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

Percentage Accurate: 44.2% → 56.5%

Time: 13.1s

Alternatives: 7

Speedup: 211.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \frac{\tan t\_0}{\sin t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 44.2% 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, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{y\_m \cdot 2}\\ \mathbf{if}\;\frac{\tan t\_0}{\sin t\_0} \leq 100:\\ \;\;\;\;\frac{1}{\cos \left(\frac{e^{0 - \log y\_m}}{e^{\log \left(\frac{2}{x\_m}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (/ x_m (* y_m 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 100.0)
     (/ 1.0 (cos (/ (exp (- 0.0 (log y_m))) (exp (log (/ 2.0 x_m))))))
     1.0)))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = x_m / (y_m * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 100.0) {
		tmp = 1.0 / cos((exp((0.0 - log(y_m))) / exp(log((2.0 / x_m)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
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 = x_m / (y_m * 2.0d0)
    if ((tan(t_0) / sin(t_0)) <= 100.0d0) then
        tmp = 1.0d0 / cos((exp((0.0d0 - log(y_m))) / exp(log((2.0d0 / x_m)))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double t_0 = x_m / (y_m * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 100.0) {
		tmp = 1.0 / Math.cos((Math.exp((0.0 - Math.log(y_m))) / Math.exp(Math.log((2.0 / x_m)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	t_0 = x_m / (y_m * 2.0)
	tmp = 0
	if (math.tan(t_0) / math.sin(t_0)) <= 100.0:
		tmp = 1.0 / math.cos((math.exp((0.0 - math.log(y_m))) / math.exp(math.log((2.0 / x_m)))))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = Float64(x_m / Float64(y_m * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 100.0)
		tmp = Float64(1.0 / cos(Float64(exp(Float64(0.0 - log(y_m))) / exp(log(Float64(2.0 / x_m))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	t_0 = x_m / (y_m * 2.0);
	tmp = 0.0;
	if ((tan(t_0) / sin(t_0)) <= 100.0)
		tmp = 1.0 / cos((exp((0.0 - log(y_m))) / exp(log((2.0 / x_m)))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 100.0], N[(1.0 / N[Cos[N[(N[Exp[N[(0.0 - N[Log[y$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[Log[N[(2.0 / x$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 100

   1. Initial program 55.1%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{\frac{1}{2} \cdot x}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{x \cdot \frac{1}{2}}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2}}{y}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2}}{y}\right)\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x \cdot \frac{1}{2}}{y}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2} \cdot x}{y}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), y\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{1}{2}\right), y\right)\right)\right) \]

      14. *-lowering-*.f64 55.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), y\right)\right)\right) \]

   5. Simplified 55.1%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{y} \cdot \frac{1}{2}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{\frac{y}{\frac{1}{2}}}\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{\frac{y}{\frac{1}{2}}}{x}}\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y \cdot \frac{1}{\frac{1}{2}}}{x}}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y \cdot 2}{x}}\right)\right)\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y \cdot 2}{x}\right)}^{-1}\right)\right)\right) \]

      7. pow-to-exp N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(e^{\log \left(\frac{y \cdot 2}{x}\right) \cdot -1}\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{y \cdot 2}{x}\right) \cdot -1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{y \cdot 2}{x}\right), -1\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y \cdot 2}{x}\right)\right), -1\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y \cdot \frac{1}{\frac{1}{2}}}{x}\right)\right), -1\right)\right)\right)\right) \]

      12. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{\frac{y}{\frac{1}{2}}}{x}\right)\right), -1\right)\right)\right)\right) \]

      13. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y}{x \cdot \frac{1}{2}}\right)\right), -1\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, \left(x \cdot \frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 23.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right) \]

   7. Applied egg-rr 23.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(e^{-1 \cdot \log \left(\frac{y}{x \cdot \frac{1}{2}}\right)}\right)\right)\right) \]

      2. exp-prod N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\log \left(\frac{y}{x \cdot \frac{1}{2}}\right)}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\log \left(\frac{\frac{y}{x}}{\frac{1}{2}}\right)}\right)\right)\right) \]

      4. log-div N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\left(\log \left(\frac{y}{x}\right) - \log \frac{1}{2}\right)}\right)\right)\right) \]

