Average Error: 31.2 → 12.6
Time: 3.0s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4.0\right) \cdot y}{x \cdot x + \left(y \cdot 4.0\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4.0\right) \le 5.256606017817734 \cdot 10^{-308}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4.0\right) \le 1.6350568935112376 \cdot 10^{+269}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4.0\right)}{x \cdot x + y \cdot \left(y \cdot 4.0\right)}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4.0\right) \le 2.4786518810190947 \cdot 10^{+298}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1.0\\ \end{array}\]
\frac{x \cdot x - \left(y \cdot 4.0\right) \cdot y}{x \cdot x + \left(y \cdot 4.0\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(y \cdot 4.0\right) \le 5.256606017817734 \cdot 10^{-308}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \cdot \left(y \cdot 4.0\right) \le 1.6350568935112376 \cdot 10^{+269}:\\
\;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4.0\right)}{x \cdot x + y \cdot \left(y \cdot 4.0\right)}\\

\mathbf{elif}\;y \cdot \left(y \cdot 4.0\right) \le 2.4786518810190947 \cdot 10^{+298}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;-1.0\\

\end{array}
double f(double x, double y) {
        double r11754125 = x;
        double r11754126 = r11754125 * r11754125;
        double r11754127 = y;
        double r11754128 = 4.0;
        double r11754129 = r11754127 * r11754128;
        double r11754130 = r11754129 * r11754127;
        double r11754131 = r11754126 - r11754130;
        double r11754132 = r11754126 + r11754130;
        double r11754133 = r11754131 / r11754132;
        return r11754133;
}
↓
double f(double x, double y) {
        double r11754134 = y;
        double r11754135 = 4.0;
        double r11754136 = r11754134 * r11754135;
        double r11754137 = r11754134 * r11754136;
        double r11754138 = 5.256606017817734e-308;
        bool r11754139 = r11754137 <= r11754138;
        double r11754140 = 1.0;
        double r11754141 = 1.6350568935112376e+269;
        bool r11754142 = r11754137 <= r11754141;
        double r11754143 = x;
        double r11754144 = r11754143 * r11754143;
        double r11754145 = r11754144 - r11754137;
        double r11754146 = r11754144 + r11754137;
        double r11754147 = r11754145 / r11754146;
        double r11754148 = 2.4786518810190947e+298;
        bool r11754149 = r11754137 <= r11754148;
        double r11754150 = -1.0;
        double r11754151 = r11754149 ? r11754140 : r11754150;
        double r11754152 = r11754142 ? r11754147 : r11754151;
        double r11754153 = r11754139 ? r11754140 : r11754152;
        return r11754153;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.2
Target31.2
Herbie12.6
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4.0\right) \cdot y}{x \cdot x + \left(y \cdot 4.0\right) \cdot y} \lt 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4.0} - \frac{\left(y \cdot y\right) \cdot 4.0}{x \cdot x + \left(y \cdot y\right) \cdot 4.0}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4.0}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4.0}{x \cdot x + \left(y \cdot y\right) \cdot 4.0}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (* y 4.0) y) < 5.256606017817734e-308 or 1.6350568935112376e+269 < (* (* y 4.0) y) < 2.4786518810190947e+298

    1. Initial program 29.5

      \[\frac{x \cdot x - \left(y \cdot 4.0\right) \cdot y}{x \cdot x + \left(y \cdot 4.0\right) \cdot y}\]

    2. Taylor expanded around inf 11.5

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

    if 5.256606017817734e-308 < (* (* y 4.0) y) < 1.6350568935112376e+269

    1. Initial program 15.5

      \[\frac{x \cdot x - \left(y \cdot 4.0\right) \cdot y}{x \cdot x + \left(y \cdot 4.0\right) \cdot y}\]

    if 2.4786518810190947e+298 < (* (* y 4.0) y)

    1. Initial program 61.7

      \[\frac{x \cdot x - \left(y \cdot 4.0\right) \cdot y}{x \cdot x + \left(y \cdot 4.0\right) \cdot y}\]

    2. Taylor expanded around 0 8.4

      \[\leadsto \color{blue}{-1.0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4.0\right) \le 5.256606017817734 \cdot 10^{-308}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4.0\right) \le 1.6350568935112376 \cdot 10^{+269}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4.0\right)}{x \cdot x + y \cdot \left(y \cdot 4.0\right)}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4.0\right) \le 2.4786518810190947 \cdot 10^{+298}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1.0\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))