Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A

Percentage Accurate: 76.6% → 99.5%

Time: 18.5s

Alternatives: 19

Speedup: 1.5×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\left(\frac{8}{3} \cdot t_0\right) \cdot t_0}{\sin x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / sin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = (((8.0d0 / 3.0d0) * t_0) * t_0) / sin(x)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / Math.sin(x);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return (((8.0 / 3.0) * t_0) * t_0) / math.sin(x)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = (((8.0 / 3.0) * t_0) * t_0) / sin(x);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Initial Program: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\left(\frac{8}{3} \cdot t_0\right) \cdot t_0}{\sin x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / sin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = (((8.0d0 / 3.0d0) * t_0) * t_0) / sin(x)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / Math.sin(x);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return (((8.0 / 3.0) * t_0) * t_0) / math.sin(x)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = (((8.0 / 3.0) * t_0) * t_0) / sin(x);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(x \cdot 0.5\right)}^{2}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{t_0}}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} \cdot \frac{t_0}{0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (sin (* x 0.5)) 2.0)))
   (if (<= x -5e-8)
     (/ 2.6666666666666665 (/ (sin x) t_0))
     (if (<= x 4.5e-8)
       (/ (* x 0.25) 0.375)
       (* (/ 1.0 (sin x)) (/ t_0 0.375))))))

C

double code(double x) {
	double t_0 = pow(sin((x * 0.5)), 2.0);
	double tmp;
	if (x <= -5e-8) {
		tmp = 2.6666666666666665 / (sin(x) / t_0);
	} else if (x <= 4.5e-8) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (1.0 / sin(x)) * (t_0 / 0.375);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x * 0.5d0)) ** 2.0d0
    if (x <= (-5d-8)) then
        tmp = 2.6666666666666665d0 / (sin(x) / t_0)
    else if (x <= 4.5d-8) then
        tmp = (x * 0.25d0) / 0.375d0
    else
        tmp = (1.0d0 / sin(x)) * (t_0 / 0.375d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.sin((x * 0.5)), 2.0);
	double tmp;
	if (x <= -5e-8) {
		tmp = 2.6666666666666665 / (Math.sin(x) / t_0);
	} else if (x <= 4.5e-8) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (1.0 / Math.sin(x)) * (t_0 / 0.375);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.pow(math.sin((x * 0.5)), 2.0)
	tmp = 0
	if x <= -5e-8:
		tmp = 2.6666666666666665 / (math.sin(x) / t_0)
	elif x <= 4.5e-8:
		tmp = (x * 0.25) / 0.375
	else:
		tmp = (1.0 / math.sin(x)) * (t_0 / 0.375)
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5)) ^ 2.0
	tmp = 0.0
	if (x <= -5e-8)
		tmp = Float64(2.6666666666666665 / Float64(sin(x) / t_0));
	elseif (x <= 4.5e-8)
		tmp = Float64(Float64(x * 0.25) / 0.375);
	else
		tmp = Float64(Float64(1.0 / sin(x)) * Float64(t_0 / 0.375));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5)) ^ 2.0;
	tmp = 0.0;
	if (x <= -5e-8)
		tmp = 2.6666666666666665 / (sin(x) / t_0);
	elseif (x <= 4.5e-8)
		tmp = (x * 0.25) / 0.375;
	else
		tmp = (1.0 / sin(x)) * (t_0 / 0.375);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -5e-8], N[(2.6666666666666665 / N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-8], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(1.0 / N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / 0.375), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999998e-8

   1. Initial program 99.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right) \cdot \sin \left(x \cdot 0.5\right)} \]

      2. *-commutative 99.2%

         \[\leadsto \color{blue}{\left(\frac{2.6666666666666665}{\sin x} \cdot \sin \left(x \cdot 0.5\right)\right)} \cdot \sin \left(x \cdot 0.5\right) \]

      3. associate-/r/ 99.1%

         \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sin \left(x \cdot 0.5\right) \]

      4. associate-/r/ 99.2%

         \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}{\sin \left(x \cdot 0.5\right)}}} \]

      5. associate-/l/ 99.2%

         \[\leadsto \frac{2.6666666666666665}{\color{blue}{\frac{\sin x}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}} \]

