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

Percentage Accurate: 44.1% → 56.5%

Time: 14.0s

Alternatives: 6

Speedup: 244.0×

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: 44.1% 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: 56.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt{0.5} \cdot \left(\frac{x\_m}{y\_m} \cdot \sqrt{0.5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+162)
   (/ 1.0 (cos (* (sqrt 0.5) (* (/ x_m y_m) (sqrt 0.5)))))
   1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+162) {
		tmp = 1.0 / cos((sqrt(0.5) * ((x_m / y_m) * sqrt(0.5))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 5d+162) then
        tmp = 1.0d0 / cos((sqrt(0.5d0) * ((x_m / y_m) * sqrt(0.5d0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+162) {
		tmp = 1.0 / Math.cos((Math.sqrt(0.5) * ((x_m / y_m) * Math.sqrt(0.5))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 5e+162:
		tmp = 1.0 / math.cos((math.sqrt(0.5) * ((x_m / y_m) * math.sqrt(0.5))))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+162)
		tmp = Float64(1.0 / cos(Float64(sqrt(0.5) * Float64(Float64(x_m / y_m) * sqrt(0.5)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 5e+162)
		tmp = 1.0 / cos((sqrt(0.5) * ((x_m / y_m) * sqrt(0.5))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+162], N[(1.0 / N[Cos[N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.9999999999999997e162

   1. Initial program 52.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 64.9

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

   5. Simplified 64.9%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. lift-/.f64 N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-pow N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

   7. Applied egg-rr 38.7%

      \[\leadsto \frac{1}{\cos \color{blue}{\left({\left({\left(\frac{x}{y \cdot 2}\right)}^{0.5}\right)}^{2}\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. unpow1/2 N/A

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

      4. sqrt-pow2 N/A

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

      5. metadata-eval N/A

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

      6. unpow1 64.9

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

      7. rem-exp-log N/A

         \[\leadsto \frac{1}{\cos \color{blue}{\left(e^{\log \left(\frac{x}{y \cdot 2}\right)}\right)}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{1}{\cos \left(e^{\log \color{blue}{\left(\frac{x}{y \cdot 2}\right)}}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{\cos \left(e^{\log \left(\frac{x}{\color{blue}{y \cdot 2}}\right)}\right)} \]

      10. associate-/r* N/A

          \[\leadsto \frac{1}{\cos \left(e^{\log \color{blue}{\left(\frac{\frac{x}{y}}{2}\right)}}\right)} \]

      11. log-div N/A

          \[\leadsto \frac{1}{\cos \left(e^{\color{blue}{\log \left(\frac{x}{y}\right) - \log 2}}\right)} \]

      12. exp-diff N/A

          \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{e^{\log \left(\frac{x}{y}\right)}}{e^{\log 2}}\right)}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{e^{\log \left(\frac{x}{y}\right)}}{e^{\log 2}}\right)}} \]

      14. lower-exp.f64 N/A

          \[\leadsto \frac{1}{\cos \left(\frac{\color{blue}{e^{\log \left(\frac{x}{y}\right)}}}{e^{\log 2}}\right)} \]

      15. lower-log.f64 N/A

          \[\leadsto \frac{1}{\cos \left(\frac{e^{\color{blue}{\log \left(\frac{x}{y}\right)}}}{e^{\log 2}}\right)} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{1}{\cos \left(\frac{e^{\log \color{blue}{\left(\frac{x}{y}\right)}}}{e^{\log 2}}\right)} \]

      17. lower-exp.f64 N/A

          \[\leadsto \frac{1}{\cos \left(\frac{e^{\log \left(\frac{x}{y}\right)}}{\color{blue}{e^{\log 2}}}\right)} \]

      18. lower-log.f64 39.6

          \[\leadsto \frac{1}{\cos \left(\frac{e^{\log \left(\frac{x}{y}\right)}}{e^{\color{blue}{\log 2}}}\right)} \]

