Average Error: 18.2 → 0.4
Time: 7.0s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.083471465588124433:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.083471465588124433:\\
\;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{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 r349123 = 1.0;
        double r349124 = x;
        double r349125 = y;
        double r349126 = r349124 - r349125;
        double r349127 = r349123 - r349125;
        double r349128 = r349126 / r349127;
        double r349129 = r349123 - r349128;
        double r349130 = log(r349129);
        double r349131 = r349123 - r349130;
        return r349131;
}
↓
double f(double x, double y) {
        double r349132 = x;
        double r349133 = y;
        double r349134 = r349132 - r349133;
        double r349135 = 1.0;
        double r349136 = r349135 - r349133;
        double r349137 = r349134 / r349136;
        double r349138 = 0.08347146558812443;
        bool r349139 = r349137 <= r349138;
        double r349140 = 1.0;
        double r349141 = r349140 / r349136;
        double r349142 = r349134 * r349141;
        double r349143 = r349135 - r349142;
        double r349144 = log(r349143);
        double r349145 = r349135 - r349144;
        double r349146 = 2.0;
        double r349147 = pow(r349133, r349146);
        double r349148 = r349132 / r349147;
        double r349149 = r349140 / r349133;
        double r349150 = r349148 - r349149;
        double r349151 = r349132 / r349133;
        double r349152 = fma(r349135, r349150, r349151);
        double r349153 = log(r349152);
        double r349154 = r349135 - r349153;
        double r349155 = r349139 ? r349145 : r349154;
        return r349155;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.2
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.08347146558812443

    1. Initial program 0.0

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

    3. Applied div-inv0.0

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

    if 0.08347146558812443 < (/ (- x y) (- 1.0 y))

    1. Initial program 60.5

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

    2. Taylor expanded around inf 1.2

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

    3. Simplified1.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.083471465588124433:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))))