Average Error: 22.7 → 8.0
Time: 4.2s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -31862854206393840 \lor \neg \left(y \le 1.22155351051059022 \cdot 10^{62}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\sqrt[3]{y \cdot y - 1 \cdot 1}}{\sqrt[3]{y - 1}} \cdot \sqrt[3]{y + 1}} \cdot \frac{y}{\sqrt[3]{y + 1}}, x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -31862854206393840 \lor \neg \left(y \le 1.22155351051059022 \cdot 10^{62}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r754884 = 1.0;
        double r754885 = x;
        double r754886 = r754884 - r754885;
        double r754887 = y;
        double r754888 = r754886 * r754887;
        double r754889 = r754887 + r754884;
        double r754890 = r754888 / r754889;
        double r754891 = r754884 - r754890;
        return r754891;
}
↓
double f(double x, double y) {
        double r754892 = y;
        double r754893 = -3.186285420639384e+16;
        bool r754894 = r754892 <= r754893;
        double r754895 = 1.2215535105105902e+62;
        bool r754896 = r754892 <= r754895;
        double r754897 = !r754896;
        bool r754898 = r754894 || r754897;
        double r754899 = x;
        double r754900 = r754899 / r754892;
        double r754901 = 1.0;
        double r754902 = r754901 / r754892;
        double r754903 = r754902 - r754901;
        double r754904 = fma(r754900, r754903, r754899);
        double r754905 = 1.0;
        double r754906 = r754892 * r754892;
        double r754907 = r754901 * r754901;
        double r754908 = r754906 - r754907;
        double r754909 = cbrt(r754908);
        double r754910 = r754892 - r754901;
        double r754911 = cbrt(r754910);
        double r754912 = r754909 / r754911;
        double r754913 = r754892 + r754901;
        double r754914 = cbrt(r754913);
        double r754915 = r754912 * r754914;
        double r754916 = r754905 / r754915;
        double r754917 = r754892 / r754914;
        double r754918 = r754916 * r754917;
        double r754919 = r754899 - r754901;
        double r754920 = fma(r754918, r754919, r754901);
        double r754921 = r754898 ? r754904 : r754920;
        return r754921;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.7
Target0.2
Herbie8.0
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.186285420639384e+16 or 1.2215535105105902e+62 < y

    1. Initial program 47.6

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

    2. Simplified29.3

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

    3. Taylor expanded around inf 14.6

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

    4. Simplified14.6

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

    if -3.186285420639384e+16 < y < 1.2215535105105902e+62

    1. Initial program 2.9

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

    2. Simplified2.6

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

    4. Applied add-cube-cbrt2.7

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

    5. Applied *-un-lft-identity2.7

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

    6. Applied times-frac2.7

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

    8. Applied flip-+2.7

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

    9. Applied cbrt-div2.7

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

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -31862854206393840 \lor \neg \left(y \le 1.22155351051059022 \cdot 10^{62}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\sqrt[3]{y \cdot y - 1 \cdot 1}}{\sqrt[3]{y - 1}} \cdot \sqrt[3]{y + 1}} \cdot \frac{y}{\sqrt[3]{y + 1}}, x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))