      5. pow-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{{\left(e^{-1}\right)}^{\log \left(\frac{y}{x}\right)}}{{\left(e^{-1}\right)}^{\log \frac{1}{2}}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left({\left(e^{-1}\right)}^{\log \left(\frac{y}{x}\right)}\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(e^{-1}\right), \log \left(\frac{y}{x}\right)\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \log \left(\frac{y}{x}\right)\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\left(\frac{y}{x}\right)\right)\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right), \mathsf{pow.f64}\left(\left(e^{-1}\right), \log \frac{1}{2}\right)\right)\right)\right) \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \log \frac{1}{2}\right)\right)\right)\right) \]

      13. log-lowering-log.f64 24.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 24.3%

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

       1. pow-sub N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\left(\log \left(\frac{y}{x}\right) - \log \frac{1}{2}\right)}\right)\right)\right) \]

       2. diff-log N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\log \left(\frac{\frac{y}{x}}{\frac{1}{2}}\right)}\right)\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\log \left(\frac{y}{x \cdot \frac{1}{2}}\right)}\right)\right)\right) \]

       4. log-div N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\left(\log y - \log \left(x \cdot \frac{1}{2}\right)\right)}\right)\right)\right) \]

       5. pow-sub N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{{\left(e^{-1}\right)}^{\log y}}{{\left(e^{-1}\right)}^{\log \left(x \cdot \frac{1}{2}\right)}}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left({\left(e^{-1}\right)}^{\log y}\right), \left({\left(e^{-1}\right)}^{\log \left(x \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

       7. pow-to-exp N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(e^{\log \left(e^{-1}\right) \cdot \log y}\right), \left({\left(e^{-1}\right)}^{\log \left(x \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

       8. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(e^{-1}\right) \cdot \log y\right)\right), \left({\left(e^{-1}\right)}^{\log \left(x \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

       9. rem-log-exp N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot \log y\right)\right), \left({\left(e^{-1}\right)}^{\log \left(x \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \log y\right)\right), \left({\left(e^{-1}\right)}^{\log \left(x \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

       11. log-lowering-log.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{log.f64}\left(y\right)\right)\right), \left({\left(e^{-1}\right)}^{\log \left(x \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

       12. pow-exp N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{log.f64}\left(y\right)\right)\right), \left(e^{-1 \cdot \log \left(x \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

       13. neg-mul-1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{log.f64}\left(y\right)\right)\right), \left(e^{\mathsf{neg}\left(\log \left(x \cdot \frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

       14. exp-lowering-exp.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{log.f64}\left(y\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\log \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right)\right) \]

       15. neg-log N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{log.f64}\left(y\right)\right)\right), \mathsf{exp.f64}\left(\log \left(\frac{1}{x \cdot \frac{1}{2}}\right)\right)\right)\right)\right) \]

       16. log-lowering-log.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{log.f64}\left(y\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{log.f64}\left(\left(\frac{1}{x \cdot \frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{log.f64}\left(y\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{log.f64}\left(\left(\frac{1}{\frac{1}{2} \cdot x}\right)\right)\right)\right)\right)\right) \]

       18. associate-/r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{log.f64}\left(y\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{log.f64}\left(\left(\frac{\frac{1}{\frac{1}{2}}}{x}\right)\right)\right)\right)\right)\right) \]

       19. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{log.f64}\left(y\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{log.f64}\left(\left(\frac{2}{x}\right)\right)\right)\right)\right)\right) \]

       20. /-lowering-/.f64 9.2%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{log.f64}\left(y\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, x\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 9.2%

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

   if 100 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 0.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 52.9%

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

4. Final simplification 20.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 100:\\ \;\;\;\;\frac{1}{\cos \left(\frac{e^{0 - \log y}}{e^{\log \left(\frac{2}{x}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+269}:\\ \;\;\;\;\frac{1}{\cos \left(e^{\log \left(\frac{x\_m}{y\_m}\right) - \log 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 1e+269)
   (/ 1.0 (cos (exp (- (log (/ x_m y_m)) (log 2.0)))))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+269) {
		tmp = 1.0 / cos(exp((log((x_m / y_m)) - log(2.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
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+269) then
        tmp = 1.0d0 / cos(exp((log((x_m / y_m)) - log(2.0d0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+269) {
		tmp = 1.0 / Math.cos(Math.exp((Math.log((x_m / y_m)) - Math.log(2.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 1e+269:
		tmp = 1.0 / math.cos(math.exp((math.log((x_m / y_m)) - math.log(2.0))))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 1e+269)
		tmp = Float64(1.0 / cos(exp(Float64(log(Float64(x_m / y_m)) - log(2.0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 1e+269)
		tmp = 1.0 / cos(exp((log((x_m / y_m)) - log(2.0))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+269], N[(1.0 / N[Cos[N[Exp[N[(N[Log[N[(x$95$m / y$95$m), $MachinePrecision]], $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1e269

