Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Percentage Accurate: 50.9% → 99.9%

Time: 12.7s

Precision: binary64

Cost: 26688

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(x, 2 \cdot y\right)\\ {\left(\frac{x}{t_0}\right)}^{4} - {\left(\frac{2 \cdot y}{t_0}\right)}^{4} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot x (* 2.0 y))))
   (- (pow (/ x t_0) 4.0) (pow (/ (* 2.0 y) t_0) 4.0))))

C

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 t_0 = hypot(x, (2.0 * y));
	return pow((x / t_0), 4.0) - pow(((2.0 * y) / t_0), 4.0);
}

Java

public static double code(double x, double y) {
	return ((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y));
}

↓

public static double code(double x, double y) {
	double t_0 = Math.hypot(x, (2.0 * y));
	return Math.pow((x / t_0), 4.0) - Math.pow(((2.0 * y) / t_0), 4.0);
}

Python

def code(x, y):
	return ((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y))

↓

def code(x, y):
	t_0 = math.hypot(x, (2.0 * y))
	return math.pow((x / t_0), 4.0) - math.pow(((2.0 * y) / t_0), 4.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * x) - Float64(Float64(y * 4.0) * y)) / Float64(Float64(x * x) + Float64(Float64(y * 4.0) * y)))
end

↓

function code(x, y)
	t_0 = hypot(x, Float64(2.0 * y))
	return Float64((Float64(x / t_0) ^ 4.0) - (Float64(Float64(2.0 * y) / t_0) ^ 4.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y));
end

↓

function tmp = code(x, y)
	t_0 = hypot(x, (2.0 * y));
	tmp = ((x / t_0) ^ 4.0) - (((2.0 * y) / t_0) ^ 4.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(2.0 * y), $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[Power[N[(x / t$95$0), $MachinePrecision], 4.0], $MachinePrecision] - N[Power[N[(N[(2.0 * y), $MachinePrecision] / t$95$0), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]]

Target

Original	50.9%
Target	51.4%
Herbie	99.9%

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

Derivation

1. Initial program 54.3%

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

2. Applied egg-rr 54.3%

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

   [Start] 54.3%	\[ \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   div-sub [=>] 54.3%	\[ \color{blue}{\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]
   fma-def [=>] 54.3%	\[ \frac{x \cdot x}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   *-commutative [=>] 54.3%	\[ \frac{x \cdot x}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   *-commutative [=>] 54.3%	\[ \frac{x \cdot x}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)} - \frac{\color{blue}{y \cdot \left(y \cdot 4\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   fma-def [=>] 54.3%	\[ \frac{x \cdot x}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)} - \frac{y \cdot \left(y \cdot 4\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]
   *-commutative [=>] 54.3%	\[ \frac{x \cdot x}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)} - \frac{y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

3. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left({\left(\frac{x}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2}\right)}^{2} + \left(-{\left({\left(\frac{y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2}\right)}^{2}\right)} \]
   Step-by-step derivation

   [Start] 54.3%	\[ \frac{x \cdot x}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)} - \frac{y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)} \]
   add-sqr-sqrt [=>] 28.5%	\[ \color{blue}{\sqrt{\frac{x \cdot x}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)} - \frac{y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \cdot \sqrt{\frac{x \cdot x}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)} - \frac{y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}}} \]
   pow2 [=>] 28.5%	\[ \color{blue}{{\left(\sqrt{\frac{x \cdot x}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)} - \frac{y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}}\right)}^{2}} \]

4. Simplified 99.6%

   \[\leadsto \color{blue}{{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{4} - {\left(\frac{2 \cdot y}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{4}} \]
   Step-by-step derivation

   [Start] 99.6%	\[ {\left({\left(\frac{x}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2}\right)}^{2} + \left(-{\left({\left(\frac{y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2}\right)}^{2}\right) \]
   unsub-neg [=>] 99.6%	\[ \color{blue}{{\left({\left(\frac{x}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2}\right)}^{2} - {\left({\left(\frac{y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2}\right)}^{2}} \]
   unpow2 [=>] 99.6%	\[ \color{blue}{{\left(\frac{x}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2} \cdot {\left(\frac{x}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2}} - {\left({\left(\frac{y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2}\right)}^{2} \]
   pow-sqr [=>] 99.6%	\[ \color{blue}{{\left(\frac{x}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{\left(2 \cdot 2\right)}} - {\left({\left(\frac{y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2}\right)}^{2} \]
   *-commutative [=>] 99.6%	\[ {\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot y}\right)}\right)}^{\left(2 \cdot 2\right)} - {\left({\left(\frac{y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2}\right)}^{2} \]
   metadata-eval [=>] 99.6%	\[ {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{\color{blue}{4}} - {\left({\left(\frac{y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2}\right)}^{2} \]
   unpow2 [=>] 99.6%	\[ {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{4} - \color{blue}{{\left(\frac{y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2} \cdot {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2}} \]
   pow-sqr [=>] 99.6%	\[ {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{4} - \color{blue}{{\left(\frac{y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{\left(2 \cdot 2\right)}} \]
   *-commutative [=>] 99.6%	\[ {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{4} - {\left(\frac{\color{blue}{2 \cdot y}}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{\left(2 \cdot 2\right)} \]
   *-commutative [=>] 99.6%	\[ {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{4} - {\left(\frac{2 \cdot y}{\mathsf{hypot}\left(x, \color{blue}{2 \cdot y}\right)}\right)}^{\left(2 \cdot 2\right)} \]
   metadata-eval [=>] 99.6%	\[ {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{4} - {\left(\frac{2 \cdot y}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{\color{blue}{4}} \]

5. Final simplification 99.6%

   \[\leadsto {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{4} - {\left(\frac{2 \cdot y}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{4} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	26688

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(x, 2 \cdot y\right)\\ {\left(\frac{x}{t_0}\right)}^{4} - {\left(\frac{2 \cdot y}{t_0}\right)}^{4} \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	20864

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(x, 2 \cdot y\right)\\ \left(\frac{x}{t_0} + 2 \cdot \frac{y}{t_0}\right) \cdot \frac{x - 2 \cdot y}{t_0} \end{array} \]

Alternative 3
Accuracy	81.3%
Cost	20616

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \mathsf{fma}\left(x, x, t_0\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+212}:\\ \;\;\;\;\frac{x \cdot x}{t_1} - \frac{t_0}{t_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{4} - {\left(\frac{y \cdot -2}{x}\right)}^{4}\\ \end{array} \]

Alternative 4
Accuracy	81.3%
Cost	14920

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \mathsf{fma}\left(x, x, t_0\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+212}:\\ \;\;\;\;\frac{x \cdot x}{t_1} - \frac{t_0}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 + \left(-1 + {\left(\frac{y}{x}\right)}^{2}\right), -8, 1\right)\\ \end{array} \]

Alternative 5
Accuracy	81.3%
Cost	13960

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+212}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 + \left(-1 + {\left(\frac{y}{x}\right)}^{2}\right), -8, 1\right)\\ \end{array} \]

Alternative 6
Accuracy	81.3%
Cost	7496

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+212}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot \frac{y}{x}, -8, 1\right)\\ \end{array} \]

Alternative 7
Accuracy	81.0%
Cost	7236

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+212}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	78.4%
Cost	1744

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{if}\;x \leq -2.95 \cdot 10^{+110}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-76}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+59}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Accuracy	71.5%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+58}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Accuracy	50.6%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023271 
(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))))