Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5

Average Error: 35.1 → 28.0

Time: 16.5s

Precision: binary64

Cost: 6848

Math

\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

↓

\[\frac{1}{\cos \left(\frac{0.5}{\frac{y}{x}}\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))

↓

(FPCore (x y) :precision binary64 (/ 1.0 (cos (/ 0.5 (/ y x)))))

C

double code(double x, double y) {
	return tan((x / (y * 2.0))) / sin((x / (y * 2.0)));
}

↓

double code(double x, double y) {
	return 1.0 / cos((0.5 / (y / x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = tan((x / (y * 2.0d0))) / sin((x / (y * 2.0d0)))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / cos((0.5d0 / (y / x)))
end function

Java

public static double code(double x, double y) {
	return Math.tan((x / (y * 2.0))) / Math.sin((x / (y * 2.0)));
}

↓

public static double code(double x, double y) {
	return 1.0 / Math.cos((0.5 / (y / x)));
}

Python

def code(x, y):
	return math.tan((x / (y * 2.0))) / math.sin((x / (y * 2.0)))

↓

def code(x, y):
	return 1.0 / math.cos((0.5 / (y / x)))

Julia

function code(x, y)
	return Float64(tan(Float64(x / Float64(y * 2.0))) / sin(Float64(x / Float64(y * 2.0))))
end

↓

function code(x, y)
	return Float64(1.0 / cos(Float64(0.5 / Float64(y / x))))
end

MATLAB

function tmp = code(x, y)
	tmp = tan((x / (y * 2.0))) / sin((x / (y * 2.0)));
end

↓

function tmp = code(x, y)
	tmp = 1.0 / cos((0.5 / (y / x)));
end

Wolfram

code[x_, y_] := N[(N[Tan[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(1.0 / N[Cos[N[(0.5 / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Target

Original	35.1
Target	28.4
Herbie	28.0

\[\begin{array}{l} \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Derivation

1. Initial program 35.1

   \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

2. Taylor expanded in x around inf 28.0

   \[\leadsto \color{blue}{\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}} \]

3. Applied egg-rr 28.0

   \[\leadsto \frac{1}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos \left(0.5 \cdot \frac{x}{y}\right)\right)\right)}} \]

4. Taylor expanded in x around inf 28.0

   \[\leadsto \frac{1}{\color{blue}{\cos \left(0.5 \cdot \frac{x}{y}\right)}} \]

5. Simplified 28.0

   \[\leadsto \frac{1}{\color{blue}{\cos \left(\frac{0.5}{y} \cdot x\right)}} \]
   Proof

   [Start] 28.0	\[ \frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)} \]
   associate-*r/ [=>] 28.0	\[ \frac{1}{\cos \color{blue}{\left(\frac{0.5 \cdot x}{y}\right)}} \]
   associate-/l* [=>] 28.0	\[ \frac{1}{\cos \color{blue}{\left(\frac{0.5}{\frac{y}{x}}\right)}} \]
   associate-/r/ [=>] 28.0	\[ \frac{1}{\cos \color{blue}{\left(\frac{0.5}{y} \cdot x\right)}} \]

6. Applied egg-rr 28.0

   \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{0.5}{\frac{y}{x}}\right)}} \]

7. Final simplification 28.0

   \[\leadsto \frac{1}{\cos \left(\frac{0.5}{\frac{y}{x}}\right)} \]

Alternatives

Alternative 1
Error	28.0
Cost	6848

\[\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)} \]

Alternative 2
Error	28.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023031 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< y -1.2303690911306994e+114) 1.0 (if (< y -9.102852406811914e-222) (/ (sin (/ x (* y 2.0))) (* (sin (/ x (* y 2.0))) (log (exp (cos (/ x (* y 2.0))))))) 1.0))

  (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))