Average Error: 17.8 → 0.1
Time: 15.1s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -109794463.93071939051151275634765625 \lor \neg \left(y \le 24629234.9915754683315753936767578125\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -109794463.93071939051151275634765625 \lor \neg \left(y \le 24629234.9915754683315753936767578125\right):\\
\;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\

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

\end{array}
double f(double x, double y) {
        double r363927 = 1.0;
        double r363928 = x;
        double r363929 = y;
        double r363930 = r363928 - r363929;
        double r363931 = r363927 - r363929;
        double r363932 = r363930 / r363931;
        double r363933 = r363927 - r363932;
        double r363934 = log(r363933);
        double r363935 = r363927 - r363934;
        return r363935;
}
↓
double f(double x, double y) {
        double r363936 = y;
        double r363937 = -109794463.93071939;
        bool r363938 = r363936 <= r363937;
        double r363939 = 24629234.99157547;
        bool r363940 = r363936 <= r363939;
        double r363941 = !r363940;
        bool r363942 = r363938 || r363941;
        double r363943 = 1.0;
        double r363944 = x;
        double r363945 = 2.0;
        double r363946 = pow(r363936, r363945);
        double r363947 = r363944 / r363946;
        double r363948 = 1.0;
        double r363949 = r363948 / r363936;
        double r363950 = r363947 - r363949;
        double r363951 = r363943 * r363950;
        double r363952 = r363944 / r363936;
        double r363953 = r363951 + r363952;
        double r363954 = log(r363953);
        double r363955 = r363943 - r363954;
        double r363956 = r363944 - r363936;
        double r363957 = r363943 - r363936;
        double r363958 = cbrt(r363957);
        double r363959 = r363958 * r363958;
        double r363960 = r363956 / r363959;
        double r363961 = r363960 / r363958;
        double r363962 = r363943 - r363961;
        double r363963 = log(r363962);
        double r363964 = r363943 - r363963;
        double r363965 = r363942 ? r363955 : r363964;
        return r363965;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original17.8
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 < -109794463.93071939 or 24629234.99157547 < y

    1. Initial program 46.1

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

    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

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

    if -109794463.93071939 < y < 24629234.99157547

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -109794463.93071939051151275634765625 \lor \neg \left(y \le 24629234.9915754683315753936767578125\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(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))))))