Average Error: 31.9 → 14.3
Time: 1.9s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 4.6888524311529329 \cdot 10^{-150}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.03257274616522845 \cdot 10^{-117}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.80478009492825009 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 7.769551355955981 \cdot 10^{204}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
\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 4.6888524311529329 \cdot 10^{-150}:\\
\;\;\;\;1\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.03257274616522845 \cdot 10^{-117}:\\
\;\;\;\;\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 3.80478009492825009 \cdot 10^{-5}:\\
\;\;\;\;1\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 7.769551355955981 \cdot 10^{204}:\\
\;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\

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

\end{array}
double f(double x, double y) {
        double r600167 = x;
        double r600168 = r600167 * r600167;
        double r600169 = y;
        double r600170 = 4.0;
        double r600171 = r600169 * r600170;
        double r600172 = r600171 * r600169;
        double r600173 = r600168 - r600172;
        double r600174 = r600168 + r600172;
        double r600175 = r600173 / r600174;
        return r600175;
}
↓
double f(double x, double y) {
        double r600176 = y;
        double r600177 = 4.0;
        double r600178 = r600176 * r600177;
        double r600179 = r600178 * r600176;
        double r600180 = 4.688852431152933e-150;
        bool r600181 = r600179 <= r600180;
        double r600182 = 1.0;
        double r600183 = 3.0325727461652284e-117;
        bool r600184 = r600179 <= r600183;
        double r600185 = x;
        double r600186 = r600185 * r600185;
        double r600187 = r600186 - r600179;
        double r600188 = r600186 + r600179;
        double r600189 = r600187 / r600188;
        double r600190 = 3.80478009492825e-05;
        bool r600191 = r600179 <= r600190;
        double r600192 = 7.769551355955981e+204;
        bool r600193 = r600179 <= r600192;
        double r600194 = -1.0;
        double r600195 = r600193 ? r600189 : r600194;
        double r600196 = r600191 ? r600182 : r600195;
        double r600197 = r600184 ? r600189 : r600196;
        double r600198 = r600181 ? r600182 : r600197;
        return r600198;
}

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
Herbie14.3
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.974323384962678118:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (* y 4.0) y) < 4.688852431152933e-150 or 3.0325727461652284e-117 < (* (* y 4.0) y) < 3.80478009492825e-05

    1. Initial program 24.3

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

    2. Taylor expanded around inf 15.4

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

    if 4.688852431152933e-150 < (* (* y 4.0) y) < 3.0325727461652284e-117 or 3.80478009492825e-05 < (* (* y 4.0) y) < 7.769551355955981e+204

    1. Initial program 15.8

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

    if 7.769551355955981e+204 < (* (* y 4.0) y)

    1. Initial program 52.1

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

    2. Taylor expanded around 0 11.8

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

  4. Final simplification14.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 4.6888524311529329 \cdot 10^{-150}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.03257274616522845 \cdot 10^{-117}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.80478009492825009 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 7.769551355955981 \cdot 10^{204}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +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))))