   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 inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{\frac{1}{2} \cdot x}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{x \cdot \frac{1}{2}}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2}}{y}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2}}{y}\right)\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x \cdot \frac{1}{2}}{y}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2} \cdot x}{y}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), y\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{1}{2}\right), y\right)\right)\right) \]

      14. *-lowering-*.f64 58.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), y\right)\right)\right) \]

   5. Simplified 58.6%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{y} \cdot \frac{1}{2}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{\frac{y}{\frac{1}{2}}}\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{\frac{y}{\frac{1}{2}}}{x}}\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y \cdot \frac{1}{\frac{1}{2}}}{x}}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y \cdot 2}{x}}\right)\right)\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y \cdot 2}{x}\right)}^{-1}\right)\right)\right) \]

      7. pow-to-exp N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(e^{\log \left(\frac{y \cdot 2}{x}\right) \cdot -1}\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{y \cdot 2}{x}\right) \cdot -1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{y \cdot 2}{x}\right), -1\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y \cdot 2}{x}\right)\right), -1\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y \cdot \frac{1}{\frac{1}{2}}}{x}\right)\right), -1\right)\right)\right)\right) \]

      12. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{\frac{y}{\frac{1}{2}}}{x}\right)\right), -1\right)\right)\right)\right) \]

      13. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y}{x \cdot \frac{1}{2}}\right)\right), -1\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, \left(x \cdot \frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 24.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right) \]

   7. Applied egg-rr 24.8%

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

      1. rem-log-exp N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(e^{\log \left(\frac{y}{x \cdot \frac{1}{2}}\right) \cdot -1}\right)\right)\right)\right) \]

      2. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left({\left(\frac{y}{x \cdot \frac{1}{2}}\right)}^{-1}\right)\right)\right)\right) \]

      3. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{1}{\frac{y}{x \cdot \frac{1}{2}}}\right)\right)\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{1}{\frac{\frac{y}{x}}{\frac{1}{2}}}\right)\right)\right)\right) \]

      5. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{1}{\frac{y}{x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{x}{y} \cdot \frac{1}{2}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{x}{y} \cdot \frac{1}{2}\right)\right)\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{\frac{x}{y}}{2}\right)\right)\right)\right) \]

      9. log-div N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{x}{y}\right) - \log 2\right)\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\log \left(\frac{x}{y}\right), \log 2\right)\right)\right)\right) \]

      11. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{y}\right)\right), \log 2\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), \log 2\right)\right)\right)\right) \]

      13. log-lowering-log.f64 32.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right), \mathsf{log.f64}\left(2\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 32.8%

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

   if 1e269 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 0.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 10.9%

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

Alternative 3: 57.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := {\left(\frac{x\_m}{y\_m}\right)}^{0.5}\\ \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+222}:\\ \;\;\;\;\frac{1}{\cos \left(t\_0 \cdot \frac{t\_0}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (pow (/ x_m y_m) 0.5)))
   (if (<= (/ x_m (* y_m 2.0)) 1e+222) (/ 1.0 (cos (* t_0 (/ t_0 2.0)))) 1.0)))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = pow((x_m / y_m), 0.5);
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+222) {
		tmp = 1.0 / cos((t_0 * (t_0 / 2.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
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 = (x_m / y_m) ** 0.5d0
    if ((x_m / (y_m * 2.0d0)) <= 1d+222) then
        tmp = 1.0d0 / cos((t_0 * (t_0 / 2.0d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double t_0 = Math.pow((x_m / y_m), 0.5);
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+222) {
		tmp = 1.0 / Math.cos((t_0 * (t_0 / 2.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	t_0 = math.pow((x_m / y_m), 0.5)
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 1e+222:
		tmp = 1.0 / math.cos((t_0 * (t_0 / 2.0)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = Float64(x_m / y_m) ^ 0.5
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 1e+222)
		tmp = Float64(1.0 / cos(Float64(t_0 * Float64(t_0 / 2.0))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	t_0 = (x_m / y_m) ^ 0.5;
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 1e+222)
		tmp = 1.0 / cos((t_0 * (t_0 / 2.0)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[Power[N[(x$95$m / y$95$m), $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+222], N[(1.0 / N[Cos[N[(t$95$0 * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1e222

