Average Error: 31.9 → 14.7
Time: 2.0s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 6.7475079204883884 \cdot 10^{-234}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3356228.6077285837:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 6.70654513150967042 \cdot 10^{125}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
\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}\;\left(y \cdot 4\right) \cdot y \le 6.7475079204883884 \cdot 10^{-234}:\\
\;\;\;\;1\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3356228.6077285837:\\
\;\;\;\;\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \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 6.70654513150967042 \cdot 10^{125}:\\
\;\;\;\;1\\

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

\end{array}
double f(double x, double y) {
        double r683977 = x;
        double r683978 = r683977 * r683977;
        double r683979 = y;
        double r683980 = 4.0;
        double r683981 = r683979 * r683980;
        double r683982 = r683981 * r683979;
        double r683983 = r683978 - r683982;
        double r683984 = r683978 + r683982;
        double r683985 = r683983 / r683984;
        return r683985;
}
↓
double f(double x, double y) {
        double r683986 = y;
        double r683987 = 4.0;
        double r683988 = r683986 * r683987;
        double r683989 = r683988 * r683986;
        double r683990 = 6.747507920488388e-234;
        bool r683991 = r683989 <= r683990;
        double r683992 = 1.0;
        double r683993 = 3356228.6077285837;
        bool r683994 = r683989 <= r683993;
        double r683995 = x;
        double r683996 = r683995 * r683995;
        double r683997 = r683996 - r683989;
        double r683998 = r683996 + r683989;
        double r683999 = r683997 / r683998;
        double r684000 = cbrt(r683999);
        double r684001 = r684000 * r684000;
        double r684002 = r684001 * r684000;
        double r684003 = 6.7065451315096704e+125;
        bool r684004 = r683989 <= r684003;
        double r684005 = -1.0;
        double r684006 = r684004 ? r683992 : r684005;
        double r684007 = r683994 ? r684002 : r684006;
        double r684008 = r683991 ? r683992 : r684007;
        return r684008;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.9
Target31.6
Herbie14.7
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.974323384962678118:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (* y 4.0) y) < 6.747507920488388e-234 or 3356228.6077285837 < (* (* y 4.0) y) < 6.7065451315096704e+125

    1. Initial program 24.7

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

    2. Taylor expanded around inf 16.0

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

    if 6.747507920488388e-234 < (* (* y 4.0) y) < 3356228.6077285837

    1. Initial program 15.9

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

    3. Applied add-cube-cbrt15.9

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

    if 6.7065451315096704e+125 < (* (* y 4.0) y)

    1. Initial program 47.2

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

    2. Taylor expanded around 0 12.7

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

  4. Final simplification14.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 6.7475079204883884 \cdot 10^{-234}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3356228.6077285837:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 6.70654513150967042 \cdot 10^{125}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(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))))