Average Error: 22.5 → 7.7
Time: 16.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -42972738058162806784:\\ \;\;\;\;x - \left(\frac{x \cdot 1}{y} - \frac{\frac{x \cdot 1}{y}}{y}\right)\\ \mathbf{elif}\;y \le 2375613545737851462610892407037814112256:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{\frac{\sqrt[3]{y \cdot y - 1 \cdot 1} \cdot \sqrt[3]{y + 1}}{\sqrt[3]{y - 1}}}}{\sqrt[3]{y + 1}}, x - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{x \cdot 1}{y} - \frac{\frac{x \cdot 1}{y}}{y}\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -42972738058162806784:\\
\;\;\;\;x - \left(\frac{x \cdot 1}{y} - \frac{\frac{x \cdot 1}{y}}{y}\right)\\

\mathbf{elif}\;y \le 2375613545737851462610892407037814112256:\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{\frac{\sqrt[3]{y \cdot y - 1 \cdot 1} \cdot \sqrt[3]{y + 1}}{\sqrt[3]{y - 1}}}}{\sqrt[3]{y + 1}}, x - 1, 1\right)\\

\mathbf{else}:\\
\;\;\;\;x - \left(\frac{x \cdot 1}{y} - \frac{\frac{x \cdot 1}{y}}{y}\right)\\

\end{array}
double f(double x, double y) {
        double r27082045 = 1.0;
        double r27082046 = x;
        double r27082047 = r27082045 - r27082046;
        double r27082048 = y;
        double r27082049 = r27082047 * r27082048;
        double r27082050 = r27082048 + r27082045;
        double r27082051 = r27082049 / r27082050;
        double r27082052 = r27082045 - r27082051;
        return r27082052;
}
↓
double f(double x, double y) {
        double r27082053 = y;
        double r27082054 = -4.297273805816281e+19;
        bool r27082055 = r27082053 <= r27082054;
        double r27082056 = x;
        double r27082057 = 1.0;
        double r27082058 = r27082056 * r27082057;
        double r27082059 = r27082058 / r27082053;
        double r27082060 = r27082059 / r27082053;
        double r27082061 = r27082059 - r27082060;
        double r27082062 = r27082056 - r27082061;
        double r27082063 = 2.3756135457378515e+39;
        bool r27082064 = r27082053 <= r27082063;
        double r27082065 = r27082053 * r27082053;
        double r27082066 = r27082057 * r27082057;
        double r27082067 = r27082065 - r27082066;
        double r27082068 = cbrt(r27082067);
        double r27082069 = r27082053 + r27082057;
        double r27082070 = cbrt(r27082069);
        double r27082071 = r27082068 * r27082070;
        double r27082072 = r27082053 - r27082057;
        double r27082073 = cbrt(r27082072);
        double r27082074 = r27082071 / r27082073;
        double r27082075 = r27082053 / r27082074;
        double r27082076 = r27082075 / r27082070;
        double r27082077 = r27082056 - r27082057;
        double r27082078 = fma(r27082076, r27082077, r27082057);
        double r27082079 = r27082064 ? r27082078 : r27082062;
        double r27082080 = r27082055 ? r27082062 : r27082079;
        return r27082080;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.5
Target0.2
Herbie7.7
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.297273805816281e+19 or 2.3756135457378515e+39 < y

    1. Initial program 47.6

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

    2. Simplified29.7

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

    4. Applied add-cube-cbrt30.5

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

    5. Applied associate-/l*30.5

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

    6. Taylor expanded around inf 14.7

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

    7. Simplified14.7

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

    if -4.297273805816281e+19 < y < 2.3756135457378515e+39

    1. Initial program 1.9

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

    2. Simplified1.7

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

    4. Applied add-cube-cbrt1.8

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

    5. Applied associate-/r*1.8

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

    7. Applied flip-+1.8

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

    8. Applied cbrt-div1.8

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

    9. Applied associate-*l/1.8

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

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -42972738058162806784:\\ \;\;\;\;x - \left(\frac{x \cdot 1}{y} - \frac{\frac{x \cdot 1}{y}}{y}\right)\\ \mathbf{elif}\;y \le 2375613545737851462610892407037814112256:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{\frac{\sqrt[3]{y \cdot y - 1 \cdot 1} \cdot \sqrt[3]{y + 1}}{\sqrt[3]{y - 1}}}}{\sqrt[3]{y + 1}}, x - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{x \cdot 1}{y} - \frac{\frac{x \cdot 1}{y}}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))