Average Error: 31.5 → 12.7
Time: 6.6s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 3.414623317947593663745953537661023590748 \cdot 10^{-192}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.810003643719863336668138253262668476651 \cdot 10^{67}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 1.756338886358467686795570653182400240677 \cdot 10^{115}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 9.415768721852470917873482124314739876564 \cdot 10^{215}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
\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 3.414623317947593663745953537661023590748 \cdot 10^{-192}:\\
\;\;\;\;1\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.810003643719863336668138253262668476651 \cdot 10^{67}:\\
\;\;\;\;\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 1.756338886358467686795570653182400240677 \cdot 10^{115}:\\
\;\;\;\;1\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 9.415768721852470917873482124314739876564 \cdot 10^{215}:\\
\;\;\;\;\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 r456619 = x;
        double r456620 = r456619 * r456619;
        double r456621 = y;
        double r456622 = 4.0;
        double r456623 = r456621 * r456622;
        double r456624 = r456623 * r456621;
        double r456625 = r456620 - r456624;
        double r456626 = r456620 + r456624;
        double r456627 = r456625 / r456626;
        return r456627;
}
↓
double f(double x, double y) {
        double r456628 = y;
        double r456629 = 4.0;
        double r456630 = r456628 * r456629;
        double r456631 = r456630 * r456628;
        double r456632 = 3.4146233179475937e-192;
        bool r456633 = r456631 <= r456632;
        double r456634 = 1.0;
        double r456635 = 3.8100036437198633e+67;
        bool r456636 = r456631 <= r456635;
        double r456637 = x;
        double r456638 = r456637 * r456637;
        double r456639 = r456638 + r456631;
        double r456640 = r456638 / r456639;
        double r456641 = r456631 / r456639;
        double r456642 = r456640 - r456641;
        double r456643 = 1.7563388863584677e+115;
        bool r456644 = r456631 <= r456643;
        double r456645 = 9.415768721852471e+215;
        bool r456646 = r456631 <= r456645;
        double r456647 = 1.0;
        double r456648 = -r456647;
        double r456649 = r456646 ? r456642 : r456648;
        double r456650 = r456644 ? r456634 : r456649;
        double r456651 = r456636 ? r456642 : r456650;
        double r456652 = r456633 ? r456634 : r456651;
        return r456652;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.5
Target31.2
Herbie12.7
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.9743233849626781184483093056769575923681:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (* y 4.0) y) < 3.4146233179475937e-192 or 3.8100036437198633e+67 < (* (* y 4.0) y) < 1.7563388863584677e+115

    1. Initial program 26.4

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

    2. Taylor expanded around inf 13.0

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

    if 3.4146233179475937e-192 < (* (* y 4.0) y) < 3.8100036437198633e+67 or 1.7563388863584677e+115 < (* (* y 4.0) y) < 9.415768721852471e+215

    1. Initial program 15.0

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

    3. Applied div-sub15.0

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

    if 9.415768721852471e+215 < (* (* y 4.0) y)

    1. Initial program 52.5

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

    3. Applied div-sub52.5

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

    4. Taylor expanded around 0 10.3

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

  4. Final simplification12.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 3.414623317947593663745953537661023590748 \cdot 10^{-192}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.810003643719863336668138253262668476651 \cdot 10^{67}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 1.756338886358467686795570653182400240677 \cdot 10^{115}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 9.415768721852470917873482124314739876564 \cdot 10^{215}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))))