Average Error: 22.7 → 0.2
Time: 4.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -102786700.59358019 \lor \neg \left(y \le 138811322.44829124\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y + 1}{y}} - \left(\frac{1}{\frac{y + 1}{y}} - 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -102786700.59358019 \lor \neg \left(y \le 138811322.44829124\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r692854 = 1.0;
        double r692855 = x;
        double r692856 = r692854 - r692855;
        double r692857 = y;
        double r692858 = r692856 * r692857;
        double r692859 = r692857 + r692854;
        double r692860 = r692858 / r692859;
        double r692861 = r692854 - r692860;
        return r692861;
}
↓
double f(double x, double y) {
        double r692862 = y;
        double r692863 = -102786700.59358019;
        bool r692864 = r692862 <= r692863;
        double r692865 = 138811322.44829124;
        bool r692866 = r692862 <= r692865;
        double r692867 = !r692866;
        bool r692868 = r692864 || r692867;
        double r692869 = 1.0;
        double r692870 = 1.0;
        double r692871 = r692870 / r692862;
        double r692872 = x;
        double r692873 = r692872 / r692862;
        double r692874 = r692871 - r692873;
        double r692875 = fma(r692869, r692874, r692872);
        double r692876 = r692862 + r692869;
        double r692877 = r692876 / r692862;
        double r692878 = r692872 / r692877;
        double r692879 = r692869 / r692877;
        double r692880 = r692879 - r692869;
        double r692881 = r692878 - r692880;
        double r692882 = r692868 ? r692875 : r692881;
        return r692882;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.7
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -102786700.59358019 or 138811322.44829124 < y

    1. Initial program 46.2

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

    2. Simplified29.2

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

    4. Applied *-un-lft-identity29.2

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

    5. Applied add-cube-cbrt30.0

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

    6. Applied times-frac30.0

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

    7. Simplified30.0

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

    9. Applied fma-udef30.0

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

    10. Simplified29.3

      \[\leadsto \color{blue}{\frac{x - 1}{\frac{y + 1}{y}}} + 1\]

    11. Taylor expanded around inf 0.2

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

    12. Simplified0.2

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

    if -102786700.59358019 < y < 138811322.44829124

    1. Initial program 0.1

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

    2. Simplified0.1

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

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

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

    5. Applied add-cube-cbrt0.5

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

    6. Applied times-frac0.5

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

    7. Simplified0.5

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

    9. Applied fma-udef0.5

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

    10. Simplified0.2

      \[\leadsto \color{blue}{\frac{x - 1}{\frac{y + 1}{y}}} + 1\]
    11. Using strategy rm

    12. Applied div-sub0.2

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

    13. Applied associate-+l-0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -102786700.59358019 \lor \neg \left(y \le 138811322.44829124\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y + 1}{y}} - \left(\frac{1}{\frac{y + 1}{y}} - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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))))