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

Percentage Accurate: 44.6% → 57.7%

Time: 14.8s

Alternatives: 7

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: 44.6% 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: 57.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+153}:\\ \;\;\;\;\frac{1}{\cos \left(\left({y\_m}^{-0.125} \cdot \left({2}^{-0.125} \cdot {\left(y\_m \cdot 2\right)}^{-0.375}\right)\right) \cdot \left(x\_m \cdot {\left(y\_m \cdot 2\right)}^{-0.5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 1e+153)
   (/
    1.0
    (cos
     (*
      (* (pow y_m -0.125) (* (pow 2.0 -0.125) (pow (* y_m 2.0) -0.375)))
      (* x_m (pow (* y_m 2.0) -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)) <= 1e+153) {
		tmp = 1.0 / cos(((pow(y_m, -0.125) * (pow(2.0, -0.125) * pow((y_m * 2.0), -0.375))) * (x_m * pow((y_m * 2.0), -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)) <= 1d+153) then
        tmp = 1.0d0 / cos((((y_m ** (-0.125d0)) * ((2.0d0 ** (-0.125d0)) * ((y_m * 2.0d0) ** (-0.375d0)))) * (x_m * ((y_m * 2.0d0) ** (-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)) <= 1e+153) {
		tmp = 1.0 / Math.cos(((Math.pow(y_m, -0.125) * (Math.pow(2.0, -0.125) * Math.pow((y_m * 2.0), -0.375))) * (x_m * Math.pow((y_m * 2.0), -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)) <= 1e+153:
		tmp = 1.0 / math.cos(((math.pow(y_m, -0.125) * (math.pow(2.0, -0.125) * math.pow((y_m * 2.0), -0.375))) * (x_m * math.pow((y_m * 2.0), -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)) <= 1e+153)
		tmp = Float64(1.0 / cos(Float64(Float64((y_m ^ -0.125) * Float64((2.0 ^ -0.125) * (Float64(y_m * 2.0) ^ -0.375))) * Float64(x_m * (Float64(y_m * 2.0) ^ -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)) <= 1e+153)
		tmp = 1.0 / cos((((y_m ^ -0.125) * ((2.0 ^ -0.125) * ((y_m * 2.0) ^ -0.375))) * (x_m * ((y_m * 2.0) ^ -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], 1e+153], N[(1.0 / N[Cos[N[(N[(N[Power[y$95$m, -0.125], $MachinePrecision] * N[(N[Power[2.0, -0.125], $MachinePrecision] * N[Power[N[(y$95$m * 2.0), $MachinePrecision], -0.375], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * N[Power[N[(y$95$m * 2.0), $MachinePrecision], -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))) < 1e153

   1. Initial program 46.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. *-inverses N/A

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

      9. remove-double-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-cos.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 28.0

          \[\leadsto \frac{1}{\cos \color{blue}{\left({\left(y \cdot 2\right)}^{-0.5} \cdot \left({\left(y \cdot 2\right)}^{-0.5} \cdot x\right)\right)}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 28.0

          \[\leadsto \frac{1}{\cos \left({\left(y \cdot 2\right)}^{-0.5} \cdot \color{blue}{\left(x \cdot {\left(y \cdot 2\right)}^{-0.5}\right)}\right)} \]

   6. Applied rewrites 28.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\cos \left({\left(y \cdot 2\right)}^{\color{blue}{\left(\frac{-1}{4} + \frac{-1}{4}\right)}} \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \frac{1}{\cos \left(\color{blue}{\left({\left(y \cdot 2\right)}^{\frac{-1}{4}} \cdot {\left(y \cdot 2\right)}^{\frac{-1}{4}}\right)} \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      4. sqr-pow N/A

         \[\leadsto \frac{1}{\cos \left(\left(\color{blue}{\left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot {\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)}\right)} \cdot {\left(y \cdot 2\right)}^{\frac{-1}{4}}\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{1}{\cos \left(\left(\left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot {\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)}\right) \cdot \color{blue}{{\left(y \cdot 2\right)}^{\frac{-1}{4}}}\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{1}{\cos \left(\color{blue}{\left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot {\left(y \cdot 2\right)}^{\frac{-1}{4}}\right)\right)} \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\cos \left(\left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \left({\left(y \cdot 2\right)}^{\color{blue}{\frac{-1}{8}}} \cdot {\left(y \cdot 2\right)}^{\frac{-1}{4}}\right)\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{\cos \left(\left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \left({\left(y \cdot 2\right)}^{\frac{-1}{8}} \cdot \color{blue}{{\left(y \cdot 2\right)}^{\frac{-1}{4}}}\right)\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      9. pow-prod-up N/A

