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

Percentage Accurate: 44.6% → 56.1%

Time: 17.7s

Alternatives: 5

Speedup: 211.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \frac{\tan t_0}{\sin t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0)))) (/ (tan t_0) (sin t_0))))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return tan(t_0) / sin(t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = x / (y * 2.0d0)
    code = tan(t_0) / sin(t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return Math.tan(t_0) / Math.sin(t_0);
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	return math.tan(t_0) / math.sin(t_0)

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	return Float64(tan(t_0) / sin(t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = x / (y * 2.0);
	tmp = tan(t_0) / sin(t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 44.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \frac{\tan t_0}{\sin t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0)))) (/ (tan t_0) (sin t_0))))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return tan(t_0) / sin(t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = x / (y * 2.0d0)
    code = tan(t_0) / sin(t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return Math.tan(t_0) / Math.sin(t_0);
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	return math.tan(t_0) / math.sin(t_0)

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	return Float64(tan(t_0) / sin(t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = x / (y * 2.0);
	tmp = tan(t_0) / sin(t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 56.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left({e}^{\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)} + -1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (log1p (+ (pow E (/ 1.0 (cos (* x (/ 0.5 y))))) -1.0)))

C

double code(double x, double y) {
	return log1p((pow(((double) M_E), (1.0 / cos((x * (0.5 / y))))) + -1.0));
}

Java

public static double code(double x, double y) {
	return Math.log1p((Math.pow(Math.E, (1.0 / Math.cos((x * (0.5 / y))))) + -1.0));
}

Python

def code(x, y):
	return math.log1p((math.pow(math.e, (1.0 / math.cos((x * (0.5 / y))))) + -1.0))

Julia

function code(x, y)
	return log1p(Float64((exp(1) ^ Float64(1.0 / cos(Float64(x * Float64(0.5 / y))))) + -1.0))
end

Wolfram

code[x_, y_] := N[Log[1 + N[(N[Power[E, N[(1.0 / N[Cos[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.5%

   \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
2. Step-by-step derivation

   1. log1p-expm1-u 41.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right)} \]

   2. div-inv 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\tan \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right)\right) \]

   3. tan-quot 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   4. associate-*l/ 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}}\right)\right) \]

   5. pow1 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   6. inv-pow 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1} \cdot \color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{-1}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   7. pow-prod-up 54.4%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\left(1 + -1\right)}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   8. metadata-eval 54.4%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\color{blue}{0}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   9. metadata-eval 54.4%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{1}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   10. div-inv 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}}\right)\right) \]

   11. *-commutative 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)}\right)\right) \]

   12. associate-/r* 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)}\right)\right) \]

   13. metadata-eval 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{\color{blue}{0.5}}{y}\right)}\right)\right) \]

3. Applied egg-rr 54.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right)} \]
4. Step-by-step derivation

   1. expm1-udef 54.5%

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

5. Applied egg-rr 54.5%

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

   1. *-un-lft-identity 54.5%

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

   2. exp-prod 54.5%

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

   3. exp-1-e 54.5%

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

7. Applied egg-rr 54.5%

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

8. Final simplification 54.5%

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

Alternative 2: 56.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\left(\left(1 + e^{\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}}\right) + -1\right) + -1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (log1p (+ (+ (+ 1.0 (exp (/ 1.0 (cos (* x (/ 0.5 y)))))) -1.0) -1.0)))

C

double code(double x, double y) {
	return log1p((((1.0 + exp((1.0 / cos((x * (0.5 / y)))))) + -1.0) + -1.0));
}

Java

public static double code(double x, double y) {
	return Math.log1p((((1.0 + Math.exp((1.0 / Math.cos((x * (0.5 / y)))))) + -1.0) + -1.0));
}

Python

def code(x, y):
	return math.log1p((((1.0 + math.exp((1.0 / math.cos((x * (0.5 / y)))))) + -1.0) + -1.0))

Julia

function code(x, y)
	return log1p(Float64(Float64(Float64(1.0 + exp(Float64(1.0 / cos(Float64(x * Float64(0.5 / y)))))) + -1.0) + -1.0))
end

Wolfram

code[x_, y_] := N[Log[1 + N[(N[(N[(1.0 + N[Exp[N[(1.0 / N[Cos[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.5%

   \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
2. Step-by-step derivation

   1. log1p-expm1-u 41.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right)} \]

   2. div-inv 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\tan \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right)\right) \]

   3. tan-quot 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   4. associate-*l/ 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}}\right)\right) \]

   5. pow1 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   6. inv-pow 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1} \cdot \color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{-1}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   7. pow-prod-up 54.4%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\left(1 + -1\right)}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   8. metadata-eval 54.4%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\color{blue}{0}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   9. metadata-eval 54.4%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{1}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   10. div-inv 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}}\right)\right) \]

   11. *-commutative 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)}\right)\right) \]

   12. associate-/r* 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)}\right)\right) \]

   13. metadata-eval 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{\color{blue}{0.5}}{y}\right)}\right)\right) \]

3. Applied egg-rr 54.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right)} \]
4. Step-by-step derivation

   1. expm1-udef 54.5%

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

5. Applied egg-rr 54.5%

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

   1. log1p-expm1-u 54.5%

      \[\leadsto \mathsf{log1p}\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right)}} - 1\right) \]

   2. expm1-def 54.5%

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

   3. log1p-udef 54.5%

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

   4. add-exp-log 54.5%

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

   5. associate-+r- 54.5%

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

7. Applied egg-rr 54.5%

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

8. Final simplification 54.5%

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

Alternative 3: 56.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (log1p (expm1 (/ 1.0 (cos (* x (/ 0.5 y)))))))

