Average Error: 31.4 → 13.8
Time: 8.9s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.75486451228188590222826906220099481589 \cdot 10^{122}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -4.332313917220748901311195815363194649965 \cdot 10^{80}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le -9211266202426181632:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{x \cdot x}{\left(y \cdot 4\right) \cdot y + x \cdot x} - \frac{\left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y + x \cdot x}\right) \cdot \left(\frac{x \cdot x}{\left(y \cdot 4\right) \cdot y + x \cdot x} - \frac{\left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y + x \cdot x}\right)\right) \cdot \left(\frac{x \cdot x}{\left(y \cdot 4\right) \cdot y + x \cdot x} - \frac{\left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y + x \cdot x}\right)}\\ \mathbf{elif}\;x \le 2.139551810604105186057739335770012586275 \cdot 10^{-153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 7.373518315070099961643182713875384049377 \cdot 10^{127}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{x \cdot x}{\left(y \cdot 4\right) \cdot y + x \cdot x} - \frac{\left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y + x \cdot x}\right) \cdot \left(\frac{x \cdot x}{\left(y \cdot 4\right) \cdot y + x \cdot x} - \frac{\left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y + x \cdot x}\right)\right) \cdot \left(\frac{x \cdot x}{\left(y \cdot 4\right) \cdot y + x \cdot x} - \frac{\left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y + x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\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:\\
\;\;\;\;\sqrt[3]{\left(\left(\frac{x \cdot x}{\left(y \cdot 4\right) \cdot y + x \cdot x} - \frac{\left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y + x \cdot x}\right) \cdot \left(\frac{x \cdot x}{\left(y \cdot 4\right) \cdot y + x \cdot x} - \frac{\left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y + x \cdot x}\right)\right) \cdot \left(\frac{x \cdot x}{\left(y \cdot 4\right) \cdot y + x \cdot x} - \frac{\left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y + x \cdot x}\right)}\\

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

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

\end{array}
double f(double x, double y) {
        double r31553303 = x;
        double r31553304 = r31553303 * r31553303;
        double r31553305 = y;
        double r31553306 = 4.0;
        double r31553307 = r31553305 * r31553306;
        double r31553308 = r31553307 * r31553305;
        double r31553309 = r31553304 - r31553308;
        double r31553310 = r31553304 + r31553308;
        double r31553311 = r31553309 / r31553310;
        return r31553311;
}
↓
double f(double x, double y) {
        double r31553312 = x;
        double r31553313 = -4.754864512281886e+122;
        bool r31553314 = r31553312 <= r31553313;
        double r31553315 = 1.0;
        double r31553316 = -4.332313917220749e+80;
        bool r31553317 = r31553312 <= r31553316;
        double r31553318 = -1.0;
        double r31553319 = -9.211266202426182e+18;
        bool r31553320 = r31553312 <= r31553319;
        double r31553321 = r31553312 * r31553312;
        double r31553322 = y;
        double r31553323 = 4.0;
        double r31553324 = r31553322 * r31553323;
        double r31553325 = r31553324 * r31553322;
        double r31553326 = r31553325 + r31553321;
        double r31553327 = r31553321 / r31553326;
        double r31553328 = r31553325 / r31553326;
        double r31553329 = r31553327 - r31553328;
        double r31553330 = r31553329 * r31553329;
        double r31553331 = r31553330 * r31553329;
        double r31553332 = cbrt(r31553331);
        double r31553333 = 2.1395518106041052e-153;
        bool r31553334 = r31553312 <= r31553333;
        double r31553335 = 7.3735183150701e+127;
        bool r31553336 = r31553312 <= r31553335;
        double r31553337 = r31553336 ? r31553332 : r31553315;
        double r31553338 = r31553334 ? r31553318 : r31553337;
        double r31553339 = r31553320 ? r31553332 : r31553338;
        double r31553340 = r31553317 ? r31553318 : r31553339;
        double r31553341 = r31553314 ? r31553315 : r31553340;
        return r31553341;
}

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 3 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 or 2.1395518106041052e-153 < x < 7.3735183150701e+127

    1. Initial program 15.3

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

    3. Applied div-sub15.3

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

    5. Applied add-cbrt-cube15.3

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

  4. Final simplification13.8

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

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))))