         \[\leadsto \frac{1}{\cos \left(\left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot \color{blue}{{\left(y \cdot 2\right)}^{\left(\frac{-1}{8} + \frac{-1}{4}\right)}}\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{\cos \left(\left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot {\left(y \cdot 2\right)}^{\color{blue}{\frac{-3}{8}}}\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{1}{\cos \left(\left({\left(y \cdot 2\right)}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot {\left(y \cdot 2\right)}^{\color{blue}{\left(\frac{\frac{-3}{4}}{2}\right)}}\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\cos \left(\left({\color{blue}{\left(y \cdot 2\right)}}^{\left(\frac{\frac{-1}{4}}{2}\right)} \cdot {\left(y \cdot 2\right)}^{\left(\frac{\frac{-3}{4}}{2}\right)}\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{1}{\cos \left(\left({\left(y \cdot 2\right)}^{\color{blue}{\frac{-1}{8}}} \cdot {\left(y \cdot 2\right)}^{\left(\frac{\frac{-3}{4}}{2}\right)}\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      14. unpow-prod-down N/A

          \[\leadsto \frac{1}{\cos \left(\left(\color{blue}{\left({y}^{\frac{-1}{8}} \cdot {2}^{\frac{-1}{8}}\right)} \cdot {\left(y \cdot 2\right)}^{\left(\frac{\frac{-3}{4}}{2}\right)}\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      15. associate-*l* N/A

          \[\leadsto \frac{1}{\cos \left(\color{blue}{\left({y}^{\frac{-1}{8}} \cdot \left({2}^{\frac{-1}{8}} \cdot {\left(y \cdot 2\right)}^{\left(\frac{\frac{-3}{4}}{2}\right)}\right)\right)} \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{\cos \left(\color{blue}{\left({y}^{\frac{-1}{8}} \cdot \left({2}^{\frac{-1}{8}} \cdot {\left(y \cdot 2\right)}^{\left(\frac{\frac{-3}{4}}{2}\right)}\right)\right)} \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      17. lower-pow.f64 N/A

          \[\leadsto \frac{1}{\cos \left(\left(\color{blue}{{y}^{\frac{-1}{8}}} \cdot \left({2}^{\frac{-1}{8}} \cdot {\left(y \cdot 2\right)}^{\left(\frac{\frac{-3}{4}}{2}\right)}\right)\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{1}{\cos \left(\left({y}^{\frac{-1}{8}} \cdot \color{blue}{\left({2}^{\frac{-1}{8}} \cdot {\left(y \cdot 2\right)}^{\left(\frac{\frac{-3}{4}}{2}\right)}\right)}\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      19. lower-pow.f64 N/A

          \[\leadsto \frac{1}{\cos \left(\left({y}^{\frac{-1}{8}} \cdot \left(\color{blue}{{2}^{\frac{-1}{8}}} \cdot {\left(y \cdot 2\right)}^{\left(\frac{\frac{-3}{4}}{2}\right)}\right)\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      20. lower-pow.f64 N/A

          \[\leadsto \frac{1}{\cos \left(\left({y}^{\frac{-1}{8}} \cdot \left({2}^{\frac{-1}{8}} \cdot \color{blue}{{\left(y \cdot 2\right)}^{\left(\frac{\frac{-3}{4}}{2}\right)}}\right)\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{\frac{-1}{2}}\right)\right)} \]

      21. metadata-eval 28.2

          \[\leadsto \frac{1}{\cos \left(\left({y}^{-0.125} \cdot \left({2}^{-0.125} \cdot {\left(y \cdot 2\right)}^{\color{blue}{-0.375}}\right)\right) \cdot \left(x \cdot {\left(y \cdot 2\right)}^{-0.5}\right)\right)} \]

   8. Applied rewrites 28.2%

      \[\leadsto \frac{1}{\cos \left(\color{blue}{\left({y}^{-0.125} \cdot \left({2}^{-0.125} \cdot {\left(y \cdot 2\right)}^{-0.375}\right)\right)} \cdot \left(x \cdot {\left(y \cdot 2\right)}^{-0.5}\right)\right)} \]

