Average Error: 31.3 → 14.7
Time: 8.3s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 0.0:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 8.14624205965243804 \cdot 10^{-197}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 1.89336738182365018 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 2.080267309844611 \cdot 10^{-17}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 5.0547066207328322 \cdot 10^{101}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 2.6000500965086023 \cdot 10^{199}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
\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}\;\left(y \cdot 4\right) \cdot y \le 0.0:\\
\;\;\;\;1\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 8.14624205965243804 \cdot 10^{-197}:\\
\;\;\;\;\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}\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 1.89336738182365018 \cdot 10^{-164}:\\
\;\;\;\;1\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 2.080267309844611 \cdot 10^{-17}:\\
\;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y}}\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 5.0547066207328322 \cdot 10^{101}:\\
\;\;\;\;1\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 2.6000500965086023 \cdot 10^{199}:\\
\;\;\;\;\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}\\

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

\end{array}
double f(double x, double y) {
        double r517580 = x;
        double r517581 = r517580 * r517580;
        double r517582 = y;
        double r517583 = 4.0;
        double r517584 = r517582 * r517583;
        double r517585 = r517584 * r517582;
        double r517586 = r517581 - r517585;
        double r517587 = r517581 + r517585;
        double r517588 = r517586 / r517587;
        return r517588;
}
↓
double f(double x, double y) {
        double r517589 = y;
        double r517590 = 4.0;
        double r517591 = r517589 * r517590;
        double r517592 = r517591 * r517589;
        double r517593 = 0.0;
        bool r517594 = r517592 <= r517593;
        double r517595 = 1.0;
        double r517596 = 8.146242059652438e-197;
        bool r517597 = r517592 <= r517596;
        double r517598 = x;
        double r517599 = r517598 * r517598;
        double r517600 = r517599 + r517592;
        double r517601 = r517599 / r517600;
        double r517602 = r517592 / r517600;
        double r517603 = r517601 - r517602;
        double r517604 = 1.89336738182365e-164;
        bool r517605 = r517592 <= r517604;
        double r517606 = 2.080267309844611e-17;
        bool r517607 = r517592 <= r517606;
        double r517608 = r517600 / r517592;
        double r517609 = r517595 / r517608;
        double r517610 = r517601 - r517609;
        double r517611 = 5.054706620732832e+101;
        bool r517612 = r517592 <= r517611;
        double r517613 = 2.6000500965086023e+199;
        bool r517614 = r517592 <= r517613;
        double r517615 = 1.0;
        double r517616 = -r517615;
        double r517617 = r517614 ? r517603 : r517616;
        double r517618 = r517612 ? r517595 : r517617;
        double r517619 = r517607 ? r517610 : r517618;
        double r517620 = r517605 ? r517595 : r517619;
        double r517621 = r517597 ? r517603 : r517620;
        double r517622 = r517594 ? r517595 : r517621;
        return r517622;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.3
Target31.1
Herbie14.7
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.974323384962678118:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (* (* y 4.0) y) < 0.0 or 8.146242059652438e-197 < (* (* y 4.0) y) < 1.89336738182365e-164 or 2.080267309844611e-17 < (* (* y 4.0) y) < 5.054706620732832e+101

    1. Initial program 25.7

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

    2. Taylor expanded around inf 16.9

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

    if 0.0 < (* (* y 4.0) y) < 8.146242059652438e-197 or 5.054706620732832e+101 < (* (* y 4.0) y) < 2.6000500965086023e+199

    1. Initial program 16.4

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

    3. Applied div-sub16.4

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

    if 1.89336738182365e-164 < (* (* y 4.0) y) < 2.080267309844611e-17

    1. Initial program 15.7

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

    3. Applied div-sub15.7

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

    5. Applied clear-num15.7

      \[\leadsto \frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \color{blue}{\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y}}}\]

    if 2.6000500965086023e+199 < (* (* y 4.0) y)

    1. Initial program 50.7

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

    3. Applied div-sub50.7

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

    4. Taylor expanded around 0 11.1

      \[\leadsto \color{blue}{-1}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification14.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 0.0:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 8.14624205965243804 \cdot 10^{-197}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 1.89336738182365018 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 2.080267309844611 \cdot 10^{-17}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 5.0547066207328322 \cdot 10^{101}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 2.6000500965086023 \cdot 10^{199}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))