      6. pow2 99.2%

         \[\leadsto \frac{2.6666666666666665}{\frac{\sin x}{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   if -4.9999999999999998e-8 < x < 4.49999999999999993e-8

   1. Initial program 63.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

      2. *-commutative 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

      3. *-commutative 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)} \]

      4. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      5. add-sqr-sqrt 53.5%

         \[\leadsto \color{blue}{\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}} \]

      6. pow2 53.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}\right)}^{2}} \]

   5. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 53.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      2. swap-sqr 36.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      3. unpow2 36.4%

         \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \]

      4. add-sqr-sqrt 63.2%

         \[\leadsto {\sin \left(x \cdot 0.5\right)}^{2} \cdot \color{blue}{\frac{2.6666666666666665}{\sin x}} \]

      5. metadata-eval 63.2%

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

      6. associate-/r* 63.2%

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

      7. *-commutative 63.2%

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

      8. div-inv 63.2%

         \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]

      9. associate-/r* 63.4%

         \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]

   if 4.49999999999999993e-8 < x

   1. Initial program 99.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.1%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

      2. *-commutative 99.1%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

      3. clear-num 99.1%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}} \]

      4. un-div-inv 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}} \]

      5. *-un-lft-identity 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\color{blue}{1 \cdot \sin x}}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}} \]

      6. times-frac 99.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{1}{2.6666666666666665} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      7. metadata-eval 99.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.375} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   5. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{0.375 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{0.375 \cdot \sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      2. *-commutative 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\color{blue}{\sin x \cdot 0.375}}{\sin \left(x \cdot 0.5\right)}} \]

      3. associate-/l* 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x \cdot 0.375}} \]

      4. unpow2 99.1%

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x \cdot 0.375} \]

      5. *-un-lft-identity 99.1%

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

      6. times-frac 99.3%

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

   7. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{1}{\sin x} \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8} \lor \neg \left(x \leq 2 \cdot 10^{-14}\right):\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5e-8) (not (<= x 2e-14)))
   (* 2.6666666666666665 (/ (pow (sin (* x 0.5)) 2.0) (sin x)))
   (/ (* x 0.25) 0.375)))

C

double code(double x) {
	double tmp;
	if ((x <= -5e-8) || !(x <= 2e-14)) {
		tmp = 2.6666666666666665 * (pow(sin((x * 0.5)), 2.0) / sin(x));
	} else {
		tmp = (x * 0.25) / 0.375;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5d-8)) .or. (.not. (x <= 2d-14))) then
        tmp = 2.6666666666666665d0 * ((sin((x * 0.5d0)) ** 2.0d0) / sin(x))
    else
        tmp = (x * 0.25d0) / 0.375d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -5e-8) || !(x <= 2e-14)) {
		tmp = 2.6666666666666665 * (Math.pow(Math.sin((x * 0.5)), 2.0) / Math.sin(x));
	} else {
		tmp = (x * 0.25) / 0.375;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -5e-8) or not (x <= 2e-14):
		tmp = 2.6666666666666665 * (math.pow(math.sin((x * 0.5)), 2.0) / math.sin(x))
	else:
		tmp = (x * 0.25) / 0.375
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5e-8) || !(x <= 2e-14))
		tmp = Float64(2.6666666666666665 * Float64((sin(Float64(x * 0.5)) ^ 2.0) / sin(x)));
	else
		tmp = Float64(Float64(x * 0.25) / 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -5e-8) || ~((x <= 2e-14)))
		tmp = 2.6666666666666665 * ((sin((x * 0.5)) ^ 2.0) / sin(x));
	else
		tmp = (x * 0.25) / 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5e-8], N[Not[LessEqual[x, 2e-14]], $MachinePrecision]], N[(2.6666666666666665 * N[(N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e-8 or 2e-14 < x

   1. Initial program 99.1%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Step-by-step derivation

      1. associate-/l* 99.1%

         \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      2. metadata-eval 99.1%

         \[\leadsto \frac{\color{blue}{2.6666666666666665} \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   5. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x} \cdot 2.6666666666666665} \]

   if -4.9999999999999998e-8 < x < 2e-14

   1. Initial program 63.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

      2. *-commutative 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

      3. *-commutative 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)} \]

      4. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      5. add-sqr-sqrt 53.5%

         \[\leadsto \color{blue}{\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}} \]

