Average Error: 18.2 → 0.3
Time: 17.7s
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 r265210 = 1.0;
        double r265211 = x;
        double r265212 = y;
        double r265213 = r265211 - r265212;
        double r265214 = r265210 - r265212;
        double r265215 = r265213 / r265214;
        double r265216 = r265210 - r265215;
        double r265217 = log(r265216);
        double r265218 = r265210 - r265217;
        return r265218;
}
↓
double f(double x, double y) {
        double r265219 = 1.0;
        double r265220 = x;
        double r265221 = y;
        double r265222 = r265220 - r265221;
        double r265223 = r265219 - r265221;
        double r265224 = r265222 / r265223;
        double r265225 = r265219 - r265224;
        double r265226 = 4.669598041573408e-13;
        bool r265227 = r265225 <= r265226;
        double r265228 = 2.0;
        double r265229 = pow(r265221, r265228);
        double r265230 = r265220 / r265229;
        double r265231 = r265220 / r265221;
        double r265232 = fma(r265219, r265230, r265231);
        double r265233 = r265219 / r265221;
        double r265234 = r265232 - r265233;
        double r265235 = log(r265234);
        double r265236 = r265219 - r265235;
        double r265237 = sqrt(r265225);
        double r265238 = log(r265237);
        double r265239 = r265238 + r265238;
        double r265240 = r265219 - r265239;
        double r265241 = r265227 ? r265236 : r265240;
        return r265241;
}

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))))))