   if 1e153 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 12.8%

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

Alternative 2: 57.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 5 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\cos \left({\left(y\_m \cdot 2\right)}^{-0.5} \cdot \left(x\_m \cdot {\left(e^{\log \left(y\_m \cdot 2\right)}\right)}^{-0.5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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+154)
   (/
    1.0
    (cos
     (* (pow (* y_m 2.0) -0.5) (* x_m (pow (exp (log (* y_m 2.0))) -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+154) {
		tmp = 1.0 / cos((pow((y_m * 2.0), -0.5) * (x_m * pow(exp(log((y_m * 2.0))), -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+154) then
        tmp = 1.0d0 / cos((((y_m * 2.0d0) ** (-0.5d0)) * (x_m * (exp(log((y_m * 2.0d0))) ** (-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+154) {
		tmp = 1.0 / Math.cos((Math.pow((y_m * 2.0), -0.5) * (x_m * Math.pow(Math.exp(Math.log((y_m * 2.0))), -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+154:
		tmp = 1.0 / math.cos((math.pow((y_m * 2.0), -0.5) * (x_m * math.pow(math.exp(math.log((y_m * 2.0))), -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+154)
		tmp = Float64(1.0 / cos(Float64((Float64(y_m * 2.0) ^ -0.5) * Float64(x_m * (exp(log(Float64(y_m * 2.0))) ^ -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+154)
		tmp = 1.0 / cos((((y_m * 2.0) ^ -0.5) * (x_m * (exp(log((y_m * 2.0))) ^ -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+154], N[(1.0 / N[Cos[N[(N[Power[N[(y$95$m * 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(x$95$m * N[Power[N[Exp[N[Log[N[(y$95$m * 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], -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))) < 5.00000000000000004e154

   1. Initial program 46.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. *-inverses N/A

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

      9. remove-double-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-cos.f64 62.0

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

   4. Applied rewrites 62.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 27.7

          \[\leadsto \frac{1}{\cos \color{blue}{\left({\left(y \cdot 2\right)}^{-0.5} \cdot \left({\left(y \cdot 2\right)}^{-0.5} \cdot x\right)\right)}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 27.7

          \[\leadsto \frac{1}{\cos \left({\left(y \cdot 2\right)}^{-0.5} \cdot \color{blue}{\left(x \cdot {\left(y \cdot 2\right)}^{-0.5}\right)}\right)} \]

   6. Applied rewrites 27.7%

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

      1. rem-exp-log N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-log.f64 27.9

         \[\leadsto \frac{1}{\cos \left({\left(y \cdot 2\right)}^{-0.5} \cdot \left(x \cdot {\left(e^{\color{blue}{\log \left(y \cdot 2\right)}}\right)}^{-0.5}\right)\right)} \]

   8. Applied rewrites 27.9%

      \[\leadsto \frac{1}{\cos \left({\left(y \cdot 2\right)}^{-0.5} \cdot \left(x \cdot {\color{blue}{\left(e^{\log \left(y \cdot 2\right)}\right)}}^{-0.5}\right)\right)} \]

   if 5.00000000000000004e154 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 5.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 13.4%

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

Alternative 3: 57.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(y\_m \cdot 2\right)}^{-0.5}\\ \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\frac{1}{\cos \left(t\_0 \cdot \left(x\_m \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (pow (* y_m 2.0) -0.5)))
   (if (<= (/ x_m (* y_m 2.0)) 2e+142) (/ 1.0 (cos (* t_0 (* x_m t_0)))) 1.0)))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double t_0 = pow((y_m * 2.0), -0.5);
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+142) {
		tmp = 1.0 / cos((t_0 * (x_m * t_0)));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (y_m * 2.0d0) ** (-0.5d0)
    if ((x_m / (y_m * 2.0d0)) <= 2d+142) then
        tmp = 1.0d0 / cos((t_0 * (x_m * t_0)))
    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 t_0 = Math.pow((y_m * 2.0), -0.5);
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+142) {
		tmp = 1.0 / Math.cos((t_0 * (x_m * t_0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	t_0 = math.pow((y_m * 2.0), -0.5)
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 2e+142:
		tmp = 1.0 / math.cos((t_0 * (x_m * t_0)))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	t_0 = Float64(y_m * 2.0) ^ -0.5
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 2e+142)
		tmp = Float64(1.0 / cos(Float64(t_0 * Float64(x_m * t_0))));
	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)
	t_0 = (y_m * 2.0) ^ -0.5;
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 2e+142)
		tmp = 1.0 / cos((t_0 * (x_m * t_0)));
	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_] := Block[{t$95$0 = N[Power[N[(y$95$m * 2.0), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2e+142], N[(1.0 / N[Cos[N[(t$95$0 * N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 2.0000000000000001e142

   1. Initial program 47.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. *-inverses N/A

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

      9. remove-double-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-cos.f64 63.1

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

   4. Applied rewrites 63.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 28.3

          \[\leadsto \frac{1}{\cos \color{blue}{\left({\left(y \cdot 2\right)}^{-0.5} \cdot \left({\left(y \cdot 2\right)}^{-0.5} \cdot x\right)\right)}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 28.3

          \[\leadsto \frac{1}{\cos \left({\left(y \cdot 2\right)}^{-0.5} \cdot \color{blue}{\left(x \cdot {\left(y \cdot 2\right)}^{-0.5}\right)}\right)} \]

   6. Applied rewrites 28.3%

      \[\leadsto \frac{1}{\cos \color{blue}{\left({\left(y \cdot 2\right)}^{-0.5} \cdot \left(x \cdot {\left(y \cdot 2\right)}^{-0.5}\right)\right)}} \]

   if 2.0000000000000001e142 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 13.4%

