Average Error: 18.1 → 0.2
Time: 6.1s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.999825418925996412:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(1 \cdot \frac{x}{{y}^{2}} + \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.999825418925996412:\\
\;\;\;\;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(\left(1 \cdot \frac{x}{{y}^{2}} + \frac{x}{y}\right) - \frac{1}{y}\right)\\

\end{array}
double f(double x, double y) {
        double r357294 = 1.0;
        double r357295 = x;
        double r357296 = y;
        double r357297 = r357295 - r357296;
        double r357298 = r357294 - r357296;
        double r357299 = r357297 / r357298;
        double r357300 = r357294 - r357299;
        double r357301 = log(r357300);
        double r357302 = r357294 - r357301;
        return r357302;
}
↓
double f(double x, double y) {
        double r357303 = x;
        double r357304 = y;
        double r357305 = r357303 - r357304;
        double r357306 = 1.0;
        double r357307 = r357306 - r357304;
        double r357308 = r357305 / r357307;
        double r357309 = 0.9998254189259964;
        bool r357310 = r357308 <= r357309;
        double r357311 = 1.0;
        double r357312 = cbrt(r357307);
        double r357313 = r357312 * r357312;
        double r357314 = r357311 / r357313;
        double r357315 = r357305 / r357312;
        double r357316 = r357314 * r357315;
        double r357317 = r357306 - r357316;
        double r357318 = log(r357317);
        double r357319 = r357306 - r357318;
        double r357320 = 2.0;
        double r357321 = pow(r357304, r357320);
        double r357322 = r357303 / r357321;
        double r357323 = r357306 * r357322;
        double r357324 = r357303 / r357304;
        double r357325 = r357323 + r357324;
        double r357326 = r357306 / r357304;
        double r357327 = r357325 - r357326;
        double r357328 = log(r357327);
        double r357329 = r357306 - r357328;
        double r357330 = r357310 ? r357319 : r357329;
        return r357330;
}

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
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9998254189259964

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied *-un-lft-identity0.1

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

    5. Applied times-frac0.1

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

    if 0.9998254189259964 < (/ (- x y) (- 1.0 y))

    1. Initial program 61.9

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

    3. Applied add-cube-cbrt56.8

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

    4. Applied *-un-lft-identity56.8

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

    5. Applied times-frac56.7

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

    6. Taylor expanded around inf 0.5

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

    7. Simplified0.5

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.999825418925996412:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(1 \cdot \frac{x}{{y}^{2}} + \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(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))))))