      6. pow2 53.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}\right)}^{2}} \]

   5. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 53.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      2. swap-sqr 36.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      3. unpow2 36.4%

         \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \]

      4. add-sqr-sqrt 63.2%

         \[\leadsto {\sin \left(x \cdot 0.5\right)}^{2} \cdot \color{blue}{\frac{2.6666666666666665}{\sin x}} \]

      5. metadata-eval 63.2%

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

      6. associate-/r* 63.2%

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

      7. *-commutative 63.2%

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

      8. div-inv 63.2%

         \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]

      9. associate-/r* 63.4%

         \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8} \lor \neg \left(x \leq 2 \cdot 10^{-14}\right):\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8} \lor \neg \left(x \leq 2 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{2.6666666666666665}{\sin x} \cdot {\sin \left(x \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5e-8) (not (<= x 2e-14)))
   (* (/ 2.6666666666666665 (sin x)) (pow (sin (* x 0.5)) 2.0))
   (/ (* x 0.25) 0.375)))

C

double code(double x) {
	double tmp;
	if ((x <= -5e-8) || !(x <= 2e-14)) {
		tmp = (2.6666666666666665 / sin(x)) * pow(sin((x * 0.5)), 2.0);
	} else {
		tmp = (x * 0.25) / 0.375;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5d-8)) .or. (.not. (x <= 2d-14))) then
        tmp = (2.6666666666666665d0 / sin(x)) * (sin((x * 0.5d0)) ** 2.0d0)
    else
        tmp = (x * 0.25d0) / 0.375d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -5e-8) || !(x <= 2e-14)) {
		tmp = (2.6666666666666665 / Math.sin(x)) * Math.pow(Math.sin((x * 0.5)), 2.0);
	} else {
		tmp = (x * 0.25) / 0.375;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -5e-8) or not (x <= 2e-14):
		tmp = (2.6666666666666665 / math.sin(x)) * math.pow(math.sin((x * 0.5)), 2.0)
	else:
		tmp = (x * 0.25) / 0.375
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5e-8) || !(x <= 2e-14))
		tmp = Float64(Float64(2.6666666666666665 / sin(x)) * (sin(Float64(x * 0.5)) ^ 2.0));
	else
		tmp = Float64(Float64(x * 0.25) / 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -5e-8) || ~((x <= 2e-14)))
		tmp = (2.6666666666666665 / sin(x)) * (sin((x * 0.5)) ^ 2.0);
	else
		tmp = (x * 0.25) / 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5e-8], N[Not[LessEqual[x, 2e-14]], $MachinePrecision]], N[(N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e-8 or 2e-14 < x

   1. Initial program 99.1%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Step-by-step derivation

      1. associate-/l* 99.1%

         \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      2. metadata-eval 99.1%

         \[\leadsto \frac{\color{blue}{2.6666666666666665} \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]
   4. Step-by-step derivation

      1. associate-/r/ 99.1%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)} \]

      2. *-commutative 99.1%

         \[\leadsto \frac{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}}{\sin x} \cdot \sin \left(x \cdot 0.5\right) \]

      3. associate-*r/ 99.1%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \cdot \sin \left(x \cdot 0.5\right) \]

      4. *-commutative 99.1%

         \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]

      5. associate-*r* 99.1%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{2.6666666666666665}{\sin x}} \]

      6. pow2 99.1%

         \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \frac{2.6666666666666665}{\sin x} \]

   5. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}} \]

   if -4.9999999999999998e-8 < x < 2e-14

   1. Initial program 63.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

      2. *-commutative 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

      3. *-commutative 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)} \]

      4. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      5. add-sqr-sqrt 53.5%

         \[\leadsto \color{blue}{\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}} \]

      6. pow2 53.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}\right)}^{2}} \]

   5. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 53.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      2. swap-sqr 36.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      3. unpow2 36.4%

         \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \]

      4. add-sqr-sqrt 63.2%

         \[\leadsto {\sin \left(x \cdot 0.5\right)}^{2} \cdot \color{blue}{\frac{2.6666666666666665}{\sin x}} \]

      5. metadata-eval 63.2%

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

      6. associate-/r* 63.2%

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

      7. *-commutative 63.2%

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

      8. div-inv 63.2%

         \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]

