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

Percentage Accurate: 44.1% → 56.5%

Time: 14.4s

Alternatives: 6

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: 44.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := \log \left(2 \cdot \frac{x\_m}{y\_m}\right)\\ t_1 := \frac{x\_m}{y\_m \cdot 2}\\ \mathbf{if}\;\frac{\tan t\_1}{\sin t\_1} \leq 1.75:\\ \;\;\;\;\frac{1}{\cos \left(e^{\left(t\_0 \cdot \log \left(\frac{x\_m}{\frac{y\_m}{0.5}}\right)\right) \cdot \frac{1}{t\_0}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (log (* 2.0 (/ x_m y_m)))) (t_1 (/ x_m (* y_m 2.0))))
   (if (<= (/ (tan t_1) (sin t_1)) 1.75)
     (/ 1.0 (cos (exp (* (* t_0 (log (/ x_m (/ y_m 0.5)))) (/ 1.0 t_0)))))
     1.0)))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double t_0 = log((2.0 * (x_m / y_m)));
	double t_1 = x_m / (y_m * 2.0);
	double tmp;
	if ((tan(t_1) / sin(t_1)) <= 1.75) {
		tmp = 1.0 / cos(exp(((t_0 * log((x_m / (y_m / 0.5)))) * (1.0 / t_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) :: t_1
    real(8) :: tmp
    t_0 = log((2.0d0 * (x_m / y_m)))
    t_1 = x_m / (y_m * 2.0d0)
    if ((tan(t_1) / sin(t_1)) <= 1.75d0) then
        tmp = 1.0d0 / cos(exp(((t_0 * log((x_m / (y_m / 0.5d0)))) * (1.0d0 / t_0))))
    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.log((2.0 * (x_m / y_m)));
	double t_1 = x_m / (y_m * 2.0);
	double tmp;
	if ((Math.tan(t_1) / Math.sin(t_1)) <= 1.75) {
		tmp = 1.0 / Math.cos(Math.exp(((t_0 * Math.log((x_m / (y_m / 0.5)))) * (1.0 / t_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.log((2.0 * (x_m / y_m)))
	t_1 = x_m / (y_m * 2.0)
	tmp = 0
	if (math.tan(t_1) / math.sin(t_1)) <= 1.75:
		tmp = 1.0 / math.cos(math.exp(((t_0 * math.log((x_m / (y_m / 0.5)))) * (1.0 / t_0))))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	t_0 = log(Float64(2.0 * Float64(x_m / y_m)))
	t_1 = Float64(x_m / Float64(y_m * 2.0))
	tmp = 0.0
	if (Float64(tan(t_1) / sin(t_1)) <= 1.75)
		tmp = Float64(1.0 / cos(exp(Float64(Float64(t_0 * log(Float64(x_m / Float64(y_m / 0.5)))) * Float64(1.0 / t_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 = log((2.0 * (x_m / y_m)));
	t_1 = x_m / (y_m * 2.0);
	tmp = 0.0;
	if ((tan(t_1) / sin(t_1)) <= 1.75)
		tmp = 1.0 / cos(exp(((t_0 * log((x_m / (y_m / 0.5)))) * (1.0 / t_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[Log[N[(2.0 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$1], $MachinePrecision] / N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], 1.75], N[(1.0 / N[Cos[N[Exp[N[(N[(t$95$0 * N[Log[N[(x$95$m / N[(y$95$m / 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$0), $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))))) < 1.75

   1. Initial program 65.0%

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

      1. clear-num N/A

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

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

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

      3. tan-quot N/A

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

      4. associate-/r/ N/A

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

      5. *-inverses N/A

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

      6. remove-double-neg N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. neg-mul-1 N/A

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

      11. remove-double-neg N/A

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

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

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

      13. associate-/r* N/A

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

      14. clear-num N/A

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

      15. associate-/l/ N/A

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

      16. associate-/r* N/A

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

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

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

      18. metadata-eval N/A

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

      19. /-lowering-/.f64 65.4%

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

   4. Applied egg-rr 65.4%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

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

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

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

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

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

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

      7. remove-double-div N/A

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

      8. associate-/r/ N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. associate-/r* N/A

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

      13. clear-num N/A

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

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

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-lowering-/.f64 30.3%

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

   6. Applied egg-rr 30.3%

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

      1. rem-log-exp N/A

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

      2. exp-to-pow N/A

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

      3. inv-pow N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. clear-num N/A

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

      7. associate-/r* N/A

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

      8. log-div N/A

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

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

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

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

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

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

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

      12. log-lowering-log.f64 30.6%

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

   8. Applied egg-rr 30.6%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

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

   10. Applied egg-rr 30.6%

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

   if 1.75 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 3.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 39.2%

