Average Error: 31.4 → 13.8
Time: 10.1s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.75486451228188590222826906220099481589 \cdot 10^{122}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -4.332313917220748901311195815363194649965 \cdot 10^{80}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le -9211266202426181632:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;x \le 2.139551810604105186057739335770012586275 \cdot 10^{-153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 7.373518315070099961643182713875384049377 \cdot 10^{127}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right) \cdot \left(\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \cdot \left(\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\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 \le -4.75486451228188590222826906220099481589 \cdot 10^{122}:\\
\;\;\;\;1\\

\mathbf{elif}\;x \le -4.332313917220748901311195815363194649965 \cdot 10^{80}:\\
\;\;\;\;-1\\

\mathbf{elif}\;x \le -9211266202426181632:\\
\;\;\;\;\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 \le 2.139551810604105186057739335770012586275 \cdot 10^{-153}:\\
\;\;\;\;-1\\

\mathbf{elif}\;x \le 7.373518315070099961643182713875384049377 \cdot 10^{127}:\\
\;\;\;\;\sqrt[3]{\left(\left(\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}\right) \cdot \left(\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}\right)\right) \cdot \left(\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}\right)}\\

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

\end{array}
double f(double x, double y) {
        double r47587477 = x;
        double r47587478 = r47587477 * r47587477;
        double r47587479 = y;
        double r47587480 = 4.0;
        double r47587481 = r47587479 * r47587480;
        double r47587482 = r47587481 * r47587479;
        double r47587483 = r47587478 - r47587482;
        double r47587484 = r47587478 + r47587482;
        double r47587485 = r47587483 / r47587484;
        return r47587485;
}
↓
double f(double x, double y) {
        double r47587486 = x;
        double r47587487 = -4.754864512281886e+122;
        bool r47587488 = r47587486 <= r47587487;
        double r47587489 = 1.0;
        double r47587490 = -4.332313917220749e+80;
        bool r47587491 = r47587486 <= r47587490;
        double r47587492 = -1.0;
        double r47587493 = -9.211266202426182e+18;
        bool r47587494 = r47587486 <= r47587493;
        double r47587495 = r47587486 * r47587486;
        double r47587496 = y;
        double r47587497 = 4.0;
        double r47587498 = r47587496 * r47587497;
        double r47587499 = r47587498 * r47587496;
        double r47587500 = r47587495 + r47587499;
        double r47587501 = r47587495 / r47587500;
        double r47587502 = r47587499 / r47587500;
        double r47587503 = r47587501 - r47587502;
        double r47587504 = 2.1395518106041052e-153;
        bool r47587505 = r47587486 <= r47587504;
        double r47587506 = 7.3735183150701e+127;
        bool r47587507 = r47587486 <= r47587506;
        double r47587508 = r47587503 * r47587503;
        double r47587509 = r47587508 * r47587503;
        double r47587510 = cbrt(r47587509);
        double r47587511 = r47587507 ? r47587510 : r47587489;
        double r47587512 = r47587505 ? r47587492 : r47587511;
        double r47587513 = r47587494 ? r47587503 : r47587512;
        double r47587514 = r47587491 ? r47587492 : r47587513;
        double r47587515 = r47587488 ? r47587489 : r47587514;
        return r47587515;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.4
Target31.1
Herbie13.8
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.9743233849626781184483093056769575923681:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -4.754864512281886e+122 or 7.3735183150701e+127 < x

    1. Initial program 56.1

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

    2. Taylor expanded around inf 9.9

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

    if -4.754864512281886e+122 < x < -4.332313917220749e+80 or -9.211266202426182e+18 < x < 2.1395518106041052e-153

    1. Initial program 24.5

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

    2. Taylor expanded around 0 15.6

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

    if -4.332313917220749e+80 < x < -9.211266202426182e+18

    1. Initial program 17.5

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

    3. Applied div-sub17.5

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

    if 2.1395518106041052e-153 < x < 7.3735183150701e+127

    1. Initial program 14.8

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

    3. Applied div-sub14.8

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

    5. Applied add-cbrt-cube14.8

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right) \cdot \left(\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \cdot \left(\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification13.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.75486451228188590222826906220099481589 \cdot 10^{122}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -4.332313917220748901311195815363194649965 \cdot 10^{80}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le -9211266202426181632:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;x \le 2.139551810604105186057739335770012586275 \cdot 10^{-153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 7.373518315070099961643182713875384049377 \cdot 10^{127}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right) \cdot \left(\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \cdot \left(\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))))