      9. associate-/r* 63.4%

         \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8} \lor \neg \left(x \leq 2 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{2.6666666666666665}{\sin x} \cdot {\sin \left(x \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8} \lor \neg \left(x \leq 2 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5e-8) (not (<= x 2e-14)))
   (/ 2.6666666666666665 (/ (sin x) (pow (sin (* x 0.5)) 2.0)))
   (/ (* x 0.25) 0.375)))

C

double code(double x) {
	double tmp;
	if ((x <= -5e-8) || !(x <= 2e-14)) {
		tmp = 2.6666666666666665 / (sin(x) / pow(sin((x * 0.5)), 2.0));
	} else {
		tmp = (x * 0.25) / 0.375;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5d-8)) .or. (.not. (x <= 2d-14))) then
        tmp = 2.6666666666666665d0 / (sin(x) / (sin((x * 0.5d0)) ** 2.0d0))
    else
        tmp = (x * 0.25d0) / 0.375d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -5e-8) || !(x <= 2e-14)) {
		tmp = 2.6666666666666665 / (Math.sin(x) / Math.pow(Math.sin((x * 0.5)), 2.0));
	} else {
		tmp = (x * 0.25) / 0.375;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -5e-8) or not (x <= 2e-14):
		tmp = 2.6666666666666665 / (math.sin(x) / math.pow(math.sin((x * 0.5)), 2.0))
	else:
		tmp = (x * 0.25) / 0.375
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5e-8) || !(x <= 2e-14))
		tmp = Float64(2.6666666666666665 / Float64(sin(x) / (sin(Float64(x * 0.5)) ^ 2.0)));
	else
		tmp = Float64(Float64(x * 0.25) / 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -5e-8) || ~((x <= 2e-14)))
		tmp = 2.6666666666666665 / (sin(x) / (sin((x * 0.5)) ^ 2.0));
	else
		tmp = (x * 0.25) / 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5e-8], N[Not[LessEqual[x, 2e-14]], $MachinePrecision]], N[(2.6666666666666665 / N[(N[Sin[x], $MachinePrecision] / N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e-8 or 2e-14 < x

   1. Initial program 99.1%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right) \cdot \sin \left(x \cdot 0.5\right)} \]

      2. *-commutative 99.1%

         \[\leadsto \color{blue}{\left(\frac{2.6666666666666665}{\sin x} \cdot \sin \left(x \cdot 0.5\right)\right)} \cdot \sin \left(x \cdot 0.5\right) \]

      3. associate-/r/ 99.0%

         \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sin \left(x \cdot 0.5\right) \]

      4. associate-/r/ 99.1%

         \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}{\sin \left(x \cdot 0.5\right)}}} \]

      5. associate-/l/ 99.2%

         \[\leadsto \frac{2.6666666666666665}{\color{blue}{\frac{\sin x}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}} \]

      6. pow2 99.2%

         \[\leadsto \frac{2.6666666666666665}{\frac{\sin x}{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   if -4.9999999999999998e-8 < x < 2e-14

   1. Initial program 63.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

      2. *-commutative 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

      3. *-commutative 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)} \]

      4. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      5. add-sqr-sqrt 53.5%

         \[\leadsto \color{blue}{\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}} \]

      6. pow2 53.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}\right)}^{2}} \]

   5. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 53.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      2. swap-sqr 36.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      3. unpow2 36.4%

         \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \]

      4. add-sqr-sqrt 63.2%

         \[\leadsto {\sin \left(x \cdot 0.5\right)}^{2} \cdot \color{blue}{\frac{2.6666666666666665}{\sin x}} \]

      5. metadata-eval 63.2%

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

      6. associate-/r* 63.2%

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

      7. *-commutative 63.2%

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

      8. div-inv 63.2%

         \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]

      9. associate-/r* 63.4%

         \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8} \lor \neg \left(x \leq 2 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \end{array} \]

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(x \cdot 0.5\right)}^{2}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{t_0}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665 \cdot t_0}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (sin (* x 0.5)) 2.0)))
   (if (<= x -5e-8)
     (/ 2.6666666666666665 (/ (sin x) t_0))
     (if (<= x 5e-10)
       (/ (* x 0.25) 0.375)
       (/ (* 2.6666666666666665 t_0) (sin x))))))