   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. 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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{\frac{1}{2} \cdot x}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{x \cdot \frac{1}{2}}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2}}{y}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2}}{y}\right)\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x \cdot \frac{1}{2}}{y}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2} \cdot x}{y}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), y\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{1}{2}\right), y\right)\right)\right) \]

      14. *-lowering-*.f64 59.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), y\right)\right)\right) \]

   5. Simplified 59.7%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{y} \cdot \frac{1}{2}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{\frac{y}{\frac{1}{2}}}\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{\frac{y}{\frac{1}{2}}}{x}}\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y \cdot \frac{1}{\frac{1}{2}}}{x}}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y \cdot 2}{x}}\right)\right)\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y \cdot 2}{x}\right)}^{-1}\right)\right)\right) \]

      7. pow-to-exp N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(e^{\log \left(\frac{y \cdot 2}{x}\right) \cdot -1}\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{y \cdot 2}{x}\right) \cdot -1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{y \cdot 2}{x}\right), -1\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y \cdot 2}{x}\right)\right), -1\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y \cdot \frac{1}{\frac{1}{2}}}{x}\right)\right), -1\right)\right)\right)\right) \]

      12. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{\frac{y}{\frac{1}{2}}}{x}\right)\right), -1\right)\right)\right)\right) \]

      13. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y}{x \cdot \frac{1}{2}}\right)\right), -1\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, \left(x \cdot \frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 25.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right) \]

   7. Applied egg-rr 25.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(e^{-1 \cdot \log \left(\frac{y}{x \cdot \frac{1}{2}}\right)}\right)\right)\right) \]

      2. exp-prod N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\log \left(\frac{y}{x \cdot \frac{1}{2}}\right)}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\log \left(\frac{\frac{y}{x}}{\frac{1}{2}}\right)}\right)\right)\right) \]

      4. log-div N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\left(\log \left(\frac{y}{x}\right) - \log \frac{1}{2}\right)}\right)\right)\right) \]

      5. pow-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{{\left(e^{-1}\right)}^{\log \left(\frac{y}{x}\right)}}{{\left(e^{-1}\right)}^{\log \frac{1}{2}}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left({\left(e^{-1}\right)}^{\log \left(\frac{y}{x}\right)}\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(e^{-1}\right), \log \left(\frac{y}{x}\right)\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \log \left(\frac{y}{x}\right)\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\left(\frac{y}{x}\right)\right)\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right), \mathsf{pow.f64}\left(\left(e^{-1}\right), \log \frac{1}{2}\right)\right)\right)\right) \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \log \frac{1}{2}\right)\right)\right)\right) \]

      13. log-lowering-log.f64 25.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 25.4%

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

       1. sqr-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{{\left(e^{-1}\right)}^{\left(\frac{\log \left(\frac{y}{x}\right)}{2}\right)} \cdot {\left(e^{-1}\right)}^{\left(\frac{\log \left(\frac{y}{x}\right)}{2}\right)}}{{\left(e^{-1}\right)}^{\log \frac{1}{2}}}\right)\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\left(\frac{\log \left(\frac{y}{x}\right)}{2}\right)} \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{\log \left(\frac{y}{x}\right)}{2}\right)}}{{\left(e^{-1}\right)}^{\log \frac{1}{2}}}\right)\right)\right) \]

       3. pow-exp N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\left(\frac{\log \left(\frac{y}{x}\right)}{2}\right)} \cdot \frac{e^{-1 \cdot \frac{\log \left(\frac{y}{x}\right)}{2}}}{{\left(e^{-1}\right)}^{\log \frac{1}{2}}}\right)\right)\right) \]

