Average Error: 31.8 → 14.1
Time: 2.6s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.4362841072323272 \cdot 10^{150}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -3.8396732637812812 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}{\frac{1}{x \cdot x - \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;y \le 1.41284087263746274 \cdot 10^{-110}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 4.54642378775940041 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}{\frac{1}{x \cdot x - \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;y \le 2.97781259123946345 \cdot 10^{57}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
\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 -3.4362841072323272 \cdot 10^{150}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -3.8396732637812812 \cdot 10^{-74}:\\
\;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}{\frac{1}{x \cdot x - \left(y \cdot 4\right) \cdot y}}\\

\mathbf{elif}\;y \le 1.41284087263746274 \cdot 10^{-110}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 4.54642378775940041 \cdot 10^{-26}:\\
\;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}{\frac{1}{x \cdot x - \left(y \cdot 4\right) \cdot y}}\\

\mathbf{elif}\;y \le 2.97781259123946345 \cdot 10^{57}:\\
\;\;\;\;1\\

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

\end{array}
double code(double x, double y) {
	return (((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y)));
}
↓
double code(double x, double y) {
	double temp;
	if ((y <= -3.436284107232327e+150)) {
		temp = -1.0;
	} else {
		double temp_1;
		if ((y <= -3.839673263781281e-74)) {
			temp_1 = ((1.0 / fma(x, x, ((y * 4.0) * y))) / (1.0 / ((x * x) - ((y * 4.0) * y))));
		} else {
			double temp_2;
			if ((y <= 1.4128408726374627e-110)) {
				temp_2 = 1.0;
			} else {
				double temp_3;
				if ((y <= 4.5464237877594004e-26)) {
					temp_3 = ((1.0 / fma(x, x, ((y * 4.0) * y))) / (1.0 / ((x * x) - ((y * 4.0) * y))));
				} else {
					double temp_4;
					if ((y <= 2.9778125912394635e+57)) {
						temp_4 = 1.0;
					} else {
						temp_4 = -1.0;
					}
					temp_3 = temp_4;
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

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
Herbie14.1
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.974323384962678118:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -3.436284107232327e+150 or 2.9778125912394635e+57 < y

    1. Initial program 51.9

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

    2. Taylor expanded around 0 12.1

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

    if -3.436284107232327e+150 < y < -3.839673263781281e-74 or 1.4128408726374627e-110 < y < 4.5464237877594004e-26

    1. Initial program 15.5

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

    3. Applied clear-num15.5

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x - \left(y \cdot 4\right) \cdot y}}}\]

    4. Simplified15.5

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}{x \cdot x - \left(y \cdot 4\right) \cdot y}}}\]
    5. Using strategy rm

    6. Applied div-inv15.6

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right) \cdot \frac{1}{x \cdot x - \left(y \cdot 4\right) \cdot y}}}\]

    7. Applied associate-/r*15.5

      \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}{\frac{1}{x \cdot x - \left(y \cdot 4\right) \cdot y}}}\]

    if -3.839673263781281e-74 < y < 1.4128408726374627e-110 or 4.5464237877594004e-26 < y < 2.9778125912394635e+57

    1. Initial program 24.8

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

    2. Taylor expanded around inf 14.9

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

  4. Final simplification14.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.4362841072323272 \cdot 10^{150}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -3.8396732637812812 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}{\frac{1}{x \cdot x - \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;y \le 1.41284087263746274 \cdot 10^{-110}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 4.54642378775940041 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}{\frac{1}{x \cdot x - \left(y \cdot 4\right) \cdot y}}\\ \mathbf{elif}\;y \le 2.97781259123946345 \cdot 10^{57}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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))))