   9. Applied egg-rr 39.6%

      \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{e^{\log \left(\frac{x}{y}\right)}}{e^{\log 2}}\right)}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{1}{\cos \left(\frac{e^{\log \color{blue}{\left(\frac{x}{y}\right)}}}{e^{\log 2}}\right)} \]

       2. rem-exp-log N/A

          \[\leadsto \frac{1}{\cos \left(\frac{\color{blue}{\frac{x}{y}}}{e^{\log 2}}\right)} \]

       3. rem-exp-log N/A

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

       4. lift-/.f64 N/A

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

       5. clear-num N/A

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

       6. lift-/.f64 N/A

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

       7. associate-/l/ N/A

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

       8. associate-/r* N/A

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

       9. metadata-eval N/A

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

       10. rem-square-sqrt N/A

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

       11. sqrt-undiv N/A

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

       12. lift-sqrt.f64 N/A

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

       13. lift-sqrt.f64 N/A

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

       14. sqrt-undiv N/A

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

       15. lift-sqrt.f64 N/A

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

       16. lift-sqrt.f64 N/A

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

       17. lift-/.f64 N/A

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

       18. div-inv N/A

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

       19. associate-*l* N/A

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

       20. *-commutative N/A

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

   11. Applied egg-rr 65.3%

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

   if 4.9999999999999997e162 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 14.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\frac{1}{\cos \left(\sqrt{0.5} \cdot \left(\frac{x}{y} \cdot \sqrt{0.5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{\frac{y\_m}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+162) (/ 1.0 (cos (/ 0.5 (/ y_m x_m)))) 1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+162) {
		tmp = 1.0 / cos((0.5 / (y_m / x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 5d+162) then
        tmp = 1.0d0 / cos((0.5d0 / (y_m / x_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+162) {
		tmp = 1.0 / Math.cos((0.5 / (y_m / x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 5e+162:
		tmp = 1.0 / math.cos((0.5 / (y_m / x_m)))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+162)
		tmp = Float64(1.0 / cos(Float64(0.5 / Float64(y_m / x_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 5e+162)
		tmp = 1.0 / cos((0.5 / (y_m / x_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+162], N[(1.0 / N[Cos[N[(0.5 / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.9999999999999997e162

   1. Initial program 52.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 64.9

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

   5. Simplified 64.9%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-/r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 64.6

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

   7. Applied egg-rr 64.6%

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

   if 4.9999999999999997e162 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 14.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 56.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{x\_m \cdot 0.5}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 5e+162) (/ 1.0 (cos (/ (* x_m 0.5) y_m))) 1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+162) {
		tmp = 1.0 / cos(((x_m * 0.5) / y_m));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 5d+162) then
        tmp = 1.0d0 / cos(((x_m * 0.5d0) / y_m))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 5e+162) {
		tmp = 1.0 / Math.cos(((x_m * 0.5) / y_m));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 5e+162:
		tmp = 1.0 / math.cos(((x_m * 0.5) / y_m))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 5e+162)
		tmp = Float64(1.0 / cos(Float64(Float64(x_m * 0.5) / y_m)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 5e+162)
		tmp = 1.0 / cos(((x_m * 0.5) / y_m));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 5e+162], N[(1.0 / N[Cos[N[(N[(x$95$m * 0.5), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 4.9999999999999997e162

   1. Initial program 52.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 64.9

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

   5. Simplified 64.9%

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

   if 4.9999999999999997e162 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 14.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 56.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+44}:\\ \;\;\;\;\frac{1}{\cos \left(x\_m \cdot \frac{0.5}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 1e+44) (/ 1.0 (cos (* x_m (/ 0.5 y_m)))) 1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+44) {
		tmp = 1.0 / cos((x_m * (0.5 / y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 1d+44) then
        tmp = 1.0d0 / cos((x_m * (0.5d0 / y_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+44) {
		tmp = 1.0 / Math.cos((x_m * (0.5 / y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 1e+44:
		tmp = 1.0 / math.cos((x_m * (0.5 / y_m)))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 1e+44)
		tmp = Float64(1.0 / cos(Float64(x_m * Float64(0.5 / y_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 1e+44)
		tmp = 1.0 / cos((x_m * (0.5 / y_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+44], N[(1.0 / N[Cos[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1.0000000000000001e44