       4. pow-exp N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\left(\frac{\log \left(\frac{y}{x}\right)}{2}\right)} \cdot \frac{e^{-1 \cdot \frac{\log \left(\frac{y}{x}\right)}{2}}}{e^{-1 \cdot \log \frac{1}{2}}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left({\left(e^{-1}\right)}^{\left(\frac{\log \left(\frac{y}{x}\right)}{2}\right)}\right), \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{y}{x}\right)}{2}}}{e^{-1 \cdot \log \frac{1}{2}}}\right)\right)\right)\right) \]

   11. Applied egg-rr 33.4%

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

   if 1e222 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 2.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 12.3%

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

Alternative 4: 57.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{1}{y\_m} \cdot {\left(\frac{\frac{2}{\frac{x\_m}{\frac{y\_m}{0.5}}}}{\frac{y\_m}{0.5}}\right)}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 2e+99)
   (/
    1.0
    (cos
     (* (/ 1.0 y_m) (pow (/ (/ 2.0 (/ x_m (/ y_m 0.5))) (/ y_m 0.5)) -1.0))))
   1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+99) {
		tmp = 1.0 / cos(((1.0 / y_m) * pow(((2.0 / (x_m / (y_m / 0.5))) / (y_m / 0.5)), -1.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
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+99) then
        tmp = 1.0d0 / cos(((1.0d0 / y_m) * (((2.0d0 / (x_m / (y_m / 0.5d0))) / (y_m / 0.5d0)) ** (-1.0d0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+99) {
		tmp = 1.0 / Math.cos(((1.0 / y_m) * Math.pow(((2.0 / (x_m / (y_m / 0.5))) / (y_m / 0.5)), -1.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 2e+99:
		tmp = 1.0 / math.cos(((1.0 / y_m) * math.pow(((2.0 / (x_m / (y_m / 0.5))) / (y_m / 0.5)), -1.0)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 2e+99)
		tmp = Float64(1.0 / cos(Float64(Float64(1.0 / y_m) * (Float64(Float64(2.0 / Float64(x_m / Float64(y_m / 0.5))) / Float64(y_m / 0.5)) ^ -1.0))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 2e+99)
		tmp = 1.0 / cos(((1.0 / y_m) * (((2.0 / (x_m / (y_m / 0.5))) / (y_m / 0.5)) ^ -1.0)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2e+99], N[(1.0 / N[Cos[N[(N[(1.0 / y$95$m), $MachinePrecision] * N[Power[N[(N[(2.0 / N[(x$95$m / N[(y$95$m / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$95$m / 0.5), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1.9999999999999999e99

   1. Initial program 49.2%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{\frac{1}{2} \cdot x}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{x \cdot \frac{1}{2}}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2}}{y}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2}}{y}\right)\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x \cdot \frac{1}{2}}{y}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2} \cdot x}{y}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), y\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{1}{2}\right), y\right)\right)\right) \]

      14. *-lowering-*.f64 63.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), y\right)\right)\right) \]

   5. Simplified 63.5%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{y} \cdot \frac{1}{2}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{\frac{y}{\frac{1}{2}}}\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{\frac{y}{\frac{1}{2}}}{x}}\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y \cdot \frac{1}{\frac{1}{2}}}{x}}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y \cdot 2}{x}}\right)\right)\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y \cdot 2}{x}\right)}^{-1}\right)\right)\right) \]

      7. pow-to-exp N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(e^{\log \left(\frac{y \cdot 2}{x}\right) \cdot -1}\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{y \cdot 2}{x}\right) \cdot -1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{y \cdot 2}{x}\right), -1\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y \cdot 2}{x}\right)\right), -1\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y \cdot \frac{1}{\frac{1}{2}}}{x}\right)\right), -1\right)\right)\right)\right) \]

      12. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{\frac{y}{\frac{1}{2}}}{x}\right)\right), -1\right)\right)\right)\right) \]

      13. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y}{x \cdot \frac{1}{2}}\right)\right), -1\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, \left(x \cdot \frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 26.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right) \]

   7. Applied egg-rr 26.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(e^{-1 \cdot \log \left(\frac{y}{x \cdot \frac{1}{2}}\right)}\right)\right)\right) \]

      2. exp-prod N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\log \left(\frac{y}{x \cdot \frac{1}{2}}\right)}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\log \left(\frac{\frac{y}{x}}{\frac{1}{2}}\right)}\right)\right)\right) \]

