Average Error: 17.6 → 0.2
Time: 24.0s
Precision: 64
\[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -55233269.11289919:\\ \;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \mathbf{elif}\;y \le 51693858.328214146:\\ \;\;\;\;1.0 - \log \left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0}, \frac{-\sqrt[3]{x - y}}{1.0 - y} \cdot \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0}, \frac{-\sqrt[3]{x - y}}{1.0 - y} \cdot \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0}, \frac{-\sqrt[3]{x - y}}{1.0 - y} \cdot \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right)\right)}\right) + \mathsf{fma}\left(\frac{-\sqrt[3]{x - y}}{1.0 - y}, \sqrt[3]{x - y} \cdot \sqrt[3]{x - y}, \frac{\sqrt[3]{x - y}}{1.0 - y} \cdot \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]
1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -55233269.11289919:\\
\;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\

\mathbf{elif}\;y \le 51693858.328214146:\\
\;\;\;\;1.0 - \log \left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0}, \frac{-\sqrt[3]{x - y}}{1.0 - y} \cdot \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0}, \frac{-\sqrt[3]{x - y}}{1.0 - y} \cdot \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0}, \frac{-\sqrt[3]{x - y}}{1.0 - y} \cdot \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right)\right)}\right) + \mathsf{fma}\left(\frac{-\sqrt[3]{x - y}}{1.0 - y}, \sqrt[3]{x - y} \cdot \sqrt[3]{x - y}, \frac{\sqrt[3]{x - y}}{1.0 - y} \cdot \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right)\right)\right)\\

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

\end{array}
double f(double x, double y) {
        double r18158260 = 1.0;
        double r18158261 = x;
        double r18158262 = y;
        double r18158263 = r18158261 - r18158262;
        double r18158264 = r18158260 - r18158262;
        double r18158265 = r18158263 / r18158264;
        double r18158266 = r18158260 - r18158265;
        double r18158267 = log(r18158266);
        double r18158268 = r18158260 - r18158267;
        return r18158268;
}
↓
double f(double x, double y) {
        double r18158269 = y;
        double r18158270 = -55233269.11289919;
        bool r18158271 = r18158269 <= r18158270;
        double r18158272 = 1.0;
        double r18158273 = r18158272 / r18158269;
        double r18158274 = x;
        double r18158275 = r18158274 / r18158269;
        double r18158276 = r18158275 - r18158273;
        double r18158277 = fma(r18158273, r18158275, r18158276);
        double r18158278 = log(r18158277);
        double r18158279 = r18158272 - r18158278;
        double r18158280 = 51693858.328214146;
        bool r18158281 = r18158269 <= r18158280;
        double r18158282 = cbrt(r18158272);
        double r18158283 = r18158282 * r18158282;
        double r18158284 = r18158274 - r18158269;
        double r18158285 = cbrt(r18158284);
        double r18158286 = -r18158285;
        double r18158287 = r18158272 - r18158269;
        double r18158288 = r18158286 / r18158287;
        double r18158289 = r18158285 * r18158285;
        double r18158290 = r18158288 * r18158289;
        double r18158291 = fma(r18158283, r18158282, r18158290);
        double r18158292 = cbrt(r18158291);
        double r18158293 = r18158292 * r18158292;
        double r18158294 = r18158292 * r18158293;
        double r18158295 = r18158285 / r18158287;
        double r18158296 = r18158295 * r18158289;
        double r18158297 = fma(r18158288, r18158289, r18158296);
        double r18158298 = r18158294 + r18158297;
        double r18158299 = log(r18158298);
        double r18158300 = r18158272 - r18158299;
        double r18158301 = r18158281 ? r18158300 : r18158279;
        double r18158302 = r18158271 ? r18158279 : r18158301;
        return r18158302;
}

Error

Bits error versus x

Bits error versus y

Target

Original17.6
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 y < -55233269.11289919 or 51693858.328214146 < y

    1. Initial program 46.1

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

      \[\leadsto 1.0 - \log \color{blue}{\left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)}\]

    if -55233269.11289919 < y < 51693858.328214146

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied add-cube-cbrt0.2

      \[\leadsto 1.0 - \log \left(1.0 - \frac{\color{blue}{\left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \sqrt[3]{x - y}}}{1 \cdot \left(1.0 - y\right)}\right)\]

    5. Applied times-frac0.2

      \[\leadsto 1.0 - \log \left(1.0 - \color{blue}{\frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{1} \cdot \frac{\sqrt[3]{x - y}}{1.0 - y}}\right)\]

    6. Applied add-cube-cbrt0.2

      \[\leadsto 1.0 - \log \left(\color{blue}{\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}\right) \cdot \sqrt[3]{1.0}} - \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{1} \cdot \frac{\sqrt[3]{x - y}}{1.0 - y}\right)\]

    7. Applied prod-diff0.2

      \[\leadsto 1.0 - \log \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0}, -\frac{\sqrt[3]{x - y}}{1.0 - y} \cdot \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{1}\right) + \mathsf{fma}\left(-\frac{\sqrt[3]{x - y}}{1.0 - y}, \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{1}, \frac{\sqrt[3]{x - y}}{1.0 - y} \cdot \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{1}\right)\right)}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt0.2

      \[\leadsto 1.0 - \log \left(\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0}, -\frac{\sqrt[3]{x - y}}{1.0 - y} \cdot \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{1}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0}, -\frac{\sqrt[3]{x - y}}{1.0 - y} \cdot \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{1}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0}, -\frac{\sqrt[3]{x - y}}{1.0 - y} \cdot \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{1}\right)}} + \mathsf{fma}\left(-\frac{\sqrt[3]{x - y}}{1.0 - y}, \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{1}, \frac{\sqrt[3]{x - y}}{1.0 - y} \cdot \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{1}\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -55233269.11289919:\\ \;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \mathbf{elif}\;y \le 51693858.328214146:\\ \;\;\;\;1.0 - \log \left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0}, \frac{-\sqrt[3]{x - y}}{1.0 - y} \cdot \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0}, \frac{-\sqrt[3]{x - y}}{1.0 - y} \cdot \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0}, \frac{-\sqrt[3]{x - y}}{1.0 - y} \cdot \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right)\right)}\right) + \mathsf{fma}\left(\frac{-\sqrt[3]{x - y}}{1.0 - y}, \sqrt[3]{x - y} \cdot \sqrt[3]{x - y}, \frac{\sqrt[3]{x - y}}{1.0 - y} \cdot \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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))))))