Average Error: 18.0 → 0.2
Time: 5.2s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9993340856260311:\\ \;\;\;\;1 - \log \left(1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.9993340856260311:\\
\;\;\;\;1 - \log \left(1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r315903 = 1.0;
        double r315904 = x;
        double r315905 = y;
        double r315906 = r315904 - r315905;
        double r315907 = r315903 - r315905;
        double r315908 = r315906 / r315907;
        double r315909 = r315903 - r315908;
        double r315910 = log(r315909);
        double r315911 = r315903 - r315910;
        return r315911;
}
↓
double f(double x, double y) {
        double r315912 = x;
        double r315913 = y;
        double r315914 = r315912 - r315913;
        double r315915 = 1.0;
        double r315916 = r315915 - r315913;
        double r315917 = r315914 / r315916;
        double r315918 = 0.9993340856260311;
        bool r315919 = r315917 <= r315918;
        double r315920 = cbrt(r315914);
        double r315921 = r315920 * r315920;
        double r315922 = r315920 / r315916;
        double r315923 = r315921 * r315922;
        double r315924 = r315915 - r315923;
        double r315925 = log(r315924);
        double r315926 = r315915 - r315925;
        double r315927 = 2.0;
        double r315928 = pow(r315913, r315927);
        double r315929 = r315912 / r315928;
        double r315930 = 1.0;
        double r315931 = r315930 / r315913;
        double r315932 = r315929 - r315931;
        double r315933 = r315912 / r315913;
        double r315934 = fma(r315915, r315932, r315933);
        double r315935 = log(r315934);
        double r315936 = r315915 - r315935;
        double r315937 = r315919 ? r315926 : r315936;
        return r315937;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.0
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)) < 0.9993340856260311

    1. Initial program 0.0

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

    3. Applied *-un-lft-identity0.0

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied times-frac0.1

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

    6. Simplified0.1

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

    if 0.9993340856260311 < (/ (- x y) (- 1.0 y))

    1. Initial program 61.9

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9993340856260311:\\ \;\;\;\;1 - \log \left(1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(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))))))