      4. log-div N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\left(\log \left(\frac{y}{x}\right) - \log \frac{1}{2}\right)}\right)\right)\right) \]

      5. pow-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{{\left(e^{-1}\right)}^{\log \left(\frac{y}{x}\right)}}{{\left(e^{-1}\right)}^{\log \frac{1}{2}}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left({\left(e^{-1}\right)}^{\log \left(\frac{y}{x}\right)}\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(e^{-1}\right), \log \left(\frac{y}{x}\right)\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \log \left(\frac{y}{x}\right)\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\left(\frac{y}{x}\right)\right)\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right), \left({\left(e^{-1}\right)}^{\log \frac{1}{2}}\right)\right)\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right), \mathsf{pow.f64}\left(\left(e^{-1}\right), \log \frac{1}{2}\right)\right)\right)\right) \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \log \frac{1}{2}\right)\right)\right)\right) \]

      13. log-lowering-log.f64 26.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 26.3%

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

   11. Applied egg-rr 63.7%

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

   if 1.9999999999999999e99 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 4.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 11.8%

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

Alternative 5: 57.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{y\_m}{x\_m \cdot 0.5}\\ \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\cos \left({\left(t\_0 \cdot t\_0\right)}^{-0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (/ y_m (* x_m 0.5))))
   (if (<= (/ x_m (* y_m 2.0)) 2e+99)
     (/ 1.0 (cos (pow (* t_0 t_0) -0.5)))
     1.0)))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = y_m / (x_m * 0.5);
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+99) {
		tmp = 1.0 / cos(pow((t_0 * t_0), -0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
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 / (x_m * 0.5d0)
    if ((x_m / (y_m * 2.0d0)) <= 2d+99) then
        tmp = 1.0d0 / cos(((t_0 * t_0) ** (-0.5d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double t_0 = y_m / (x_m * 0.5);
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+99) {
		tmp = 1.0 / Math.cos(Math.pow((t_0 * t_0), -0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	t_0 = y_m / (x_m * 0.5)
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 2e+99:
		tmp = 1.0 / math.cos(math.pow((t_0 * t_0), -0.5))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = Float64(y_m / Float64(x_m * 0.5))
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 2e+99)
		tmp = Float64(1.0 / cos((Float64(t_0 * t_0) ^ -0.5)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	t_0 = y_m / (x_m * 0.5);
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 2e+99)
		tmp = 1.0 / cos(((t_0 * t_0) ^ -0.5));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[(y$95$m / N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2e+99], N[(1.0 / N[Cos[N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1.9999999999999999e99

   1. Initial program 49.2%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{\frac{1}{2} \cdot x}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{x \cdot \frac{1}{2}}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2}}{y}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2}}{y}\right)\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x \cdot \frac{1}{2}}{y}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2} \cdot x}{y}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), y\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{1}{2}\right), y\right)\right)\right) \]

      14. *-lowering-*.f64 63.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), y\right)\right)\right) \]

   5. Simplified 63.5%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{y} \cdot \frac{1}{2}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{\frac{y}{\frac{1}{2}}}\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{\frac{y}{\frac{1}{2}}}{x}}\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y \cdot \frac{1}{\frac{1}{2}}}{x}}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y \cdot 2}{x}}\right)\right)\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y \cdot 2}{x}\right)}^{-1}\right)\right)\right) \]

      7. pow-to-exp N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(e^{\log \left(\frac{y \cdot 2}{x}\right) \cdot -1}\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{y \cdot 2}{x}\right) \cdot -1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{y \cdot 2}{x}\right), -1\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y \cdot 2}{x}\right)\right), -1\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y \cdot \frac{1}{\frac{1}{2}}}{x}\right)\right), -1\right)\right)\right)\right) \]

      12. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{\frac{y}{\frac{1}{2}}}{x}\right)\right), -1\right)\right)\right)\right) \]

      13. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y}{x \cdot \frac{1}{2}}\right)\right), -1\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, \left(x \cdot \frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 26.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right) \]

