Average Error: 18.7 → 0.1
Time: 6.9s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -118579514.584518134593963623046875 \lor \neg \left(y \le 32110906.8722298182547092437744140625\right):\\ \;\;\;\;1 - \log \left(\left(\sqrt[3]{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}} \cdot \sqrt[3]{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right) \cdot \sqrt[3]{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -118579514.584518134593963623046875 \lor \neg \left(y \le 32110906.8722298182547092437744140625\right):\\
\;\;\;\;1 - \log \left(\left(\sqrt[3]{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}} \cdot \sqrt[3]{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right) \cdot \sqrt[3]{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\

\end{array}
double f(double x, double y) {
        double r387949 = 1.0;
        double r387950 = x;
        double r387951 = y;
        double r387952 = r387950 - r387951;
        double r387953 = r387949 - r387951;
        double r387954 = r387952 / r387953;
        double r387955 = r387949 - r387954;
        double r387956 = log(r387955);
        double r387957 = r387949 - r387956;
        return r387957;
}
↓
double f(double x, double y) {
        double r387958 = y;
        double r387959 = -118579514.58451813;
        bool r387960 = r387958 <= r387959;
        double r387961 = 32110906.87222982;
        bool r387962 = r387958 <= r387961;
        double r387963 = !r387962;
        bool r387964 = r387960 || r387963;
        double r387965 = 1.0;
        double r387966 = x;
        double r387967 = 2.0;
        double r387968 = pow(r387958, r387967);
        double r387969 = r387966 / r387968;
        double r387970 = 1.0;
        double r387971 = r387970 / r387958;
        double r387972 = r387969 - r387971;
        double r387973 = r387965 * r387972;
        double r387974 = r387966 / r387958;
        double r387975 = r387973 + r387974;
        double r387976 = cbrt(r387975);
        double r387977 = r387976 * r387976;
        double r387978 = r387977 * r387976;
        double r387979 = log(r387978);
        double r387980 = r387965 - r387979;
        double r387981 = r387966 - r387958;
        double r387982 = r387965 - r387958;
        double r387983 = r387970 / r387982;
        double r387984 = r387981 * r387983;
        double r387985 = r387965 - r387984;
        double r387986 = log(r387985);
        double r387987 = r387965 - r387986;
        double r387988 = r387964 ? r387980 : r387987;
        return r387988;
}

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
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -118579514.58451813 or 32110906.87222982 < y

    1. Initial program 47.0

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

    5. Applied add-cube-cbrt0.1

      \[\leadsto 1 - \log \color{blue}{\left(\left(\sqrt[3]{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}} \cdot \sqrt[3]{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right) \cdot \sqrt[3]{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)}\]

    if -118579514.58451813 < y < 32110906.87222982

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -118579514.584518134593963623046875 \lor \neg \left(y \le 32110906.8722298182547092437744140625\right):\\ \;\;\;\;1 - \log \left(\left(\sqrt[3]{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}} \cdot \sqrt[3]{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right) \cdot \sqrt[3]{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.619472414) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

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