Average Error: 18.2 → 0.2
Time: 20.7s
Precision: 64
\[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.8581617207853955:\\ \;\;\;\;1.0 - \left(\log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right) + \log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y}\right) - \frac{1.0}{y}\right)\\ \end{array}\]
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 0.8581617207853955:\\
\;\;\;\;1.0 - \left(\log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right) + \log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)\right)\\

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

\end{array}
double f(double x, double y) {
        double r15232132 = 1.0;
        double r15232133 = x;
        double r15232134 = y;
        double r15232135 = r15232133 - r15232134;
        double r15232136 = r15232132 - r15232134;
        double r15232137 = r15232135 / r15232136;
        double r15232138 = r15232132 - r15232137;
        double r15232139 = log(r15232138);
        double r15232140 = r15232132 - r15232139;
        return r15232140;
}
↓
double f(double x, double y) {
        double r15232141 = x;
        double r15232142 = y;
        double r15232143 = r15232141 - r15232142;
        double r15232144 = 1.0;
        double r15232145 = r15232144 - r15232142;
        double r15232146 = r15232143 / r15232145;
        double r15232147 = 0.8581617207853955;
        bool r15232148 = r15232146 <= r15232147;
        double r15232149 = r15232144 - r15232146;
        double r15232150 = sqrt(r15232149);
        double r15232151 = log(r15232150);
        double r15232152 = r15232151 + r15232151;
        double r15232153 = r15232144 - r15232152;
        double r15232154 = r15232144 / r15232142;
        double r15232155 = r15232141 / r15232142;
        double r15232156 = fma(r15232154, r15232155, r15232155);
        double r15232157 = r15232156 - r15232154;
        double r15232158 = log(r15232157);
        double r15232159 = r15232144 - r15232158;
        double r15232160 = r15232148 ? r15232153 : r15232159;
        return r15232160;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.2
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)) < 0.8581617207853955

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied log-prod0.0

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

    if 0.8581617207853955 < (/ (- x y) (- 1.0 y))

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.7

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.8581617207853955:\\ \;\;\;\;1.0 - \left(\log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right) + \log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y}\right) - \frac{1.0}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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))))))