Average Error: 31.1 → 12.6
Time: 9.7s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4.0\right) \cdot y}{x \cdot x + \left(y \cdot 4.0\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot x \le 7.1088010196971 \cdot 10^{-314}:\\ \;\;\;\;-1.0\\ \mathbf{elif}\;x \cdot x \le 9.758857908001147 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(4.0 \cdot y\right) + x \cdot x}{x \cdot x - y \cdot \left(4.0 \cdot y\right)}}\\ \mathbf{elif}\;x \cdot x \le 6.296502919195928 \cdot 10^{+44}:\\ \;\;\;\;-1.0\\ \mathbf{elif}\;x \cdot x \le 3.1031769081973586 \cdot 10^{+257}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(4.0 \cdot y\right) + x \cdot x}{x \cdot x - y \cdot \left(4.0 \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{x \cdot x - \left(y \cdot 4.0\right) \cdot y}{x \cdot x + \left(y \cdot 4.0\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot x \le 7.1088010196971 \cdot 10^{-314}:\\
\;\;\;\;-1.0\\

\mathbf{elif}\;x \cdot x \le 9.758857908001147 \cdot 10^{+18}:\\
\;\;\;\;\frac{1}{\frac{y \cdot \left(4.0 \cdot y\right) + x \cdot x}{x \cdot x - y \cdot \left(4.0 \cdot y\right)}}\\

\mathbf{elif}\;x \cdot x \le 6.296502919195928 \cdot 10^{+44}:\\
\;\;\;\;-1.0\\

\mathbf{elif}\;x \cdot x \le 3.1031769081973586 \cdot 10^{+257}:\\
\;\;\;\;\frac{1}{\frac{y \cdot \left(4.0 \cdot y\right) + x \cdot x}{x \cdot x - y \cdot \left(4.0 \cdot y\right)}}\\

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

\end{array}
double f(double x, double y) {
        double r32634531 = x;
        double r32634532 = r32634531 * r32634531;
        double r32634533 = y;
        double r32634534 = 4.0;
        double r32634535 = r32634533 * r32634534;
        double r32634536 = r32634535 * r32634533;
        double r32634537 = r32634532 - r32634536;
        double r32634538 = r32634532 + r32634536;
        double r32634539 = r32634537 / r32634538;
        return r32634539;
}
↓
double f(double x, double y) {
        double r32634540 = x;
        double r32634541 = r32634540 * r32634540;
        double r32634542 = 7.1088010196971e-314;
        bool r32634543 = r32634541 <= r32634542;
        double r32634544 = -1.0;
        double r32634545 = 9.758857908001147e+18;
        bool r32634546 = r32634541 <= r32634545;
        double r32634547 = 1.0;
        double r32634548 = y;
        double r32634549 = 4.0;
        double r32634550 = r32634549 * r32634548;
        double r32634551 = r32634548 * r32634550;
        double r32634552 = r32634551 + r32634541;
        double r32634553 = r32634541 - r32634551;
        double r32634554 = r32634552 / r32634553;
        double r32634555 = r32634547 / r32634554;
        double r32634556 = 6.296502919195928e+44;
        bool r32634557 = r32634541 <= r32634556;
        double r32634558 = 3.1031769081973586e+257;
        bool r32634559 = r32634541 <= r32634558;
        double r32634560 = r32634559 ? r32634555 : r32634547;
        double r32634561 = r32634557 ? r32634544 : r32634560;
        double r32634562 = r32634546 ? r32634555 : r32634561;
        double r32634563 = r32634543 ? r32634544 : r32634562;
        return r32634563;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.1
Target31.2
Herbie12.6
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4.0\right) \cdot y}{x \cdot x + \left(y \cdot 4.0\right) \cdot y} \lt 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4.0} - \frac{\left(y \cdot y\right) \cdot 4.0}{x \cdot x + \left(y \cdot y\right) \cdot 4.0}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4.0}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4.0}{x \cdot x + \left(y \cdot y\right) \cdot 4.0}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x x) < 7.1088010196971e-314 or 9.758857908001147e+18 < (* x x) < 6.296502919195928e+44

    1. Initial program 29.3

      \[\frac{x \cdot x - \left(y \cdot 4.0\right) \cdot y}{x \cdot x + \left(y \cdot 4.0\right) \cdot y}\]

    2. Taylor expanded around 0 10.1

      \[\leadsto \color{blue}{-1.0}\]

    if 7.1088010196971e-314 < (* x x) < 9.758857908001147e+18 or 6.296502919195928e+44 < (* x x) < 3.1031769081973586e+257

    1. Initial program 16.1

      \[\frac{x \cdot x - \left(y \cdot 4.0\right) \cdot y}{x \cdot x + \left(y \cdot 4.0\right) \cdot y}\]
    2. Using strategy rm

    3. Applied clear-num16.1

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(y \cdot 4.0\right) \cdot y}{x \cdot x - \left(y \cdot 4.0\right) \cdot y}}}\]

    if 3.1031769081973586e+257 < (* x x)

    1. Initial program 56.4

      \[\frac{x \cdot x - \left(y \cdot 4.0\right) \cdot y}{x \cdot x + \left(y \cdot 4.0\right) \cdot y}\]

    2. Taylor expanded around inf 9.6

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

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \le 7.1088010196971 \cdot 10^{-314}:\\ \;\;\;\;-1.0\\ \mathbf{elif}\;x \cdot x \le 9.758857908001147 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(4.0 \cdot y\right) + x \cdot x}{x \cdot x - y \cdot \left(4.0 \cdot y\right)}}\\ \mathbf{elif}\;x \cdot x \le 6.296502919195928 \cdot 10^{+44}:\\ \;\;\;\;-1.0\\ \mathbf{elif}\;x \cdot x \le 3.1031769081973586 \cdot 10^{+257}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(4.0 \cdot y\right) + x \cdot x}{x \cdot x - y \cdot \left(4.0 \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

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