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

Alternative 2: 56.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{y\_m \cdot 2}\\ \mathbf{if}\;\frac{\tan t\_0}{\sin t\_0} \leq 1.8:\\ \;\;\;\;\frac{1}{\cos \left(e^{\log \left(\frac{x\_m}{y\_m}\right) - \log 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (/ x_m (* y_m 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 1.8)
     (/ 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 t_0 = x_m / (y_m * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 1.8) {
		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) :: t_0
    real(8) :: tmp
    t_0 = x_m / (y_m * 2.0d0)
    if ((tan(t_0) / sin(t_0)) <= 1.8d0) 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 t_0 = x_m / (y_m * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 1.8) {
		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):
	t_0 = x_m / (y_m * 2.0)
	tmp = 0
	if (math.tan(t_0) / math.sin(t_0)) <= 1.8:
		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)
	t_0 = Float64(x_m / Float64(y_m * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 1.8)
		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)
	t_0 = x_m / (y_m * 2.0);
	tmp = 0.0;
	if ((tan(t_0) / sin(t_0)) <= 1.8)
		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_] := 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], 1.8], 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 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 1.80000000000000004

   1. Initial program 64.8%

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

      1. clear-num N/A

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

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

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

      3. tan-quot N/A

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

      4. associate-/r/ N/A

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

      5. *-inverses N/A

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

      6. remove-double-neg N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. neg-mul-1 N/A

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

      11. remove-double-neg N/A

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

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

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

      13. associate-/r* N/A

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

      14. clear-num N/A

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

      15. associate-/l/ N/A

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

      16. associate-/r* N/A

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

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

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

      18. metadata-eval N/A

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

      19. /-lowering-/.f64 65.1%

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

   4. Applied egg-rr 65.1%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

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

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

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

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

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

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

      7. remove-double-div N/A

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

      8. associate-/r/ N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. associate-/r* N/A

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

      13. clear-num N/A

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

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

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-lowering-/.f64 30.1%

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

   6. Applied egg-rr 30.1%

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

      1. rem-log-exp N/A

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

      2. exp-to-pow N/A

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

      3. inv-pow N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. clear-num N/A

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

      7. associate-/r* N/A

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

      8. log-div N/A

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

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

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

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

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

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

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

      12. log-lowering-log.f64 30.4%

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

   8. Applied egg-rr 30.4%

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

   if 1.80000000000000004 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 3.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 39.4%

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

Alternative 3: 56.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{y\_m \cdot 2}\\ \mathbf{if}\;\frac{\tan t\_0}{\sin t\_0} \leq 3.1:\\ \;\;\;\;\frac{1}{\cos \left(e^{\log \left(\frac{\frac{x\_m}{y\_m}}{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 3.1)
     (/ 1.0 (cos (exp (log (/ (/ x_m y_m) 2.0)))))
     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)) <= 3.1) {
		tmp = 1.0 / cos(exp(log(((x_m / y_m) / 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 * 2.0d0)
    if ((tan(t_0) / sin(t_0)) <= 3.1d0) then
        tmp = 1.0d0 / cos(exp(log(((x_m / y_m) / 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 = x_m / (y_m * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 3.1) {
		tmp = 1.0 / Math.cos(Math.exp(Math.log(((x_m / y_m) / 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 = x_m / (y_m * 2.0)
	tmp = 0
	if (math.tan(t_0) / math.sin(t_0)) <= 3.1:
		tmp = 1.0 / math.cos(math.exp(math.log(((x_m / y_m) / 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 / Float64(y_m * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 3.1)
		tmp = Float64(1.0 / cos(exp(log(Float64(Float64(x_m / y_m) / 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 * 2.0);
	tmp = 0.0;
	if ((tan(t_0) / sin(t_0)) <= 3.1)
		tmp = 1.0 / cos(exp(log(((x_m / y_m) / 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[(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], 3.1], N[(1.0 / N[Cos[N[Exp[N[Log[N[(N[(x$95$m / y$95$m), $MachinePrecision] / 2.0), $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))))) < 3.10000000000000009

   1. Initial program 62.5%

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

      1. clear-num N/A

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

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

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

      3. tan-quot N/A

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

      4. associate-/r/ N/A

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

      5. *-inverses N/A

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

      6. remove-double-neg N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. neg-mul-1 N/A

