Average Error: 32.3 → 13.5
Time: 3.3s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4.5422239760964022 \cdot 10^{172}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.58062656605551297 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\ \mathbf{elif}\;y \le 1.19856641460827694 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 2.05879739523357 \cdot 10^{107}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\ \mathbf{elif}\;y \le 2.49672079982528521 \cdot 10^{132}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -4.5422239760964022 \cdot 10^{172}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.58062656605551297 \cdot 10^{-97}:\\
\;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\

\mathbf{elif}\;y \le 1.19856641460827694 \cdot 10^{-62}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 2.05879739523357 \cdot 10^{107}:\\
\;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\

\mathbf{elif}\;y \le 2.49672079982528521 \cdot 10^{132}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;-1\\

\end{array}
double f(double x, double y) {
        double r722690 = x;
        double r722691 = r722690 * r722690;
        double r722692 = y;
        double r722693 = 4.0;
        double r722694 = r722692 * r722693;
        double r722695 = r722694 * r722692;
        double r722696 = r722691 - r722695;
        double r722697 = r722691 + r722695;
        double r722698 = r722696 / r722697;
        return r722698;
}
↓
double f(double x, double y) {
        double r722699 = y;
        double r722700 = -4.542223976096402e+172;
        bool r722701 = r722699 <= r722700;
        double r722702 = -1.0;
        double r722703 = -1.580626566055513e-97;
        bool r722704 = r722699 <= r722703;
        double r722705 = x;
        double r722706 = 4.0;
        double r722707 = r722699 * r722706;
        double r722708 = r722707 * r722699;
        double r722709 = fma(r722705, r722705, r722708);
        double r722710 = r722709 / r722705;
        double r722711 = r722705 / r722710;
        double r722712 = r722709 / r722699;
        double r722713 = r722707 / r722712;
        double r722714 = r722711 - r722713;
        double r722715 = 1.198566414608277e-62;
        bool r722716 = r722699 <= r722715;
        double r722717 = 1.0;
        double r722718 = 2.0587973952335703e+107;
        bool r722719 = r722699 <= r722718;
        double r722720 = 2.4967207998252852e+132;
        bool r722721 = r722699 <= r722720;
        double r722722 = r722721 ? r722717 : r722702;
        double r722723 = r722719 ? r722714 : r722722;
        double r722724 = r722716 ? r722717 : r722723;
        double r722725 = r722704 ? r722714 : r722724;
        double r722726 = r722701 ? r722702 : r722725;
        return r722726;
}

Error

Bits error versus x

Bits error versus y

Target

Original32.3
Target32.0
Herbie13.5
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.974323384962678118:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -4.542223976096402e+172 or 2.4967207998252852e+132 < y

    1. Initial program 60.7

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

    2. Taylor expanded around 0 7.6

      \[\leadsto \color{blue}{-1}\]

    if -4.542223976096402e+172 < y < -1.580626566055513e-97 or 1.198566414608277e-62 < y < 2.0587973952335703e+107

    1. Initial program 18.3

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
    2. Using strategy rm

    3. Applied div-sub18.3

      \[\leadsto \color{blue}{\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\]

    4. Simplified17.9

      \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}}} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

    5. Simplified17.8

      \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}} - \color{blue}{\frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}}\]

    if -1.580626566055513e-97 < y < 1.198566414608277e-62 or 2.0587973952335703e+107 < y < 2.4967207998252852e+132

    1. Initial program 25.8

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

    2. Taylor expanded around inf 13.5

      \[\leadsto \color{blue}{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.5422239760964022 \cdot 10^{172}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.58062656605551297 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\ \mathbf{elif}\;y \le 1.19856641460827694 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 2.05879739523357 \cdot 10^{107}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x}} - \frac{y \cdot 4}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{y}}\\ \mathbf{elif}\;y \le 2.49672079982528521 \cdot 10^{132}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4))) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4)))) 2) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))))

  (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))))