Average Error: 18.8 → 0.2
Time: 1.2m
Precision: 64
\[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.9998974867815034:\\ \;\;\;\;1.0 - \log \left(1.0 - \frac{\frac{x - y}{\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}}}{\sqrt[3]{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1.0}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]
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 0.9998974867815034:\\
\;\;\;\;1.0 - \log \left(1.0 - \frac{\frac{x - y}{\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}}}{\sqrt[3]{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 r22090803 = 1.0;
        double r22090804 = x;
        double r22090805 = y;
        double r22090806 = r22090804 - r22090805;
        double r22090807 = r22090803 - r22090805;
        double r22090808 = r22090806 / r22090807;
        double r22090809 = r22090803 - r22090808;
        double r22090810 = log(r22090809);
        double r22090811 = r22090803 - r22090810;
        return r22090811;
}
↓
double f(double x, double y) {
        double r22090812 = x;
        double r22090813 = y;
        double r22090814 = r22090812 - r22090813;
        double r22090815 = 1.0;
        double r22090816 = r22090815 - r22090813;
        double r22090817 = r22090814 / r22090816;
        double r22090818 = 0.9998974867815034;
        bool r22090819 = r22090817 <= r22090818;
        double r22090820 = cbrt(r22090816);
        double r22090821 = r22090820 * r22090820;
        double r22090822 = r22090814 / r22090821;
        double r22090823 = r22090822 / r22090820;
        double r22090824 = r22090815 - r22090823;
        double r22090825 = log(r22090824);
        double r22090826 = r22090815 - r22090825;
        double r22090827 = r22090812 / r22090813;
        double r22090828 = r22090815 / r22090813;
        double r22090829 = r22090827 - r22090828;
        double r22090830 = fma(r22090827, r22090828, r22090829);
        double r22090831 = log(r22090830);
        double r22090832 = r22090815 - r22090831;
        double r22090833 = r22090819 ? r22090826 : r22090832;
        return r22090833;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.8
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.61947241:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9998974867815034

    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-cube-cbrt0.1

      \[\leadsto 1.0 - \log \left(1.0 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}\right) \cdot \sqrt[3]{1.0 - y}}}\right)\]

    4. Applied associate-/r*0.1

      \[\leadsto 1.0 - \log \left(1.0 - \color{blue}{\frac{\frac{x - y}{\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}}}{\sqrt[3]{1.0 - y}}}\right)\]

    if 0.9998974867815034 < (/ (- x y) (- 1.0 y))

    1. Initial program 61.9

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.9998974867815034:\\ \;\;\;\;1.0 - \log \left(1.0 - \frac{\frac{x - y}{\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}}}{\sqrt[3]{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1.0}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +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))))))