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

      11. remove-double-neg N/A

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

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

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

      13. associate-/r* N/A

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

      14. clear-num N/A

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

      15. associate-/l/ N/A

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

      16. associate-/r* N/A

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

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

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

      18. metadata-eval N/A

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

      19. /-lowering-/.f64 62.7%

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

   4. Applied egg-rr 62.7%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

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

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

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

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

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

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

      7. remove-double-div N/A

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

      8. associate-/r/ N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. associate-/r* N/A

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

      13. clear-num N/A

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

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

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-lowering-/.f64 29.1%

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

   6. Applied egg-rr 29.1%

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

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

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. neg-log N/A

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

      5. clear-num N/A

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

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

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. associate-/r* N/A

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

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

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

      11. /-lowering-/.f64 30.2%

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

   8. Applied egg-rr 30.2%

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

   if 3.10000000000000009 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 2.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 42.7%

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

Alternative 4: 57.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x\_m}{y\_m \cdot 2} \leq 10^{+91}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{y\_m \cdot \frac{1}{x\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) 1e+91)
   (/ 1.0 (cos (/ 0.5 (* y_m (/ 1.0 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)) <= 1e+91) {
		tmp = 1.0 / cos((0.5 / (y_m * (1.0 / 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) :: tmp
    if ((x_m / (y_m * 2.0d0)) <= 1d+91) then
        tmp = 1.0d0 / cos((0.5d0 / (y_m * (1.0d0 / 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 tmp;
	if ((x_m / (y_m * 2.0)) <= 1e+91) {
		tmp = 1.0 / Math.cos((0.5 / (y_m * (1.0 / 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):
	tmp = 0
	if (x_m / (y_m * 2.0)) <= 1e+91:
		tmp = 1.0 / math.cos((0.5 / (y_m * (1.0 / 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)) <= 1e+91)
		tmp = Float64(1.0 / cos(Float64(0.5 / Float64(y_m * Float64(1.0 / 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)
	tmp = 0.0;
	if ((x_m / (y_m * 2.0)) <= 1e+91)
		tmp = 1.0 / cos((0.5 / (y_m * (1.0 / 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_] := If[LessEqual[N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 1e+91], N[(1.0 / N[Cos[N[(0.5 / N[(y$95$m * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 1.00000000000000008e91

   1. Initial program 55.4%

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

      1. clear-num N/A

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

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

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

      3. tan-quot N/A

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

      4. associate-/r/ N/A

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

      5. *-inverses N/A

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

      6. remove-double-neg N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. neg-mul-1 N/A

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

      11. remove-double-neg N/A

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

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

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

      13. associate-/r* N/A

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

      14. clear-num N/A

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

      15. associate-/l/ N/A

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

      16. associate-/r* N/A

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

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

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

      18. metadata-eval N/A

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

      19. /-lowering-/.f64 67.3%

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

   4. Applied egg-rr 67.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. /-lowering-/.f64 67.3%

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

   6. Applied egg-rr 67.3%

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

   if 1.00000000000000008e91 < (/.f64 x (*.f64 y #s(literal 2 binary64)))

   1. Initial program 6.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 13.2%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot 2} \leq 10^{+91}:\\ \;\;\;\;\frac{1}{\cos \left(\frac{0.5}{y \cdot \frac{1}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.4% accurate, 2.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m) :precision binary64 (/ 1.0 (cos (* x_m (/ 0.5 y_m)))))

C

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

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 / cos((x_m * (0.5d0 / y_m)))
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 / Math.cos((x_m * (0.5 / y_m)));
}

Python

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

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(1.0 / cos(Float64(x_m * Float64(0.5 / y_m))))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := N[(1.0 / N[Cos[N[(x$95$m * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.9%

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

   1. clear-num N/A

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

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

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

   3. tan-quot N/A

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

   4. associate-/r/ N/A

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

   5. *-inverses N/A

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

   6. remove-double-neg N/A

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

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

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

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

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

   9. metadata-eval N/A

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

   10. neg-mul-1 N/A

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

   11. remove-double-neg N/A

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

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

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

   13. associate-/r* N/A

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

   14. clear-num N/A

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

   15. associate-/l/ N/A

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

   16. associate-/r* N/A

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

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

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

   18. metadata-eval N/A

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

   19. /-lowering-/.f64 55.5%

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

4. Applied egg-rr 55.5%

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

   1. associate-/r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-/r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. /-lowering-/.f64 55.6%

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

6. Applied egg-rr 55.6%

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

7. Final simplification 55.6%

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

Alternative 6: 55.0% 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 45.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 54.9%

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

Developer Target 1: 55.0% 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 2024139 
(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)))))