C

double code(double x) {
	double t_0 = pow(sin((x * 0.5)), 2.0);
	double tmp;
	if (x <= -5e-8) {
		tmp = 2.6666666666666665 / (sin(x) / t_0);
	} else if (x <= 5e-10) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (2.6666666666666665 * t_0) / sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x * 0.5d0)) ** 2.0d0
    if (x <= (-5d-8)) then
        tmp = 2.6666666666666665d0 / (sin(x) / t_0)
    else if (x <= 5d-10) then
        tmp = (x * 0.25d0) / 0.375d0
    else
        tmp = (2.6666666666666665d0 * t_0) / sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.sin((x * 0.5)), 2.0);
	double tmp;
	if (x <= -5e-8) {
		tmp = 2.6666666666666665 / (Math.sin(x) / t_0);
	} else if (x <= 5e-10) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (2.6666666666666665 * t_0) / Math.sin(x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.pow(math.sin((x * 0.5)), 2.0)
	tmp = 0
	if x <= -5e-8:
		tmp = 2.6666666666666665 / (math.sin(x) / t_0)
	elif x <= 5e-10:
		tmp = (x * 0.25) / 0.375
	else:
		tmp = (2.6666666666666665 * t_0) / math.sin(x)
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5)) ^ 2.0
	tmp = 0.0
	if (x <= -5e-8)
		tmp = Float64(2.6666666666666665 / Float64(sin(x) / t_0));
	elseif (x <= 5e-10)
		tmp = Float64(Float64(x * 0.25) / 0.375);
	else
		tmp = Float64(Float64(2.6666666666666665 * t_0) / sin(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5)) ^ 2.0;
	tmp = 0.0;
	if (x <= -5e-8)
		tmp = 2.6666666666666665 / (sin(x) / t_0);
	elseif (x <= 5e-10)
		tmp = (x * 0.25) / 0.375;
	else
		tmp = (2.6666666666666665 * t_0) / sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -5e-8], N[(2.6666666666666665 / N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-10], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(2.6666666666666665 * t$95$0), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999998e-8

   1. Initial program 99.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right) \cdot \sin \left(x \cdot 0.5\right)} \]

      2. *-commutative 99.2%

         \[\leadsto \color{blue}{\left(\frac{2.6666666666666665}{\sin x} \cdot \sin \left(x \cdot 0.5\right)\right)} \cdot \sin \left(x \cdot 0.5\right) \]

      3. associate-/r/ 99.1%

         \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sin \left(x \cdot 0.5\right) \]

      4. associate-/r/ 99.2%

         \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}{\sin \left(x \cdot 0.5\right)}}} \]

      5. associate-/l/ 99.2%

         \[\leadsto \frac{2.6666666666666665}{\color{blue}{\frac{\sin x}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}} \]

      6. pow2 99.2%

         \[\leadsto \frac{2.6666666666666665}{\frac{\sin x}{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   if -4.9999999999999998e-8 < x < 5.00000000000000031e-10

   1. Initial program 63.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

      2. *-commutative 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

      3. *-commutative 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)} \]

      4. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      5. add-sqr-sqrt 53.5%

         \[\leadsto \color{blue}{\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}} \]

      6. pow2 53.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}\right)}^{2}} \]

   5. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 53.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      2. swap-sqr 36.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      3. unpow2 36.4%

         \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \]

      4. add-sqr-sqrt 63.2%

         \[\leadsto {\sin \left(x \cdot 0.5\right)}^{2} \cdot \color{blue}{\frac{2.6666666666666665}{\sin x}} \]

      5. metadata-eval 63.2%

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

      6. associate-/r* 63.2%

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

      7. *-commutative 63.2%

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

      8. div-inv 63.2%

         \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]

      9. associate-/r* 63.4%

         \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]

   if 5.00000000000000031e-10 < x

   1. Initial program 99.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right) \cdot \sin \left(x \cdot 0.5\right)} \]

      2. associate-*r/ 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \cdot \sin \left(x \cdot 0.5\right) \]

      3. *-commutative 99.1%

         \[\leadsto \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \cdot \sin \left(x \cdot 0.5\right) \]

      4. associate-*l/ 99.2%

         \[\leadsto \color{blue}{\frac{\left(2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]

