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

Average Accuracy: 44.2% → 57.1%

Time: 14.8s

Precision: binary64

Cost: 33476

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 1.8:\\ \;\;\;\;\frac{1}{\log \left(e^{1 + \cos \left(0.5 \cdot \frac{x}{y}\right)}\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot e^{{\left(\frac{x}{y}\right)}^{2} \cdot -0.0625} + -1}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 1.8)
     (/ 1.0 (+ (log (exp (+ 1.0 (cos (* 0.5 (/ x y)))))) -1.0))
     (/ 1.0 (+ (* 2.0 (exp (* (pow (/ x y) 2.0) -0.0625))) -1.0)))))

C

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

↓

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 1.8) {
		tmp = 1.0 / (log(exp((1.0 + cos((0.5 * (x / y)))))) + -1.0);
	} else {
		tmp = 1.0 / ((2.0 * exp((pow((x / y), 2.0) * -0.0625))) + -1.0);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * 2.0d0)
    if ((tan(t_0) / sin(t_0)) <= 1.8d0) then
        tmp = 1.0d0 / (log(exp((1.0d0 + cos((0.5d0 * (x / y)))))) + (-1.0d0))
    else
        tmp = 1.0d0 / ((2.0d0 * exp((((x / y) ** 2.0d0) * (-0.0625d0)))) + (-1.0d0))
    end if
    code = tmp
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) {
	double t_0 = x / (y * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 1.8) {
		tmp = 1.0 / (Math.log(Math.exp((1.0 + Math.cos((0.5 * (x / y)))))) + -1.0);
	} else {
		tmp = 1.0 / ((2.0 * Math.exp((Math.pow((x / y), 2.0) * -0.0625))) + -1.0);
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = x / (y * 2.0)
	tmp = 0
	if (math.tan(t_0) / math.sin(t_0)) <= 1.8:
		tmp = 1.0 / (math.log(math.exp((1.0 + math.cos((0.5 * (x / y)))))) + -1.0)
	else:
		tmp = 1.0 / ((2.0 * math.exp((math.pow((x / y), 2.0) * -0.0625))) + -1.0)
	return tmp

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)
	t_0 = Float64(x / Float64(y * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 1.8)
		tmp = Float64(1.0 / Float64(log(exp(Float64(1.0 + cos(Float64(0.5 * Float64(x / y)))))) + -1.0));
	else
		tmp = Float64(1.0 / Float64(Float64(2.0 * exp(Float64((Float64(x / y) ^ 2.0) * -0.0625))) + -1.0));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	t_0 = x / (y * 2.0);
	tmp = 0.0;
	if ((tan(t_0) / sin(t_0)) <= 1.8)
		tmp = 1.0 / (log(exp((1.0 + cos((0.5 * (x / y)))))) + -1.0);
	else
		tmp = 1.0 / ((2.0 * exp((((x / y) ^ 2.0) * -0.0625))) + -1.0);
	end
	tmp_2 = tmp;
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_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 1.8], N[(1.0 / N[(N[Log[N[Exp[N[(1.0 + N[Cos[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(2.0 * N[Exp[N[(N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	44.2%
Target	55.0%
Herbie	57.1%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 1.80000000000000004

   1. Initial program 62.9%

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

   2. Taylor expanded in x around inf 62.9%

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

   3. Applied egg-rr 62.9%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\cos \left(0.5 \cdot \frac{x}{y}\right)\right)} - 1}} \]

   4. Applied egg-rr 62.9%

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

   if 1.80000000000000004 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

   1. Initial program 3.2%

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

   2. Taylor expanded in x around inf 40.3%

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

   3. Applied egg-rr 40.3%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\cos \left(0.5 \cdot \frac{x}{y}\right)\right)} - 1}} \]

   4. Taylor expanded in x around 0 44.0%

      \[\leadsto \frac{1}{e^{\color{blue}{\log 2 + -0.0625 \cdot \frac{{x}^{2}}{{y}^{2}}}} - 1} \]

   5. Simplified 44.5%

      \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{fma}\left(\frac{x}{y} \cdot \frac{x}{y}, -0.0625, \log 2\right)}} - 1} \]
      Proof

      [Start] 44.0	\[ \frac{1}{e^{\log 2 + -0.0625 \cdot \frac{{x}^{2}}{{y}^{2}}} - 1} \]
      +-commutative [=>] 44.0	\[ \frac{1}{e^{\color{blue}{-0.0625 \cdot \frac{{x}^{2}}{{y}^{2}} + \log 2}} - 1} \]
      *-commutative [=>] 44.0	\[ \frac{1}{e^{\color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot -0.0625} + \log 2} - 1} \]
      fma-def [=>] 44.0	\[ \frac{1}{e^{\color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{{y}^{2}}, -0.0625, \log 2\right)}} - 1} \]
      unpow2 [=>] 44.0	\[ \frac{1}{e^{\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{{y}^{2}}, -0.0625, \log 2\right)} - 1} \]
      unpow2 [=>] 44.0	\[ \frac{1}{e^{\mathsf{fma}\left(\frac{x \cdot x}{\color{blue}{y \cdot y}}, -0.0625, \log 2\right)} - 1} \]
      times-frac [=>] 44.5	\[ \frac{1}{e^{\mathsf{fma}\left(\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -0.0625, \log 2\right)} - 1} \]

   6. Applied egg-rr 44.5%

      \[\leadsto \frac{1}{\color{blue}{e^{{\left(\frac{x}{y}\right)}^{2} \cdot -0.0625} \cdot 2} - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 1.8:\\ \;\;\;\;\frac{1}{\log \left(e^{1 + \cos \left(0.5 \cdot \frac{x}{y}\right)}\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot e^{{\left(\frac{x}{y}\right)}^{2} \cdot -0.0625} + -1}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	57.1%
Cost	33220

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 1.8:\\ \;\;\;\;\frac{1}{\log \left(e^{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot e^{{\left(\frac{x}{y}\right)}^{2} \cdot -0.0625} + -1}\\ \end{array} \]

Alternative 2
Accuracy	57.1%
Cost	27140

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 1.8:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot e^{{\left(\frac{x}{y}\right)}^{2} \cdot -0.0625} + -1}\\ \end{array} \]

Alternative 3
Accuracy	57.1%
Cost	20420

\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \mathbf{if}\;\frac{\tan t_0}{\sin t_0} \leq 1.528:\\ \;\;\;\;\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Accuracy	6.1%
Cost	64

\[-2 \]

Alternative 5
Accuracy	55.6%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023141 
(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)))))