Average Error: 18.5 → 0.1
Time: 30.1s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -34474611802046.8828125 \lor \neg \left(y \le 43744445.7007110416889190673828125\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\left(\left(-\frac{\mathsf{fma}\left(y, 1 + y, 1 \cdot 1\right)}{\frac{{1}^{3} - {y}^{3}}{x - y}}\right) + {\left(\sqrt[3]{1}\right)}^{3}\right) + \frac{x - y}{{1}^{3} - {y}^{3}} \cdot \left(\left(-\mathsf{fma}\left(y, 1 + y, 1 \cdot 1\right)\right) + \mathsf{fma}\left(y, 1 + y, 1 \cdot 1\right)\right)}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -34474611802046.8828125 \lor \neg \left(y \le 43744445.7007110416889190673828125\right):\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \left(\log \left(\sqrt{\left(\left(-\frac{\mathsf{fma}\left(y, 1 + y, 1 \cdot 1\right)}{\frac{{1}^{3} - {y}^{3}}{x - y}}\right) + {\left(\sqrt[3]{1}\right)}^{3}\right) + \frac{x - y}{{1}^{3} - {y}^{3}} \cdot \left(\left(-\mathsf{fma}\left(y, 1 + y, 1 \cdot 1\right)\right) + \mathsf{fma}\left(y, 1 + y, 1 \cdot 1\right)\right)}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r277291 = 1.0;
        double r277292 = x;
        double r277293 = y;
        double r277294 = r277292 - r277293;
        double r277295 = r277291 - r277293;
        double r277296 = r277294 / r277295;
        double r277297 = r277291 - r277296;
        double r277298 = log(r277297);
        double r277299 = r277291 - r277298;
        return r277299;
}
↓
double f(double x, double y) {
        double r277300 = y;
        double r277301 = -34474611802046.883;
        bool r277302 = r277300 <= r277301;
        double r277303 = 43744445.70071104;
        bool r277304 = r277300 <= r277303;
        double r277305 = !r277304;
        bool r277306 = r277302 || r277305;
        double r277307 = 1.0;
        double r277308 = r277307 / r277300;
        double r277309 = x;
        double r277310 = r277309 / r277300;
        double r277311 = r277310 - r277308;
        double r277312 = fma(r277308, r277310, r277311);
        double r277313 = log(r277312);
        double r277314 = r277307 - r277313;
        double r277315 = r277307 + r277300;
        double r277316 = r277307 * r277307;
        double r277317 = fma(r277300, r277315, r277316);
        double r277318 = 3.0;
        double r277319 = pow(r277307, r277318);
        double r277320 = pow(r277300, r277318);
        double r277321 = r277319 - r277320;
        double r277322 = r277309 - r277300;
        double r277323 = r277321 / r277322;
        double r277324 = r277317 / r277323;
        double r277325 = -r277324;
        double r277326 = cbrt(r277307);
        double r277327 = pow(r277326, r277318);
        double r277328 = r277325 + r277327;
        double r277329 = r277322 / r277321;
        double r277330 = -r277317;
        double r277331 = r277330 + r277317;
        double r277332 = r277329 * r277331;
        double r277333 = r277328 + r277332;
        double r277334 = sqrt(r277333);
        double r277335 = log(r277334);
        double r277336 = r277307 - r277300;
        double r277337 = r277322 / r277336;
        double r277338 = r277307 - r277337;
        double r277339 = sqrt(r277338);
        double r277340 = log(r277339);
        double r277341 = r277335 + r277340;
        double r277342 = r277307 - r277341;
        double r277343 = r277306 ? r277314 : r277342;
        return r277343;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.5
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -34474611802046.883 or 43744445.70071104 < y

    1. Initial program 47.4

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

    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

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

    if -34474611802046.883 < y < 43744445.70071104

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt0.2

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

    4. Applied log-prod0.2

      \[\leadsto 1 - \color{blue}{\left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)}\]
    5. Using strategy rm

    6. Applied flip3--0.2

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

    7. Applied associate-/r/0.2

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

    8. Applied add-cube-cbrt0.2

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

    9. Applied prod-diff0.2

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

    10. Simplified0.2

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

    11. Simplified0.2

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

    13. Applied fma-udef0.2

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

    14. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -34474611802046.8828125 \lor \neg \left(y \le 43744445.7007110416889190673828125\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\left(\left(-\frac{\mathsf{fma}\left(y, 1 + y, 1 \cdot 1\right)}{\frac{{1}^{3} - {y}^{3}}{x - y}}\right) + {\left(\sqrt[3]{1}\right)}^{3}\right) + \frac{x - y}{{1}^{3} - {y}^{3}} \cdot \left(\left(-\mathsf{fma}\left(y, 1 + y, 1 \cdot 1\right)\right) + \mathsf{fma}\left(y, 1 + y, 1 \cdot 1\right)\right)}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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))))))