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

Alternative 4: 57.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\frac{1}{\cos \left({\left(y\_m \cdot 2\right)}^{-0.5} \cdot \left(x\_m \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{y\_m}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 2e+142)
   (/
    1.0
    (cos (* (pow (* y_m 2.0) -0.5) (* x_m (* (sqrt 0.5) (sqrt (/ 1.0 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)) <= 2e+142) {
		tmp = 1.0 / cos((pow((y_m * 2.0), -0.5) * (x_m * (sqrt(0.5) * sqrt((1.0 / 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)) <= 2d+142) then
        tmp = 1.0d0 / cos((((y_m * 2.0d0) ** (-0.5d0)) * (x_m * (sqrt(0.5d0) * sqrt((1.0d0 / 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)) <= 2e+142) {
		tmp = 1.0 / Math.cos((Math.pow((y_m * 2.0), -0.5) * (x_m * (Math.sqrt(0.5) * Math.sqrt((1.0 / 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)) <= 2e+142:
		tmp = 1.0 / math.cos((math.pow((y_m * 2.0), -0.5) * (x_m * (math.sqrt(0.5) * math.sqrt((1.0 / 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)) <= 2e+142)
		tmp = Float64(1.0 / cos(Float64((Float64(y_m * 2.0) ^ -0.5) * Float64(x_m * Float64(sqrt(0.5) * sqrt(Float64(1.0 / 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)) <= 2e+142)
		tmp = 1.0 / cos((((y_m * 2.0) ^ -0.5) * (x_m * (sqrt(0.5) * sqrt((1.0 / 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], 2e+142], N[(1.0 / N[Cos[N[(N[Power[N[(y$95$m * 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(x$95$m * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[N[(1.0 / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 2.0000000000000001e142

   1. Initial program 47.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. *-inverses N/A

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

      9. remove-double-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-cos.f64 63.1

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

   4. Applied rewrites 63.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 28.3

          \[\leadsto \frac{1}{\cos \color{blue}{\left({\left(y \cdot 2\right)}^{-0.5} \cdot \left({\left(y \cdot 2\right)}^{-0.5} \cdot x\right)\right)}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 28.3

          \[\leadsto \frac{1}{\cos \left({\left(y \cdot 2\right)}^{-0.5} \cdot \color{blue}{\left(x \cdot {\left(y \cdot 2\right)}^{-0.5}\right)}\right)} \]

   6. Applied rewrites 28.3%

      \[\leadsto \frac{1}{\cos \color{blue}{\left({\left(y \cdot 2\right)}^{-0.5} \cdot \left(x \cdot {\left(y \cdot 2\right)}^{-0.5}\right)\right)}} \]

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 28.1

         \[\leadsto \frac{1}{\cos \left({\left(y \cdot 2\right)}^{-0.5} \cdot \left(x \cdot \left(\sqrt{0.5} \cdot \sqrt{\color{blue}{\frac{1}{y}}}\right)\right)\right)} \]

   9. Applied rewrites 28.1%

      \[\leadsto \frac{1}{\cos \left({\left(y \cdot 2\right)}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{\frac{1}{y}}\right)}\right)\right)} \]

   if 2.0000000000000001e142 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 13.4%

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

Alternative 5: 57.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+57}:\\ \;\;\;\;\frac{1}{\cos \left(x\_m \cdot \frac{0.5}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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+57) (/ 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+57) {
		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+57) 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+57) {
		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+57:
		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+57)
		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+57)
		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+57], 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))) < 1.00000000000000005e57

   1. Initial program 50.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. *-inverses N/A

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

      9. remove-double-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-cos.f64 67.5

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

   4. Applied rewrites 67.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-prod-up N/A

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

      13. metadata-eval N/A

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

      14. inv-pow N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 67.7

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

   6. Applied rewrites 67.7%

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

   if 1.00000000000000005e57 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 7.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 12.5%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+57}:\\ \;\;\;\;\frac{1}{\cos \left(x \cdot \frac{0.5}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.1% 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 40.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 53.9%

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

Alternative 7: 6.6% 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 40.5%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. inv-pow N/A

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

   5. sqr-pow N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-pow.f64 N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval 18.2

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

4. Applied rewrites 18.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. unpow-prod-down N/A

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

   6. associate-*l* N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. pow-prod-down N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-pow.f64 11.2

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

6. Applied rewrites 11.2%

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

7. Taylor expanded in y around -inf

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

   1. unpow2 N/A

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

   2. rem-square-sqrt N/A

      \[\leadsto -2 \cdot \color{blue}{\frac{1}{2}} \]

   3. metadata-eval 7.2

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

9. Applied rewrites 7.2%

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

Developer Target 1: 56.1% 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 2024233 
(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)))))