Average Error: 18.3 → 0.4
Time: 31.5s
Precision: binary64
Cost: 27649
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.025059855742236136:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \leq 0.025059855742236136:\\
\;\;\;\;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}:\\
\;\;\;\;1 - \log \left(\frac{x + -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 (<= (/ (- x y) (- 1.0 y)) 0.025059855742236136)
   (-
    1.0
    (log
     (-
      1.0
      (*
       (/ 1.0 (* (cbrt (- 1.0 y)) (cbrt (- 1.0 y))))
       (/ (- x y) (cbrt (- 1.0 y)))))))
   (- 1.0 (log (/ (+ x -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 (((x - y) / (1.0 - y)) <= 0.025059855742236136) {
		tmp = 1.0 - log(1.0 - ((1.0 / (cbrt(1.0 - y) * cbrt(1.0 - y))) * ((x - y) / cbrt(1.0 - y))));
	} else {
		tmp = 1.0 - log((x + -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

Original18.3
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Alternatives

Alternative 1
Error53.9
Cost52672
\[1 - \log \left(1 - \frac{\sqrt{y} + \sqrt{x}}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{\sqrt{x} - \sqrt{y}}{\sqrt[3]{1 - y}}\right)\]
Alternative 2
Error46.3
Cost39872
\[1 - \log \left(1 - \frac{\sqrt{x - y}}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{\sqrt{x - y}}{\sqrt[3]{1 - y}}\right)\]
Alternative 3
Error41.4
Cost39872
\[1 - \log \left(1 - \frac{\sqrt[3]{x - y}}{1 - \sqrt{y}} \cdot \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{1 + \sqrt{y}}\right)\]
Alternative 4
Error20.7
Cost39872
\[1 - \log \left(1 - \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{\sqrt{1 - y}} \cdot \frac{\sqrt[3]{x - y}}{\sqrt{1 - y}}\right)\]
Alternative 5
Error54.0
Cost32960
\[1 - \log \left(1 - \left(\sqrt{y} + \sqrt{x}\right) \cdot \frac{\sqrt{x} - \sqrt{y}}{1 - y}\right)\]
Alternative 6
Error18.4
Cost27072
\[1 - \left(\log \left(\sqrt[3]{1 - \frac{x - y}{1 - y}}\right) + 2 \cdot \log \left(\sqrt[3]{1 - \frac{x - y}{1 - y}}\right)\right)\]
Alternative 7
Error16.8
Cost26944
\[1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\]
Alternative 8
Error17.3
Cost26816
\[1 - \log \left(1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}\right)\]
Alternative 9
Error16.9
Cost26816
\[1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\]
Alternative 10
Error17.4
Cost26816
\[1 - \log \left(1 - \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{\frac{1 - y}{\sqrt[3]{x - y}}}\right)\]
Alternative 11
Error29.9
Cost21248
\[1 - \left(\log \left(1 - {\left(\frac{x - y}{1 - y}\right)}^{3}\right) - \log \left(1 + \frac{x - y}{1 - y} \cdot \left(1 + \frac{x - y}{1 - y}\right)\right)\right)\]
Alternative 12
Error36.2
Cost20864
\[1 - \left(\log \left(1 - \frac{x - y}{\frac{{\left(1 - y\right)}^{2}}{x - y}}\right) - \log \left(1 + \frac{x - y}{1 - y}\right)\right)\]
Alternative 13
Error20.8
Cost20288
\[1 - \log \left(1 - \frac{x - y}{\sqrt{1 - y}} \cdot \frac{1}{\sqrt{1 - y}}\right)\]
Alternative 14
Error20.9
Cost20160
\[1 - \log \left(1 + \frac{\frac{y - x}{\sqrt{1 - y}}}{\sqrt{1 - y}}\right)\]
Alternative 15
Error46.8
Cost20160
\[1 - \log \left(1 - \sqrt{x - y} \cdot \frac{\sqrt{x - y}}{1 - y}\right)\]
Alternative 16
Error41.5
Cost20160
\[1 - \log \left(1 - \frac{\frac{x - y}{1 + \sqrt{y}}}{1 - \sqrt{y}}\right)\]
Alternative 17
Error46.9
Cost20160
\[1 - \log \left(1 - \frac{\sqrt{x - y}}{\frac{1 - y}{\sqrt{x - y}}}\right)\]
Alternative 18
Error29.8
Cost19968