      5. associate-*l* 99.2%

         \[\leadsto \frac{\color{blue}{2.6666666666666665 \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}}{\sin x} \]

      6. pow2 99.2%

         \[\leadsto \frac{2.6666666666666665 \cdot \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot {\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665 \cdot {\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]

Alternative 6: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{t_0}{0.375 \cdot \frac{\sin x}{t_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ t_0 (* 0.375 (/ (sin x) t_0)))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return t_0 / (0.375 * (sin(x) / t_0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = t_0 / (0.375d0 * (sin(x) / t_0))
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return t_0 / (0.375 * (Math.sin(x) / t_0));
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return t_0 / (0.375 * (math.sin(x) / t_0))

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(t_0 / Float64(0.375 * Float64(sin(x) / t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = t_0 / (0.375 * (sin(x) / t_0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 / N[(0.375 * N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
4. Step-by-step derivation

   1. associate-*r/ 99.3%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

   2. *-commutative 99.3%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

   3. clear-num 99.1%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}} \]

   4. un-div-inv 99.3%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}} \]

   5. *-un-lft-identity 99.3%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\color{blue}{1 \cdot \sin x}}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}} \]

   6. times-frac 99.6%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{1}{2.6666666666666665} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   7. metadata-eval 99.6%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.375} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{0.375 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

6. Final simplification 99.6%

   \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{0.375 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

Alternative 7: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot -0.5\right)\\ 2.6666666666666665 \cdot \frac{t_0}{\frac{\sin x}{t_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x -0.5))))
   (* 2.6666666666666665 (/ t_0 (/ (sin x) t_0)))))

C

double code(double x) {
	double t_0 = sin((x * -0.5));
	return 2.6666666666666665 * (t_0 / (sin(x) / t_0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * (-0.5d0)))
    code = 2.6666666666666665d0 * (t_0 / (sin(x) / t_0))
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * -0.5));
	return 2.6666666666666665 * (t_0 / (Math.sin(x) / t_0));
}

Python

def code(x):
	t_0 = math.sin((x * -0.5))
	return 2.6666666666666665 * (t_0 / (math.sin(x) / t_0))

Julia

function code(x)
	t_0 = sin(Float64(x * -0.5))
	return Float64(2.6666666666666665 * Float64(t_0 / Float64(sin(x) / t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * -0.5));
	tmp = 2.6666666666666665 * (t_0 / (sin(x) / t_0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * -0.5), $MachinePrecision]], $MachinePrecision]}, N[(2.6666666666666665 * N[(t$95$0 / N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Step-by-step derivation

   1. associate-/l* 99.3%

      \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   2. associate-*r/ 99.3%

      \[\leadsto \color{blue}{\frac{8}{3} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. metadata-eval 99.3%

      \[\leadsto \color{blue}{2.6666666666666665} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   4. remove-double-neg 99.3%

      \[\leadsto 2.6666666666666665 \cdot \frac{\color{blue}{-\left(-\sin \left(x \cdot 0.5\right)\right)}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   5. sin-neg 99.3%

      \[\leadsto 2.6666666666666665 \cdot \frac{-\color{blue}{\sin \left(-x \cdot 0.5\right)}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   6. distribute-lft-neg-out 99.3%

      \[\leadsto 2.6666666666666665 \cdot \frac{-\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   7. neg-mul-1 99.3%

      \[\leadsto 2.6666666666666665 \cdot \frac{\color{blue}{-1 \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   8. *-commutative 99.3%

      \[\leadsto 2.6666666666666665 \cdot \frac{\color{blue}{\sin \left(\left(-x\right) \cdot 0.5\right) \cdot -1}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   9. associate-/l* 99.3%

      \[\leadsto 2.6666666666666665 \cdot \color{blue}{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{\frac{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}{-1}}} \]

   10. distribute-lft-neg-out 99.3%

       \[\leadsto 2.6666666666666665 \cdot \frac{\sin \color{blue}{\left(-x \cdot 0.5\right)}}{\frac{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}{-1}} \]

   11. distribute-rgt-neg-in 99.3%

       \[\leadsto 2.6666666666666665 \cdot \frac{\sin \color{blue}{\left(x \cdot \left(-0.5\right)\right)}}{\frac{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}{-1}} \]

