Average Error: 31.1 → 13.3
Time: 8.0s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot x \le 2.474217005200442472341357463282609203046 \cdot 10^{-201}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 2.354236108762997383350198288921334328983 \cdot 10^{-102}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;x \cdot x \le 5.880877246270694688154653410666269919282 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 5.631796052196258562903200319313186965612 \cdot 10^{278}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\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}\;x \cdot x \le 2.474217005200442472341357463282609203046 \cdot 10^{-201}:\\
\;\;\;\;-1\\

\mathbf{elif}\;x \cdot x \le 2.354236108762997383350198288921334328983 \cdot 10^{-102}:\\
\;\;\;\;\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}\;x \cdot x \le 5.880877246270694688154653410666269919282 \cdot 10^{-9}:\\
\;\;\;\;-1\\

\mathbf{elif}\;x \cdot x \le 5.631796052196258562903200319313186965612 \cdot 10^{278}:\\
\;\;\;\;\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 r31750626 = x;
        double r31750627 = r31750626 * r31750626;
        double r31750628 = y;
        double r31750629 = 4.0;
        double r31750630 = r31750628 * r31750629;
        double r31750631 = r31750630 * r31750628;
        double r31750632 = r31750627 - r31750631;
        double r31750633 = r31750627 + r31750631;
        double r31750634 = r31750632 / r31750633;
        return r31750634;
}
↓
double f(double x, double y) {
        double r31750635 = x;
        double r31750636 = r31750635 * r31750635;
        double r31750637 = 2.4742170052004425e-201;
        bool r31750638 = r31750636 <= r31750637;
        double r31750639 = -1.0;
        double r31750640 = 2.3542361087629974e-102;
        bool r31750641 = r31750636 <= r31750640;
        double r31750642 = y;
        double r31750643 = 4.0;
        double r31750644 = r31750642 * r31750643;
        double r31750645 = r31750644 * r31750642;
        double r31750646 = r31750636 + r31750645;
        double r31750647 = r31750636 / r31750646;
        double r31750648 = r31750645 / r31750646;
        double r31750649 = r31750647 - r31750648;
        double r31750650 = 5.880877246270695e-09;
        bool r31750651 = r31750636 <= r31750650;
        double r31750652 = 5.6317960521962586e+278;
        bool r31750653 = r31750636 <= r31750652;
        double r31750654 = 1.0;
        double r31750655 = r31750653 ? r31750649 : r31750654;
        double r31750656 = r31750651 ? r31750639 : r31750655;
        double r31750657 = r31750641 ? r31750649 : r31750656;
        double r31750658 = r31750638 ? r31750639 : r31750657;
        return r31750658;
}

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
Target30.8
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.9743233849626781184483093056769575923681:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x x) < 2.4742170052004425e-201 or 2.3542361087629974e-102 < (* x x) < 5.880877246270695e-09

    1. Initial program 24.2

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

    2. Taylor expanded around 0 13.7

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

    if 2.4742170052004425e-201 < (* x x) < 2.3542361087629974e-102 or 5.880877246270695e-09 < (* x x) < 5.6317960521962586e+278

    1. Initial program 16.1

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

    3. Applied div-sub16.1

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

    if 5.6317960521962586e+278 < (* x x)

    1. Initial program 59.5

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

    2. Taylor expanded around inf 9.3

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

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \le 2.474217005200442472341357463282609203046 \cdot 10^{-201}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 2.354236108762997383350198288921334328983 \cdot 10^{-102}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;x \cdot x \le 5.880877246270694688154653410666269919282 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 5.631796052196258562903200319313186965612 \cdot 10^{278}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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))))