Average Error: 31.2 → 13.9
Time: 13.4s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4.758167358261969805068202523877348166337 \cdot 10^{148}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.63689029313189507811324572738338926855 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x}} - \sqrt{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;y \le 1.575397687437991456648509198388754062127 \cdot 10^{-55}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 8.134021215145637850375061461407044861883 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x}} - \sqrt{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;y \le 98355.24381927796639502048492431640625:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
\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}\;y \le -4.758167358261969805068202523877348166337 \cdot 10^{148}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.63689029313189507811324572738338926855 \cdot 10^{-160}:\\
\;\;\;\;\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x}} - \sqrt{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\

\mathbf{elif}\;y \le 1.575397687437991456648509198388754062127 \cdot 10^{-55}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 8.134021215145637850375061461407044861883 \cdot 10^{-38}:\\
\;\;\;\;\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x}} - \sqrt{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\

\mathbf{elif}\;y \le 98355.24381927796639502048492431640625:\\
\;\;\;\;1\\

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

\end{array}
double f(double x, double y) {
        double r1186515 = x;
        double r1186516 = r1186515 * r1186515;
        double r1186517 = y;
        double r1186518 = 4.0;
        double r1186519 = r1186517 * r1186518;
        double r1186520 = r1186519 * r1186517;
        double r1186521 = r1186516 - r1186520;
        double r1186522 = r1186516 + r1186520;
        double r1186523 = r1186521 / r1186522;
        return r1186523;
}
↓
double f(double x, double y) {
        double r1186524 = y;
        double r1186525 = -4.75816735826197e+148;
        bool r1186526 = r1186524 <= r1186525;
        double r1186527 = 1.0;
        double r1186528 = -r1186527;
        double r1186529 = -1.636890293131895e-160;
        bool r1186530 = r1186524 <= r1186529;
        double r1186531 = 1.0;
        double r1186532 = x;
        double r1186533 = r1186532 * r1186532;
        double r1186534 = 4.0;
        double r1186535 = r1186524 * r1186534;
        double r1186536 = r1186535 * r1186524;
        double r1186537 = r1186533 + r1186536;
        double r1186538 = r1186537 / r1186533;
        double r1186539 = r1186531 / r1186538;
        double r1186540 = r1186536 / r1186537;
        double r1186541 = sqrt(r1186540);
        double r1186542 = r1186541 * r1186541;
        double r1186543 = r1186539 - r1186542;
        double r1186544 = 1.5753976874379915e-55;
        bool r1186545 = r1186524 <= r1186544;
        double r1186546 = 8.134021215145638e-38;
        bool r1186547 = r1186524 <= r1186546;
        double r1186548 = 98355.24381927797;
        bool r1186549 = r1186524 <= r1186548;
        double r1186550 = r1186549 ? r1186531 : r1186528;
        double r1186551 = r1186547 ? r1186543 : r1186550;
        double r1186552 = r1186545 ? r1186531 : r1186551;
        double r1186553 = r1186530 ? r1186543 : r1186552;
        double r1186554 = r1186526 ? r1186528 : r1186553;
        return r1186554;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.2
Target30.9
Herbie13.9
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.9743233849626781184483093056769575923681:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -4.75816735826197e+148 or 98355.24381927797 < y

    1. Initial program 47.9

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

    3. Applied div-sub47.9

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

    4. Taylor expanded around 0 13.3

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

    if -4.75816735826197e+148 < y < -1.636890293131895e-160 or 1.5753976874379915e-55 < y < 8.134021215145638e-38

    1. Initial program 16.3

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

    3. Applied div-sub16.3

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

    5. Applied add-sqr-sqrt16.3

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

    7. Applied clear-num16.3

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

    if -1.636890293131895e-160 < y < 1.5753976874379915e-55 or 8.134021215145638e-38 < y < 98355.24381927797

    1. Initial program 24.9

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

    2. Taylor expanded around inf 12.6

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

  4. Final simplification13.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.758167358261969805068202523877348166337 \cdot 10^{148}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.63689029313189507811324572738338926855 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x}} - \sqrt{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;y \le 1.575397687437991456648509198388754062127 \cdot 10^{-55}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 8.134021215145637850375061461407044861883 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x}} - \sqrt{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;y \le 98355.24381927796639502048492431640625:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))) 0.974323384962678118) (- (/ (* 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))))