C

double code(double x, double y) {
	return log1p(expm1((1.0 / cos((x * (0.5 / y))))));
}

Java

public static double code(double x, double y) {
	return Math.log1p(Math.expm1((1.0 / Math.cos((x * (0.5 / y))))));
}

Python

def code(x, y):
	return math.log1p(math.expm1((1.0 / math.cos((x * (0.5 / y))))))

Julia

function code(x, y)
	return log1p(expm1(Float64(1.0 / cos(Float64(x * Float64(0.5 / y))))))
end

Wolfram

code[x_, y_] := N[Log[1 + N[(Exp[N[(1.0 / N[Cos[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.5%

   \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
2. Step-by-step derivation

   1. log1p-expm1-u 41.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right)} \]

   2. div-inv 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\tan \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right)\right) \]

   3. tan-quot 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   4. associate-*l/ 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}}\right)\right) \]

   5. pow1 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   6. inv-pow 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1} \cdot \color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{-1}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   7. pow-prod-up 54.4%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\left(1 + -1\right)}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   8. metadata-eval 54.4%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\color{blue}{0}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   9. metadata-eval 54.4%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{1}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   10. div-inv 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}}\right)\right) \]

   11. *-commutative 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)}\right)\right) \]

   12. associate-/r* 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)}\right)\right) \]

   13. metadata-eval 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{\color{blue}{0.5}}{y}\right)}\right)\right) \]

3. Applied egg-rr 54.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right)} \]

4. Final simplification 54.5%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right) \]

Alternative 4: 56.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.5%

   \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
2. Step-by-step derivation

   1. log1p-expm1-u 41.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right)} \]

   2. div-inv 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\tan \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right)\right) \]

   3. tan-quot 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   4. associate-*l/ 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}}\right)\right) \]

   5. pow1 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1}} \cdot \frac{1}{\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   6. inv-pow 40.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{1} \cdot \color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{-1}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   7. pow-prod-up 54.4%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\left(1 + -1\right)}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   8. metadata-eval 54.4%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\sin \left(\frac{x}{y \cdot 2}\right)}^{\color{blue}{0}}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   9. metadata-eval 54.4%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{1}}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right) \]

   10. div-inv 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \color{blue}{\left(x \cdot \frac{1}{y \cdot 2}\right)}}\right)\right) \]

   11. *-commutative 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)}\right)\right) \]

   12. associate-/r* 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)}\right)\right) \]

   13. metadata-eval 54.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{\color{blue}{0.5}}{y}\right)}\right)\right) \]

3. Applied egg-rr 54.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\right)\right)} \]
4. Step-by-step derivation

   1. expm1-udef 54.5%

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

5. Applied egg-rr 54.5%

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

6. Taylor expanded in x around inf 54.4%

   \[\leadsto \color{blue}{\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}} \]
7. Step-by-step derivation

   1. *-commutative 54.4%

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

   2. metadata-eval 54.4%

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

   3. times-frac 54.4%

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

   4. *-rgt-identity 54.4%

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

   5. associate-*r/ 54.5%

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

8. Simplified 54.5%

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

9. Final simplification 54.5%

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

Alternative 5: 56.1% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 41.5%

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

2. Taylor expanded in x around 0 54.5%

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

3. Final simplification 54.5%

   \[\leadsto 1 \]

Developer target: 55.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ t_1 := \sin t_0\\ \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \log \left(e^{\cos t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))) (t_1 (sin t_0)))
   (if (< y -1.2303690911306994e+114)
     1.0
     (if (< y -9.102852406811914e-222)
       (/ t_1 (* t_1 (log (exp (cos t_0)))))
       1.0))))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = sin(t_0);
	double tmp;
	if (y < -1.2303690911306994e+114) {
		tmp = 1.0;
	} else if (y < -9.102852406811914e-222) {
		tmp = t_1 / (t_1 * log(exp(cos(t_0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (y * 2.0d0)
    t_1 = sin(t_0)
    if (y < (-1.2303690911306994d+114)) then
        tmp = 1.0d0
    else if (y < (-9.102852406811914d-222)) then
        tmp = t_1 / (t_1 * log(exp(cos(t_0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = Math.sin(t_0);
	double tmp;
	if (y < -1.2303690911306994e+114) {
		tmp = 1.0;
	} else if (y < -9.102852406811914e-222) {
		tmp = t_1 / (t_1 * Math.log(Math.exp(Math.cos(t_0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	t_1 = math.sin(t_0)
	tmp = 0
	if y < -1.2303690911306994e+114:
		tmp = 1.0
	elif y < -9.102852406811914e-222:
		tmp = t_1 / (t_1 * math.log(math.exp(math.cos(t_0))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	t_1 = sin(t_0)
	tmp = 0.0
	if (y < -1.2303690911306994e+114)
		tmp = 1.0;
	elseif (y < -9.102852406811914e-222)
		tmp = Float64(t_1 / Float64(t_1 * log(exp(cos(t_0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * 2.0);
	t_1 = sin(t_0);
	tmp = 0.0;
	if (y < -1.2303690911306994e+114)
		tmp = 1.0;
	elseif (y < -9.102852406811914e-222)
		tmp = t_1 / (t_1 * log(exp(cos(t_0))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[Less[y, -1.2303690911306994e+114], 1.0, If[Less[y, -9.102852406811914e-222], N[(t$95$1 / N[(t$95$1 * N[Log[N[Exp[N[Cos[t$95$0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]]

Reproduce

herbie shell --seed 2023230 
(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)))))