Average Error: 22.7 → 7.4
Time: 4.5s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4.38962381722942814 \cdot 10^{28} \lor \neg \left(y \le 103782153027757.69\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -4.38962381722942814 \cdot 10^{28} \lor \neg \left(y \le 103782153027757.69\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r931981 = 1.0;
        double r931982 = x;
        double r931983 = r931981 - r931982;
        double r931984 = y;
        double r931985 = r931983 * r931984;
        double r931986 = r931984 + r931981;
        double r931987 = r931985 / r931986;
        double r931988 = r931981 - r931987;
        return r931988;
}
↓
double f(double x, double y) {
        double r931989 = y;
        double r931990 = -4.389623817229428e+28;
        bool r931991 = r931989 <= r931990;
        double r931992 = 103782153027757.69;
        bool r931993 = r931989 <= r931992;
        double r931994 = !r931993;
        bool r931995 = r931991 || r931994;
        double r931996 = 1.0;
        double r931997 = x;
        double r931998 = 2.0;
        double r931999 = pow(r931989, r931998);
        double r932000 = r931997 / r931999;
        double r932001 = r931997 / r931989;
        double r932002 = r932000 - r932001;
        double r932003 = fma(r931996, r932002, r931997);
        double r932004 = r931996 - r931997;
        double r932005 = r932004 * r931989;
        double r932006 = r931989 + r931996;
        double r932007 = r932005 / r932006;
        double r932008 = r931996 - r932007;
        double r932009 = r931995 ? r932003 : r932008;
        return r932009;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.7
Target0.2
Herbie7.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.389623817229428e+28 or 103782153027757.69 < y

    1. Initial program 47.0

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

    2. Simplified29.1

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

    4. Applied add-cube-cbrt29.9

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

    5. Applied *-un-lft-identity29.9

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

    6. Applied times-frac29.9

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

    7. Taylor expanded around inf 14.1

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

    8. Simplified14.1

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

    if -4.389623817229428e+28 < y < 103782153027757.69

    1. Initial program 1.4

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.38962381722942814 \cdot 10^{28} \lor \neg \left(y \le 103782153027757.69\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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))))