Average Error: 18.3 → 0.1
Time: 53.3s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -60638491.817888684570789337158203125:\\ \;\;\;\;1 - \log \left(\frac{x}{y} \cdot \frac{1}{y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \le 4239884631250423709580859711422464:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y} \cdot \frac{1}{y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -60638491.817888684570789337158203125:\\
\;\;\;\;1 - \log \left(\frac{x}{y} \cdot \frac{1}{y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\

\mathbf{elif}\;y \le 4239884631250423709580859711422464:\\
\;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r23955940 = 1.0;
        double r23955941 = x;
        double r23955942 = y;
        double r23955943 = r23955941 - r23955942;
        double r23955944 = r23955940 - r23955942;
        double r23955945 = r23955943 / r23955944;
        double r23955946 = r23955940 - r23955945;
        double r23955947 = log(r23955946);
        double r23955948 = r23955940 - r23955947;
        return r23955948;
}
↓
double f(double x, double y) {
        double r23955949 = y;
        double r23955950 = -60638491.817888685;
        bool r23955951 = r23955949 <= r23955950;
        double r23955952 = 1.0;
        double r23955953 = x;
        double r23955954 = r23955953 / r23955949;
        double r23955955 = r23955952 / r23955949;
        double r23955956 = r23955954 * r23955955;
        double r23955957 = r23955955 - r23955954;
        double r23955958 = r23955956 - r23955957;
        double r23955959 = log(r23955958);
        double r23955960 = r23955952 - r23955959;
        double r23955961 = 4.2398846312504237e+33;
        bool r23955962 = r23955949 <= r23955961;
        double r23955963 = r23955953 - r23955949;
        double r23955964 = r23955952 - r23955949;
        double r23955965 = cbrt(r23955964);
        double r23955966 = r23955965 * r23955965;
        double r23955967 = r23955963 / r23955966;
        double r23955968 = r23955967 / r23955965;
        double r23955969 = r23955952 - r23955968;
        double r23955970 = log(r23955969);
        double r23955971 = r23955952 - r23955970;
        double r23955972 = r23955962 ? r23955971 : r23955960;
        double r23955973 = r23955951 ? r23955960 : r23955972;
        return r23955973;
}

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
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -60638491.817888685 or 4.2398846312504237e+33 < y

    1. Initial program 47.4

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

    3. Applied add-cube-cbrt43.7

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

    4. Applied associate-/r*43.7

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

    5. Taylor expanded around inf 0.1

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

    6. Simplified0.1

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

    if -60638491.817888685 < y < 4.2398846312504237e+33

    1. Initial program 0.1

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
    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 associate-/r*0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -60638491.817888684570789337158203125:\\ \;\;\;\;1 - \log \left(\frac{x}{y} \cdot \frac{1}{y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \le 4239884631250423709580859711422464:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y} \cdot \frac{1}{y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"

  :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))))))