Average Error: 32.6 → 12.9
Time: 1.9s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.759129799435656048380758639203459110241 \cdot 10^{141}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -6.751306280841241990574366329684766225048 \cdot 10^{-71}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}\\ \mathbf{elif}\;x \le 8.488500927325731364094840215005056497754 \cdot 10^{-70}:\\ \;\;\;\;\sqrt[3]{{-1}^{3}}\\ \mathbf{elif}\;x \le 5.311417125820034023983251080218725430022 \cdot 10^{136}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\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 -3.759129799435656048380758639203459110241 \cdot 10^{141}:\\
\;\;\;\;1\\

\mathbf{elif}\;x \le -6.751306280841241990574366329684766225048 \cdot 10^{-71}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}\\

\mathbf{elif}\;x \le 8.488500927325731364094840215005056497754 \cdot 10^{-70}:\\
\;\;\;\;\sqrt[3]{{-1}^{3}}\\

\mathbf{elif}\;x \le 5.311417125820034023983251080218725430022 \cdot 10^{136}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}\\

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

\end{array}
double f(double x, double y) {
        double r664893 = x;
        double r664894 = r664893 * r664893;
        double r664895 = y;
        double r664896 = 4.0;
        double r664897 = r664895 * r664896;
        double r664898 = r664897 * r664895;
        double r664899 = r664894 - r664898;
        double r664900 = r664894 + r664898;
        double r664901 = r664899 / r664900;
        return r664901;
}
↓
double f(double x, double y) {
        double r664902 = x;
        double r664903 = -3.759129799435656e+141;
        bool r664904 = r664902 <= r664903;
        double r664905 = 1.0;
        double r664906 = -6.751306280841242e-71;
        bool r664907 = r664902 <= r664906;
        double r664908 = r664902 * r664902;
        double r664909 = y;
        double r664910 = 4.0;
        double r664911 = r664909 * r664910;
        double r664912 = r664911 * r664909;
        double r664913 = r664908 - r664912;
        double r664914 = r664908 + r664912;
        double r664915 = r664913 / r664914;
        double r664916 = 3.0;
        double r664917 = pow(r664915, r664916);
        double r664918 = cbrt(r664917);
        double r664919 = 8.488500927325731e-70;
        bool r664920 = r664902 <= r664919;
        double r664921 = -1.0;
        double r664922 = pow(r664921, r664916);
        double r664923 = cbrt(r664922);
        double r664924 = 5.311417125820034e+136;
        bool r664925 = r664902 <= r664924;
        double r664926 = r664925 ? r664918 : r664905;
        double r664927 = r664920 ? r664923 : r664926;
        double r664928 = r664907 ? r664918 : r664927;
        double r664929 = r664904 ? r664905 : r664928;
        return r664929;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.6
Target32.2
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.9743233849626781184483093056769575923681:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -3.759129799435656e+141 or 5.311417125820034e+136 < x

    1. Initial program 60.2

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

    2. Taylor expanded around inf 9.1

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

    if -3.759129799435656e+141 < x < -6.751306280841242e-71 or 8.488500927325731e-70 < x < 5.311417125820034e+136

    1. Initial program 16.8

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

    3. Applied add-cbrt-cube44.0

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}}\]

    4. Applied add-cbrt-cube44.5

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)}}}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}\]

    5. Applied cbrt-undiv44.5

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)}{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}}\]

    6. Simplified16.8

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}}\]

    if -6.751306280841242e-71 < x < 8.488500927325731e-70

    1. Initial program 26.3

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

    3. Applied add-cbrt-cube53.6

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}}\]

    4. Applied add-cbrt-cube53.7

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)}}}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}\]

    5. Applied cbrt-undiv53.7

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)}{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}}\]

    6. Simplified26.3

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}}\]

    7. Taylor expanded around 0 12.0

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

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.759129799435656048380758639203459110241 \cdot 10^{141}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -6.751306280841241990574366329684766225048 \cdot 10^{-71}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}\\ \mathbf{elif}\;x \le 8.488500927325731364094840215005056497754 \cdot 10^{-70}:\\ \;\;\;\;\sqrt[3]{{-1}^{3}}\\ \mathbf{elif}\;x \le 5.311417125820034023983251080218725430022 \cdot 10^{136}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(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))))