\[1 - \log \left(\sqrt[3]{{\left(1 - \frac{x - y}{1 - y}\right)}^{3}}\right)\]
Alternative 19
Error18.4
Cost19968
\[\sqrt[3]{{\left(1 - \log \left(1 - \frac{x - y}{1 - y}\right)\right)}^{3}}\]
Alternative 20
Error37.4
Cost19904
\[1 - \log \log \left(e^{1 - \frac{x - y}{1 - y}}\right)\]
Alternative 21
Error37.4
Cost19904
\[1 - \log \left(1 - \log \left(e^{\frac{x - y}{1 - y}}\right)\right)\]
Alternative 22
Error26.5
Cost14464
\[1 - \log \left(\frac{1 - \frac{x - y}{\frac{{\left(1 - y\right)}^{2}}{x - y}}}{1 + \frac{x - y}{1 - y}}\right)\]
Alternative 23
Error20.8
Cost14080
\[1 - \log \left(1 + \frac{y - x}{1 - {y}^{3}} \cdot \left(1 + \left(y + y \cdot y\right)\right)\right)\]
Alternative 24
Error43.6
Cost13376
\[1 - \left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right)\]
Alternative 25
Error41.8
Cost13376
\[1 + \left(\log \left(1 - y\right) + \log \left(\frac{-1}{x}\right)\right)\]
Alternative 26
Error57.9
Cost13376
\[1 - \left(\log \left(\frac{-1}{1 - y}\right) + \log x\right)\]
Alternative 27
Error57.9
Cost13248
\[1 - \left(\log \left(x + -1\right) - \log y\right)\]
Alternative 28
Error37.6
Cost7872
\[1 - \left(\log \left(1 + \frac{y}{1 - y}\right) - \frac{x}{\left(1 - y\right) \cdot \left(1 + \frac{y}{1 - y}\right)}\right)\]
Alternative 29
Error18.1
Cost7488
\[1 - \log \left(1 - \frac{x - y}{1 - y \cdot y} \cdot \left(1 + y\right)\right)\]
Alternative 30
Error18.3
Cost7360
\[1 - \log \left(\frac{y}{1 - y} + \left(1 - \frac{x}{1 - y}\right)\right)\]
Alternative 31
Error17.9
Cost7232
\[1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\]
Alternative 32
Error18.3
Cost7104
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
Alternative 33
Error18.3
Cost7104
\[\log \left(\frac{e}{1 - \frac{x - y}{1 - y}}\right)\]
Alternative 34
Error38.4
Cost6976
\[1 - \log \left(1 + \frac{y}{1 - y}\right)\]
Alternative 35
Error36.1
Cost6912
\[1 - \log \left(-\frac{x}{1 - y}\right)\]
Alternative 36
Error36.1
Cost6912
\[\log \left(\frac{e}{-\frac{x}{1 - y}}\right)\]
Alternative 37
Error37.9
Cost6848
\[1 - \log \left(\frac{x + -1}{y}\right)\]
Alternative 38
Error25.2
Cost6848
\[1 - \left(y + \log \left(1 - x\right)\right)\]
Alternative 39
Error24.0
Cost6720
\[\log \left(\frac{e}{1 - x}\right)\]
Alternative 40
Error24.0
Cost6720
\[1 - \log \left(1 - x\right)\]
Alternative 41
Error36.6
Cost64
\[1\]
Alternative 42
Error62.0
Cost64
\[0\]
Alternative 43
Error60.6
Cost64
\[-1\]

Error

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.0250598557422361361

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt_binary64_80980.0

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

    4. Applied *-un-lft-identity_binary64_80630.0

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

    5. Applied times-frac_binary64_80690.0

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

    6. Simplified0.0

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

    if 0.0250598557422361361 < (/.f64 (-.f64 x y) (-.f64 1 y))

    1. Initial program 60.4

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

    2. Taylor expanded around inf 1.3

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

    3. Simplified1.3

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

    4. Simplified1.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.025059855742236136:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(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))))))