Average Error: 32.3 → 13.6
Time: 2.5s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.71073899284975746 \cdot 10^{142}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.38415960656676321 \cdot 10^{-97}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \log \left(e^{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right)\\ \mathbf{elif}\;y \le 1.1196094733529732 \cdot 10^{-55}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 28161126.972837694:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \log \left(e^{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right)\\ \mathbf{elif}\;y \le 4.01162503157803517 \cdot 10^{38}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 2.16201455188241352 \cdot 10^{107}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \log \left(e^{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right)\\ \mathbf{elif}\;y \le 4.36744780112662469 \cdot 10^{128}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
\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 -7.71073899284975746 \cdot 10^{142}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.38415960656676321 \cdot 10^{-97}:\\
\;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \log \left(e^{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right)\\

\mathbf{elif}\;y \le 1.1196094733529732 \cdot 10^{-55}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 28161126.972837694:\\
\;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \log \left(e^{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right)\\

\mathbf{elif}\;y \le 4.01162503157803517 \cdot 10^{38}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 2.16201455188241352 \cdot 10^{107}:\\
\;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \log \left(e^{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right)\\

\mathbf{elif}\;y \le 4.36744780112662469 \cdot 10^{128}:\\
\;\;\;\;1\\

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

\end{array}
double f(double x, double y) {
        double r815018 = x;
        double r815019 = r815018 * r815018;
        double r815020 = y;
        double r815021 = 4.0;
        double r815022 = r815020 * r815021;
        double r815023 = r815022 * r815020;
        double r815024 = r815019 - r815023;
        double r815025 = r815019 + r815023;
        double r815026 = r815024 / r815025;
        return r815026;
}
↓
double f(double x, double y) {
        double r815027 = y;
        double r815028 = -7.710738992849757e+142;
        bool r815029 = r815027 <= r815028;
        double r815030 = -1.0;
        double r815031 = -1.3841596065667632e-97;
        bool r815032 = r815027 <= r815031;
        double r815033 = x;
        double r815034 = r815033 * r815033;
        double r815035 = 4.0;
        double r815036 = r815027 * r815035;
        double r815037 = r815036 * r815027;
        double r815038 = r815034 + r815037;
        double r815039 = r815034 / r815038;
        double r815040 = r815037 / r815038;
        double r815041 = exp(r815040);
        double r815042 = log(r815041);
        double r815043 = r815039 - r815042;
        double r815044 = 1.1196094733529732e-55;
        bool r815045 = r815027 <= r815044;
        double r815046 = 1.0;
        double r815047 = 28161126.972837694;
        bool r815048 = r815027 <= r815047;
        double r815049 = 4.011625031578035e+38;
        bool r815050 = r815027 <= r815049;
        double r815051 = 2.1620145518824135e+107;
        bool r815052 = r815027 <= r815051;
        double r815053 = 4.367447801126625e+128;
        bool r815054 = r815027 <= r815053;
        double r815055 = r815054 ? r815046 : r815030;
        double r815056 = r815052 ? r815043 : r815055;
        double r815057 = r815050 ? r815046 : r815056;
        double r815058 = r815048 ? r815043 : r815057;
        double r815059 = r815045 ? r815046 : r815058;
        double r815060 = r815032 ? r815043 : r815059;
        double r815061 = r815029 ? r815030 : r815060;
        return r815061;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.3
Target32.0
Herbie13.6
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.974323384962678118:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -7.710738992849757e+142 or 4.367447801126625e+128 < y

    1. Initial program 59.1

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

    2. Taylor expanded around 0 8.6

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

    if -7.710738992849757e+142 < y < -1.3841596065667632e-97 or 1.1196094733529732e-55 < y < 28161126.972837694 or 4.011625031578035e+38 < y < 2.1620145518824135e+107

    1. Initial program 16.7

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

    3. Applied div-sub16.7

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

    5. Applied add-log-exp16.7

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

    if -1.3841596065667632e-97 < y < 1.1196094733529732e-55 or 28161126.972837694 < y < 4.011625031578035e+38 or 2.1620145518824135e+107 < y < 4.367447801126625e+128

    1. Initial program 25.0

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

    2. Taylor expanded around inf 14.8

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

  4. Final simplification13.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.71073899284975746 \cdot 10^{142}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.38415960656676321 \cdot 10^{-97}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \log \left(e^{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right)\\ \mathbf{elif}\;y \le 1.1196094733529732 \cdot 10^{-55}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 28161126.972837694:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \log \left(e^{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right)\\ \mathbf{elif}\;y \le 4.01162503157803517 \cdot 10^{38}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 2.16201455188241352 \cdot 10^{107}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \log \left(e^{\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right)\\ \mathbf{elif}\;y \le 4.36744780112662469 \cdot 10^{128}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))