Average Error: 18.3 → 0.1
Time: 25.5s
Precision: 64
\[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.9999999625628327:\\ \;\;\;\;\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1.0}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]
1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.9999999625628327:\\
\;\;\;\;\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)\\

\mathbf{else}:\\
\;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1.0}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r15361885 = 1.0;
        double r15361886 = x;
        double r15361887 = y;
        double r15361888 = r15361886 - r15361887;
        double r15361889 = r15361885 - r15361887;
        double r15361890 = r15361888 / r15361889;
        double r15361891 = r15361885 - r15361890;
        double r15361892 = log(r15361891);
        double r15361893 = r15361885 - r15361892;
        return r15361893;
}
↓
double f(double x, double y) {
        double r15361894 = x;
        double r15361895 = y;
        double r15361896 = r15361894 - r15361895;
        double r15361897 = 1.0;
        double r15361898 = r15361897 - r15361895;
        double r15361899 = r15361896 / r15361898;
        double r15361900 = 0.9999999625628327;
        bool r15361901 = r15361899 <= r15361900;
        double r15361902 = exp(r15361897);
        double r15361903 = r15361897 - r15361899;
        double r15361904 = r15361902 / r15361903;
        double r15361905 = log(r15361904);
        double r15361906 = r15361894 / r15361895;
        double r15361907 = r15361897 / r15361895;
        double r15361908 = r15361906 - r15361907;
        double r15361909 = fma(r15361906, r15361907, r15361908);
        double r15361910 = log(r15361909);
        double r15361911 = r15361897 - r15361910;
        double r15361912 = r15361901 ? r15361905 : r15361911;
        return r15361912;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.3
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.61947241:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9999999625628327

    1. Initial program 0.1

      \[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
    2. Using strategy rm

    3. Applied add-log-exp0.1

      \[\leadsto \color{blue}{\log \left(e^{1.0}\right)} - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]

    4. Applied diff-log0.1

      \[\leadsto \color{blue}{\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)}\]

    if 0.9999999625628327 < (/ (- x y) (- 1.0 y))

    1. Initial program 60.9

      \[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]

    2. Taylor expanded around inf 0.1

      \[\leadsto 1.0 - \log \color{blue}{\left(\left(\frac{x}{y} + 1.0 \cdot \frac{x}{{y}^{2}}\right) - 1.0 \cdot \frac{1}{y}\right)}\]

    3. Simplified0.1

      \[\leadsto 1.0 - \log \color{blue}{\left(\mathsf{fma}\left(\frac{x}{y}, \frac{1.0}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.9999999625628327:\\ \;\;\;\;\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1.0}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(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))))))