   7. Applied egg-rr 26.1%

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

      1. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y}{x \cdot \frac{1}{2}}\right)}^{-1}\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y}{x \cdot \frac{1}{2}}\right)}^{\left(\frac{-1}{2} + \frac{-1}{2}\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y}{x \cdot \frac{1}{2}}\right)}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{-1}{2}\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y}{x \cdot \frac{1}{2}}\right)}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}\right)\right)\right) \]

      5. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y}{x \cdot \frac{1}{2}}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\frac{y}{x \cdot \frac{1}{2}}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y}{x \cdot \frac{1}{2}} \cdot \frac{y}{x \cdot \frac{1}{2}}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{pow.f64}\left(\left(\frac{y}{x \cdot \frac{1}{2}} \cdot \frac{y}{x \cdot \frac{1}{2}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{y}{x \cdot \frac{1}{2}}\right), \left(\frac{y}{x \cdot \frac{1}{2}}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left(x \cdot \frac{1}{2}\right)\right), \left(\frac{y}{x \cdot \frac{1}{2}}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{y}{x \cdot \frac{1}{2}}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(y, \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      13. metadata-eval 62.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

   9. Applied egg-rr 62.2%

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

   if 1.9999999999999999e99 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 4.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 11.8%

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

Alternative 6: 57.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{\frac{y\_m}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (/ x_m (* y_m 2.0)) 2e+99) (/ 1.0 (cos (/ 0.5 (/ y_m x_m)))) 1.0))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+99) {
		tmp = 1.0 / cos((0.5 / (y_m / x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
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+99) then
        tmp = 1.0d0 / cos((0.5d0 / (y_m / x_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m / (y_m * 2.0)) <= 2e+99) {
		tmp = 1.0 / Math.cos((0.5 / (y_m / x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 2e+99:
		tmp = 1.0 / math.cos((0.5 / (y_m / x_m)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m / Float64(y_m * 2.0)) <= 2e+99)
		tmp = Float64(1.0 / cos(Float64(0.5 / Float64(y_m / x_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 2e+99)
		tmp = 1.0 / cos((0.5 / (y_m / x_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 2e+99], 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))) < 1.9999999999999999e99

   1. Initial program 49.2%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{\frac{1}{2} \cdot x}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(\frac{x \cdot \frac{1}{2}}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2}}{y}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \cos \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2} \cdot 1}{y}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(x \cdot \frac{\frac{1}{2}}{y}\right)\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x \cdot \frac{1}{2}}{y}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2} \cdot x}{y}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), y\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{1}{2}\right), y\right)\right)\right) \]

      14. *-lowering-*.f64 63.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), y\right)\right)\right) \]

   5. Simplified 63.5%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{y} \cdot \frac{1}{2}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{x}{\frac{y}{\frac{1}{2}}}\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{\frac{y}{\frac{1}{2}}}{x}}\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y \cdot \frac{1}{\frac{1}{2}}}{x}}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y \cdot 2}{x}}\right)\right)\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y \cdot 2}{x}\right)}^{-1}\right)\right)\right) \]

      7. pow-to-exp N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(e^{\log \left(\frac{y \cdot 2}{x}\right) \cdot -1}\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{y \cdot 2}{x}\right) \cdot -1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{y \cdot 2}{x}\right), -1\right)\right)\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y \cdot 2}{x}\right)\right), -1\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y \cdot \frac{1}{\frac{1}{2}}}{x}\right)\right), -1\right)\right)\right)\right) \]

      12. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{\frac{y}{\frac{1}{2}}}{x}\right)\right), -1\right)\right)\right)\right) \]

      13. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{y}{x \cdot \frac{1}{2}}\right)\right), -1\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, \left(x \cdot \frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 26.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), -1\right)\right)\right)\right) \]

   7. Applied egg-rr 26.1%

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

      1. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left({\left(\frac{y}{x \cdot \frac{1}{2}}\right)}^{-1}\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{y}{x \cdot \frac{1}{2}}}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{\frac{\frac{y}{x}}{\frac{1}{2}}}\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{\frac{1}{2}}{\frac{y}{x}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{y}{x}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 63.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(y, x\right)\right)\right)\right) \]

   9. Applied egg-rr 63.5%

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

   if 1.9999999999999999e99 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 4.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 11.8%

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

Alternative 7: 55.4% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
y_m = abs(y)
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 53.6%

   \[\leadsto \color{blue}{1} \]
6. 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 2024158 
(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)))))