Average Error: 18.5 → 0.0
Time: 9.4s
Precision: binary64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\log \left(\frac{e}{1 - x} - \frac{e \cdot y}{1 - x}\right)\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\log \left(\frac{e}{1 - x} - \frac{e \cdot y}{1 - x}\right)
(FPCore (x y) :precision binary64 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))
↓
(FPCore (x y)
 :precision binary64
 (log (- (/ E (- 1.0 x)) (/ (* E y) (- 1.0 x)))))
double code(double x, double y) {
	return 1.0 - log(1.0 - ((x - y) / (1.0 - y)));
}
↓
double code(double x, double y) {
	return log((((double) M_E) / (1.0 - x)) - ((((double) M_E) * y) / (1.0 - x)));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.5
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. Initial program 18.5

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

  3. Applied add-log-exp_binary6418.5

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

  4. Applied diff-log_binary6418.5

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

  5. Simplified18.5

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

  6. Taylor expanded around 0 0.0

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

  7. Final simplification0.0

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

Alternatives

Reproduce

herbie shell --seed 2021118 
(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))))))