   1. Initial program 57.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-cos.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 71.2

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

   5. Simplified 71.2%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. inv-pow N/A

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

      8. exp-to-pow N/A

         \[\leadsto \frac{1}{\cos \left(\color{blue}{e^{\log \left(y \cdot 2\right) \cdot -1}} \cdot x\right)} \]

      9. lift-log.f64 N/A

         \[\leadsto \frac{1}{\cos \left(e^{\color{blue}{\log \left(y \cdot 2\right)} \cdot -1} \cdot x\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\cos \left(e^{\color{blue}{-1 \cdot \log \left(y \cdot 2\right)}} \cdot x\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\cos \left(e^{\color{blue}{-1 \cdot \log \left(y \cdot 2\right)}} \cdot x\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{\cos \color{blue}{\left(e^{-1 \cdot \log \left(y \cdot 2\right)} \cdot x\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\cos \left(e^{\color{blue}{-1 \cdot \log \left(y \cdot 2\right)}} \cdot x\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\cos \left(e^{\color{blue}{\log \left(y \cdot 2\right) \cdot -1}} \cdot x\right)} \]

      15. lift-log.f64 N/A

          \[\leadsto \frac{1}{\cos \left(e^{\color{blue}{\log \left(y \cdot 2\right)} \cdot -1} \cdot x\right)} \]

      16. exp-to-pow N/A

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

      17. inv-pow N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-/r* N/A

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

      21. metadata-eval N/A

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

      22. lower-/.f64 70.8

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

   7. Applied egg-rr 70.8%

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

   if 1.0000000000000001e44 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 12.6%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+44}:\\ \;\;\;\;\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.4% accurate, 244.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m) :precision binary64 1.0)

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	return 1.0;
}

Fortran

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

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	return 1.0;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	return 1.0

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	return 1.0
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := 1.0

Derivation

1. Initial program 45.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 56.5%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Alternative 6: 6.7% accurate, 244.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ -1 \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m) :precision binary64 -1.0)

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	return -1.0;
}

Fortran

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

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	return -1.0;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	return -1.0

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	return -1.0
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp = code(x_m, y_m)
	tmp = -1.0;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := -1.0

Derivation

1. Initial program 45.6%

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

   1. lift-*.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. pow-sqr N/A

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

   8. pow-prod-down N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l/ N/A

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

   13. associate-/r/ N/A

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

   14. lower-/.f64 N/A

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

   15. div-inv N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval N/A

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

   18. lift-*.f64 N/A

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

   19. associate-*l/ N/A

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

   20. associate-/r/ N/A

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

   21. lower-/.f64 N/A

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

   22. div-inv N/A

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

   23. lower-*.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

   26. metadata-eval 14.2

       \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left({\left(\frac{y}{x \cdot 0.5} \cdot \frac{y}{x \cdot 0.5}\right)}^{\color{blue}{-0.5}}\right)} \]

4. Applied egg-rr 14.2%

   \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \color{blue}{\left({\left(\frac{y}{x \cdot 0.5} \cdot \frac{y}{x \cdot 0.5}\right)}^{-0.5}\right)}} \]

5. Taylor expanded in y around -inf

   \[\leadsto \color{blue}{-1} \]
6. Step-by-step derivation

7. Simplified 6.2%

   \[\leadsto \color{blue}{-1} \]
8. Add Preprocessing

Developer Target 1: 55.4% 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 2024212 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -1230369091130699400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) 1 (if (< y -4551426203405957/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (sin (/ x (* y 2))) (* (sin (/ x (* y 2))) (log (exp (cos (/ x (* y 2))))))) 1)))

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