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

Average Error: 31.7 → 0.1

Time: 15.2s

Precision: binary64

Cost: 39488

Math

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

↓

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

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 (* y 2.0))))
   (log (exp (- (pow (/ x t_0) 2.0) (pow (* 2.0 (/ y t_0)) 2.0))))))

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, (y * 2.0));
	return log(exp((pow((x / t_0), 2.0) - pow((2.0 * (y / t_0)), 2.0))));
}

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, (y * 2.0));
	return Math.log(Math.exp((Math.pow((x / t_0), 2.0) - Math.pow((2.0 * (y / t_0)), 2.0))));
}

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, (y * 2.0))
	return math.log(math.exp((math.pow((x / t_0), 2.0) - math.pow((2.0 * (y / t_0)), 2.0))))

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(y * 2.0))
	return log(exp(Float64((Float64(x / t_0) ^ 2.0) - (Float64(2.0 * Float64(y / t_0)) ^ 2.0))))
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, (y * 2.0));
	tmp = log(exp((((x / t_0) ^ 2.0) - ((2.0 * (y / t_0)) ^ 2.0))));
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[(y * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]}, N[Log[N[Exp[N[(N[Power[N[(x / t$95$0), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(2.0 * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Target

Original	31.7
Target	31.4
Herbie	0.1

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

Derivation

1. Initial program 31.7

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

2. Applied egg-rr 31.4

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

3. Simplified 31.4

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

   [Start] 31.4	\[ \frac{x \cdot x}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)} + \left(-\frac{y \cdot 4}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)} \cdot y\right) \]
   sub-neg [<=] 31.4	\[ \color{blue}{\frac{x \cdot x}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)} - \frac{y \cdot 4}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)} \cdot y} \]
   associate-/l* [=>] 31.0	\[ \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x}}} - \frac{y \cdot 4}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)} \cdot y \]
   associate-*l/ [=>] 31.4	\[ \frac{x}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x}} - \color{blue}{\frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   *-commutative [<=] 31.4	\[ \frac{x}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x}} - \frac{\color{blue}{y \cdot \left(y \cdot 4\right)}}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)} \]

4. Applied egg-rr 32.5

   \[\leadsto \color{blue}{\left(\frac{\sqrt{x}}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \sqrt{x} + \frac{y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right) \cdot \left(\frac{\sqrt{x}}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \sqrt{x} - \frac{y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)} \]

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.2
Cost	20864

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

Alternative 2
Error	3.7
Cost	8776

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;1 + \left(\frac{y \cdot -4}{\frac{x}{\frac{y}{x}}} + \frac{y}{x} \cdot \frac{y \cdot -4}{x}\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\frac{x}{x + \frac{t_0}{x}} + \frac{y \cdot \left(y \cdot -4\right)}{\mathsf{fma}\left(x, x, t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{\frac{y}{\frac{x}{y}}}, -1\right)\\ \end{array} \]

Alternative 3
Error	12.4
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+228}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{y \cdot -4}{\frac{x}{\frac{y}{x}}} + \frac{y}{x} \cdot \frac{y \cdot -4}{x}\right)\\ \end{array} \]

Alternative 4
Error	12.4
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{\frac{y}{\frac{x}{y}}}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+228}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{y \cdot -4}{\frac{x}{\frac{y}{x}}} + \frac{y}{x} \cdot \frac{y \cdot -4}{x}\right)\\ \end{array} \]

Alternative 5
Error	12.6
Cost	1864

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-170}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+228}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{y \cdot -4}{\frac{x}{\frac{y}{x}}} + \frac{y}{x} \cdot \frac{y \cdot -4}{x}\right)\\ \end{array} \]

Alternative 6
Error	12.6
Cost	1736

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-170}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+228}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(2 + \frac{y}{x} \cdot \left(-4 \cdot \frac{y}{x}\right)\right)\\ \end{array} \]

Alternative 7
Error	17.5
Cost	1360

\[\begin{array}{l} t_0 := 1 + y \cdot \left(\frac{y}{x} \cdot \frac{-4}{x}\right)\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{+131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+86}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-18}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(2 + \frac{y}{x} \cdot \left(-4 \cdot \frac{y}{x}\right)\right)\\ \end{array} \]

Alternative 8
Error	17.5
Cost	1234

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+131} \lor \neg \left(x \leq -1.4 \cdot 10^{+86}\right) \land \left(x \leq -2.05 \cdot 10^{-86} \lor \neg \left(x \leq 1.45 \cdot 10^{-16}\right)\right):\\ \;\;\;\;1 + y \cdot \left(\frac{y}{x} \cdot \frac{-4}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9
Error	18.0
Cost	1100

\[\begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+147}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+86}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-85}:\\ \;\;\;\;1 + \frac{-4 \cdot \left(y \cdot y\right)}{x \cdot x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-18}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	18.0
Cost	592

\[\begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+147}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -4 \cdot 10^{+85}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-85}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-17}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	32.1
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022356 
(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))))