Average Error: 18.2 → 0.3
Time: 17.4s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 - \frac{x - y}{1 - y} \le 4.669598041573408409021794795989990234375 \cdot 10^{-13}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;1 - \frac{x - y}{1 - y} \le 4.669598041573408409021794795989990234375 \cdot 10^{-13}:\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r311033 = 1.0;
        double r311034 = x;
        double r311035 = y;
        double r311036 = r311034 - r311035;
        double r311037 = r311033 - r311035;
        double r311038 = r311036 / r311037;
        double r311039 = r311033 - r311038;
        double r311040 = log(r311039);
        double r311041 = r311033 - r311040;
        return r311041;
}
↓
double f(double x, double y) {
        double r311042 = 1.0;
        double r311043 = x;
        double r311044 = y;
        double r311045 = r311043 - r311044;
        double r311046 = r311042 - r311044;
        double r311047 = r311045 / r311046;
        double r311048 = r311042 - r311047;
        double r311049 = 4.669598041573408e-13;
        bool r311050 = r311048 <= r311049;
        double r311051 = 2.0;
        double r311052 = pow(r311044, r311051);
        double r311053 = r311043 / r311052;
        double r311054 = r311043 / r311044;
        double r311055 = fma(r311042, r311053, r311054);
        double r311056 = r311042 / r311044;
        double r311057 = r311055 - r311056;
        double r311058 = log(r311057);
        double r311059 = r311042 - r311058;
        double r311060 = sqrt(r311048);
        double r311061 = log(r311060);
        double r311062 = r311061 + r311061;
        double r311063 = r311042 - r311062;
        double r311064 = r311050 ? r311059 : r311063;
        return r311064;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.2
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- 1.0 (/ (- x y) (- 1.0 y))) < 4.669598041573408e-13

    1. Initial program 63.6

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if 4.669598041573408e-13 < (- 1.0 (/ (- x y) (- 1.0 y)))

    1. Initial program 0.3

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

    3. Applied add-sqr-sqrt0.4

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

    4. Applied log-prod0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{x - y}{1 - y} \le 4.669598041573408409021794795989990234375 \cdot 10^{-13}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))))))