Average Error: 18.2 → 0.1
Time: 13.4s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -51178879.934393361 \lor \neg \left(y \le 53143023.8704946265\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(1 - \frac{{\left(\sqrt[3]{x - y}\right)}^{3}}{1 - y}\right) + \left(\frac{{\left(\sqrt[3]{x - y}\right)}^{3}}{1 - y} - \frac{{\left(\sqrt[3]{x - y}\right)}^{3}}{1 - y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -51178879.934393361 \lor \neg \left(y \le 53143023.8704946265\right):\\
\;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r351523 = 1.0;
        double r351524 = x;
        double r351525 = y;
        double r351526 = r351524 - r351525;
        double r351527 = r351523 - r351525;
        double r351528 = r351526 / r351527;
        double r351529 = r351523 - r351528;
        double r351530 = log(r351529);
        double r351531 = r351523 - r351530;
        return r351531;
}
↓
double f(double x, double y) {
        double r351532 = y;
        double r351533 = -51178879.93439336;
        bool r351534 = r351532 <= r351533;
        double r351535 = 53143023.87049463;
        bool r351536 = r351532 <= r351535;
        double r351537 = !r351536;
        bool r351538 = r351534 || r351537;
        double r351539 = 1.0;
        double r351540 = exp(r351539);
        double r351541 = x;
        double r351542 = 2.0;
        double r351543 = pow(r351532, r351542);
        double r351544 = r351541 / r351543;
        double r351545 = r351541 / r351532;
        double r351546 = fma(r351544, r351539, r351545);
        double r351547 = r351539 / r351532;
        double r351548 = r351546 - r351547;
        double r351549 = r351540 / r351548;
        double r351550 = log(r351549);
        double r351551 = r351541 - r351532;
        double r351552 = cbrt(r351551);
        double r351553 = 3.0;
        double r351554 = pow(r351552, r351553);
        double r351555 = r351539 - r351532;
        double r351556 = r351554 / r351555;
        double r351557 = r351539 - r351556;
        double r351558 = r351556 - r351556;
        double r351559 = r351557 + r351558;
        double r351560 = log(r351559);
        double r351561 = r351539 - r351560;
        double r351562 = r351538 ? r351550 : r351561;
        return r351562;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.2
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -51178879.93439336 or 53143023.87049463 < y

    1. Initial program 47.0

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

    3. Applied add-log-exp47.0

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

    4. Applied diff-log47.0

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

    5. Taylor expanded around inf 0.2

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

    6. Simplified0.2

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

    if -51178879.93439336 < y < 53143023.87049463

    1. Initial program 0.1

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

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

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied times-frac0.1

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

    6. Applied add-sqr-sqrt0.1

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

    7. Applied prod-diff0.1

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

    8. Simplified0.1

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

    9. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -51178879.934393361 \lor \neg \left(y \le 53143023.8704946265\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(1 - \frac{{\left(\sqrt[3]{x - y}\right)}^{3}}{1 - y}\right) + \left(\frac{{\left(\sqrt[3]{x - y}\right)}^{3}}{1 - y} - \frac{{\left(\sqrt[3]{x - y}\right)}^{3}}{1 - y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))))