Average Error: 31.8 → 13.4
Time: 4.1s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.4667592323261061 \cdot 10^{131}:\\ \;\;\;\;\sqrt[3]{1}\\ \mathbf{elif}\;x \le -8.0114334775173874 \cdot 10^{-117}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;x \le 1.9644387956096076 \cdot 10^{-169}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 1.06815270095652453 \cdot 10^{-15}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;x \le 2.0459804393047752 \cdot 10^{46}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1}\\ \end{array}\]
\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 -1.4667592323261061 \cdot 10^{131}:\\
\;\;\;\;\sqrt[3]{1}\\

\mathbf{elif}\;x \le -8.0114334775173874 \cdot 10^{-117}:\\
\;\;\;\;\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}\;x \le 1.9644387956096076 \cdot 10^{-169}:\\
\;\;\;\;-1\\

\mathbf{elif}\;x \le 1.06815270095652453 \cdot 10^{-15}:\\
\;\;\;\;\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}\;x \le 2.0459804393047752 \cdot 10^{46}:\\
\;\;\;\;-1\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{1}\\

\end{array}
double f(double x, double y) {
        double r583194 = x;
        double r583195 = r583194 * r583194;
        double r583196 = y;
        double r583197 = 4.0;
        double r583198 = r583196 * r583197;
        double r583199 = r583198 * r583196;
        double r583200 = r583195 - r583199;
        double r583201 = r583195 + r583199;
        double r583202 = r583200 / r583201;
        return r583202;
}
↓
double f(double x, double y) {
        double r583203 = x;
        double r583204 = -1.4667592323261061e+131;
        bool r583205 = r583203 <= r583204;
        double r583206 = 1.0;
        double r583207 = cbrt(r583206);
        double r583208 = -8.011433477517387e-117;
        bool r583209 = r583203 <= r583208;
        double r583210 = r583203 * r583203;
        double r583211 = y;
        double r583212 = 4.0;
        double r583213 = r583211 * r583212;
        double r583214 = r583213 * r583211;
        double r583215 = r583210 - r583214;
        double r583216 = r583210 + r583214;
        double r583217 = r583215 / r583216;
        double r583218 = cbrt(r583217);
        double r583219 = r583218 * r583218;
        double r583220 = r583219 * r583218;
        double r583221 = 1.9644387956096076e-169;
        bool r583222 = r583203 <= r583221;
        double r583223 = -1.0;
        double r583224 = 1.0681527009565245e-15;
        bool r583225 = r583203 <= r583224;
        double r583226 = 2.0459804393047752e+46;
        bool r583227 = r583203 <= r583226;
        double r583228 = r583227 ? r583223 : r583207;
        double r583229 = r583225 ? r583220 : r583228;
        double r583230 = r583222 ? r583223 : r583229;
        double r583231 = r583209 ? r583220 : r583230;
        double r583232 = r583205 ? r583207 : r583231;
        return r583232;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.8
Target31.5
Herbie13.4
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.974323384962678118:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.4667592323261061e+131 or 2.0459804393047752e+46 < x

    1. Initial program 51.0

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

    3. Applied add-cbrt-cube63.2

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

    4. Applied add-cbrt-cube63.6

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

    5. Applied cbrt-undiv63.6

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

    6. Simplified51.0

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

    7. Taylor expanded around inf 12.1

      \[\leadsto \sqrt[3]{{\color{blue}{1}}^{3}}\]

    if -1.4667592323261061e+131 < x < -8.011433477517387e-117 or 1.9644387956096076e-169 < x < 1.0681527009565245e-15

    1. Initial program 15.6

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

    3. Applied add-cube-cbrt15.6

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

    if -8.011433477517387e-117 < x < 1.9644387956096076e-169 or 1.0681527009565245e-15 < x < 2.0459804393047752e+46

    1. Initial program 27.4

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

    2. Taylor expanded around 0 12.6

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

  4. Final simplification13.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.4667592323261061 \cdot 10^{131}:\\ \;\;\;\;\sqrt[3]{1}\\ \mathbf{elif}\;x \le -8.0114334775173874 \cdot 10^{-117}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;x \le 1.9644387956096076 \cdot 10^{-169}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 1.06815270095652453 \cdot 10^{-15}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;x \le 2.0459804393047752 \cdot 10^{46}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(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))))