: 6.2s
Precision: binary64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -48019698.52435404:\\ \;\;\;\;\log \left(\frac{e^{1}}{\left(\frac{x}{y \cdot y} + \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \mathbf{elif}\;y \leq 43088487.23901788:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{\left(\frac{x}{y \cdot y} + \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \leq -48019698.52435404:\\
\;\;\;\;\log \left(\frac{e^{1}}{\left(\frac{x}{y \cdot y} + \frac{x}{y}\right) - \frac{1}{y}}\right)\\

\mathbf{elif}\;y \leq 43088487.23901788:\\
\;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\frac{e^{1}}{\left(\frac{x}{y \cdot y} + \frac{x}{y}\right) - \frac{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 (<= y -48019698.52435404)
   (log (/ (exp 1.0) (- (+ (/ x (* y y)) (/ x y)) (/ 1.0 y))))
   (if (<= y 43088487.23901788)
     (-
      1.0
      (log
       (-
        1.0
        (*
         (/ 1.0 (* (cbrt (- 1.0 y)) (cbrt (- 1.0 y))))
         (/ (- x y) (cbrt (- 1.0 y)))))))
     (log (/ (exp 1.0) (- (+ (/ x (* y y)) (/ 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 <= -48019698.52435404) {
		tmp = log(exp(1.0) / (((x / (y * y)) + (x / y)) - (1.0 / y)));
	} else if (y <= 43088487.23901788) {
		tmp = 1.0 - log(1.0 - ((1.0 / (cbrt(1.0 - y) * cbrt(1.0 - y))) * ((x - y) / cbrt(1.0 - y))));
	} else {
		tmp = log(exp(1.0) / (((x / (y * y)) + (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

Original18.7
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 < -48019698.524354041 or 43088487.2390178815 < y

    1. Initial program 46.7

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

    3. Applied add-log-exp_binary6446.7

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

    4. Applied diff-log_binary6446.7

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

    5. Taylor expanded around inf 0.2

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

    6. Simplified0.2

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

    if -48019698.524354041 < y < 43088487.2390178815

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt_binary640.1

      \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}\right)\]

    4. Applied *-un-lft-identity_binary640.1

      \[\leadsto 1 - \log \left(1 - \frac{\color{blue}{1 \cdot \left(x - y\right)}}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}\right)\]

    5. Applied times-frac_binary640.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -48019698.52435404:\\ \;\;\;\;\log \left(\frac{e^{1}}{\left(\frac{x}{y \cdot y} + \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \mathbf{elif}\;y \leq 43088487.23901788:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{\left(\frac{x}{y \cdot y} + \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020273 
(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))))))