Average Error: 18.2 → 0.3
Time: 1.5m
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.09185737703913710028302830323809757828712:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, \frac{-1}{1 - y} \cdot \left(x - y\right)\right) + \mathsf{fma}\left(\frac{-1}{1 - y}, x - y, \frac{1}{1 - y} \cdot \left(x - y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.09185737703913710028302830323809757828712:\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, \frac{-1}{1 - y} \cdot \left(x - y\right)\right) + \mathsf{fma}\left(\frac{-1}{1 - y}, x - y, \frac{1}{1 - y} \cdot \left(x - y\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r16771246 = 1.0;
        double r16771247 = x;
        double r16771248 = y;
        double r16771249 = r16771247 - r16771248;
        double r16771250 = r16771246 - r16771248;
        double r16771251 = r16771249 / r16771250;
        double r16771252 = r16771246 - r16771251;
        double r16771253 = log(r16771252);
        double r16771254 = r16771246 - r16771253;
        return r16771254;
}
↓
double f(double x, double y) {
        double r16771255 = x;
        double r16771256 = y;
        double r16771257 = r16771255 - r16771256;
        double r16771258 = 1.0;
        double r16771259 = r16771258 - r16771256;
        double r16771260 = r16771257 / r16771259;
        double r16771261 = 0.0918573770391371;
        bool r16771262 = r16771260 <= r16771261;
        double r16771263 = cbrt(r16771258);
        double r16771264 = r16771263 * r16771263;
        double r16771265 = -1.0;
        double r16771266 = r16771265 / r16771259;
        double r16771267 = r16771266 * r16771257;
        double r16771268 = fma(r16771264, r16771263, r16771267);
        double r16771269 = 1.0;
        double r16771270 = r16771269 / r16771259;
        double r16771271 = r16771270 * r16771257;
        double r16771272 = fma(r16771266, r16771257, r16771271);
        double r16771273 = r16771268 + r16771272;
        double r16771274 = log(r16771273);
        double r16771275 = r16771258 - r16771274;
        double r16771276 = r16771258 / r16771256;
        double r16771277 = r16771255 / r16771256;
        double r16771278 = r16771277 - r16771276;
        double r16771279 = fma(r16771276, r16771277, r16771278);
        double r16771280 = log(r16771279);
        double r16771281 = r16771258 - r16771280;
        double r16771282 = r16771262 ? r16771275 : r16771281;
        return r16771282;
}

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 (/ (- x y) (- 1.0 y)) < 0.0918573770391371

    1. Initial program 0.0

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

    3. Applied div-inv0.0

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

    4. Applied add-cube-cbrt0.0

      \[\leadsto 1 - \log \left(\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}} - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\]

    5. Applied prod-diff0.0

      \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -\frac{1}{1 - y} \cdot \left(x - y\right)\right) + \mathsf{fma}\left(-\frac{1}{1 - y}, x - y, \frac{1}{1 - y} \cdot \left(x - y\right)\right)\right)}\]

    if 0.0918573770391371 < (/ (- x y) (- 1.0 y))

    1. Initial program 60.7

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

    2. Taylor expanded around inf 1.0

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

    3. Simplified1.0

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.09185737703913710028302830323809757828712:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, \frac{-1}{1 - y} \cdot \left(x - y\right)\right) + \mathsf{fma}\left(\frac{-1}{1 - y}, x - y, \frac{1}{1 - y} \cdot \left(x - y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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))))))