Average Error: 32.1 → 12.9
Time: 3.3s
Precision: binary64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.040004113947438 \cdot 10^{-200}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 1.2864016597837409 \cdot 10^{-130}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 8.302497373615917 \cdot 10^{-66}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 3.4612524063184616 \cdot 10^{+242}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot \left(y \cdot 4\right)}{x \cdot x - y \cdot \left(y \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\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}\;x \cdot x \leq 4.040004113947438 \cdot 10^{-200}:\\
\;\;\;\;-1\\

\mathbf{elif}\;x \cdot x \leq 1.2864016597837409 \cdot 10^{-130}:\\
\;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\

\mathbf{elif}\;x \cdot x \leq 8.302497373615917 \cdot 10^{-66}:\\
\;\;\;\;-1\\

\mathbf{elif}\;x \cdot x \leq 3.4612524063184616 \cdot 10^{+242}:\\
\;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot \left(y \cdot 4\right)}{x \cdot x - y \cdot \left(y \cdot 4\right)}}\\

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

\end{array}
(FPCore (x y)
 :precision binary64
 (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 4.040004113947438e-200)
   -1.0
   (if (<= (* x x) 1.2864016597837409e-130)
     (/ (- (* x x) (* y (* y 4.0))) (+ (* x x) (* y (* y 4.0))))
     (if (<= (* x x) 8.302497373615917e-66)
       -1.0
       (if (<= (* x x) 3.4612524063184616e+242)
         (/ 1.0 (/ (+ (* x x) (* y (* y 4.0))) (- (* x x) (* y (* y 4.0)))))
         1.0)))))
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 tmp;
	if ((x * x) <= 4.040004113947438e-200) {
		tmp = -1.0;
	} else if ((x * x) <= 1.2864016597837409e-130) {
		tmp = ((x * x) - (y * (y * 4.0))) / ((x * x) + (y * (y * 4.0)));
	} else if ((x * x) <= 8.302497373615917e-66) {
		tmp = -1.0;
	} else if ((x * x) <= 3.4612524063184616e+242) {
		tmp = 1.0 / (((x * x) + (y * (y * 4.0))) / ((x * x) - (y * (y * 4.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.1
Target31.8
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (*.f64 x x) < 4.0400041139474382e-200 or 1.28640165978374089e-130 < (*.f64 x x) < 8.3024973736159172e-66

    1. Initial program 26.0

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

    2. Taylor expanded around 0 12.5

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

    if 4.0400041139474382e-200 < (*.f64 x x) < 1.28640165978374089e-130

    1. Initial program 17.7

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

    if 8.3024973736159172e-66 < (*.f64 x x) < 3.46125240631846158e242

    1. Initial program 16.2

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

    3. Applied clear-num_binary64_1439916.2

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

    4. Simplified16.2

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

    if 3.46125240631846158e242 < (*.f64 x x)

    1. Initial program 55.3

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

    2. Taylor expanded around inf 10.0

      \[\leadsto \color{blue}{1}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.040004113947438 \cdot 10^{-200}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 1.2864016597837409 \cdot 10^{-130}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 8.302497373615917 \cdot 10^{-66}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 3.4612524063184616 \cdot 10^{+242}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot \left(y \cdot 4\right)}{x \cdot x - y \cdot \left(y \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :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)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

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