   12. metadata-eval 99.3%

       \[\leadsto 2.6666666666666665 \cdot \frac{\sin \left(x \cdot \color{blue}{-0.5}\right)}{\frac{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}{-1}} \]

   13. associate-/l/ 99.3%

       \[\leadsto 2.6666666666666665 \cdot \frac{\sin \left(x \cdot -0.5\right)}{\color{blue}{\frac{\sin x}{-1 \cdot \sin \left(x \cdot 0.5\right)}}} \]

   14. neg-mul-1 99.3%

       \[\leadsto 2.6666666666666665 \cdot \frac{\sin \left(x \cdot -0.5\right)}{\frac{\sin x}{\color{blue}{-\sin \left(x \cdot 0.5\right)}}} \]

   15. sin-neg 99.3%

       \[\leadsto 2.6666666666666665 \cdot \frac{\sin \left(x \cdot -0.5\right)}{\frac{\sin x}{\color{blue}{\sin \left(-x \cdot 0.5\right)}}} \]

   16. distribute-lft-neg-out 99.3%

       \[\leadsto 2.6666666666666665 \cdot \frac{\sin \left(x \cdot -0.5\right)}{\frac{\sin x}{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}} \]

   17. distribute-lft-neg-out 99.3%

       \[\leadsto 2.6666666666666665 \cdot \frac{\sin \left(x \cdot -0.5\right)}{\frac{\sin x}{\sin \color{blue}{\left(-x \cdot 0.5\right)}}} \]

   18. distribute-rgt-neg-in 99.3%

       \[\leadsto 2.6666666666666665 \cdot \frac{\sin \left(x \cdot -0.5\right)}{\frac{\sin x}{\sin \color{blue}{\left(x \cdot \left(-0.5\right)\right)}}} \]

   19. metadata-eval 99.3%

       \[\leadsto 2.6666666666666665 \cdot \frac{\sin \left(x \cdot -0.5\right)}{\frac{\sin x}{\sin \left(x \cdot \color{blue}{-0.5}\right)}} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{\sin \left(x \cdot -0.5\right)}{\frac{\sin x}{\sin \left(x \cdot -0.5\right)}}} \]

4. Final simplification 99.3%

   \[\leadsto 2.6666666666666665 \cdot \frac{\sin \left(x \cdot -0.5\right)}{\frac{\sin x}{\sin \left(x \cdot -0.5\right)}} \]

Alternative 8: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t_0 \cdot \left(t_0 \cdot \frac{2.6666666666666665}{\sin x}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (* t_0 (* t_0 (/ 2.6666666666666665 (sin x))))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return t_0 * (t_0 * (2.6666666666666665 / sin(x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = t_0 * (t_0 * (2.6666666666666665d0 / sin(x)))
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return t_0 * (t_0 * (2.6666666666666665 / Math.sin(x)));
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return t_0 * (t_0 * (2.6666666666666665 / math.sin(x)))

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(t_0 * Float64(t_0 * Float64(2.6666666666666665 / sin(x))))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = t_0 * (t_0 * (2.6666666666666665 / sin(x)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 * N[(t$95$0 * N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]

4. Final simplification 99.3%

   \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right) \]

Alternative 9: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;\frac{\frac{0.5 + -0.5 \cdot \cos x}{0.375}}{\sin x}\\ \mathbf{elif}\;x \leq 0.00012:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} \cdot \frac{0.5 - \frac{\cos x}{2}}{0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.000145)
   (/ (/ (+ 0.5 (* -0.5 (cos x))) 0.375) (sin x))
   (if (<= x 0.00012)
     (/ (* x 0.25) 0.375)
     (* (/ 1.0 (sin x)) (/ (- 0.5 (/ (cos x) 2.0)) 0.375)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.000145) {
		tmp = ((0.5 + (-0.5 * cos(x))) / 0.375) / sin(x);
	} else if (x <= 0.00012) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (1.0 / sin(x)) * ((0.5 - (cos(x) / 2.0)) / 0.375);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.000145d0)) then
        tmp = ((0.5d0 + ((-0.5d0) * cos(x))) / 0.375d0) / sin(x)
    else if (x <= 0.00012d0) then
        tmp = (x * 0.25d0) / 0.375d0
    else
        tmp = (1.0d0 / sin(x)) * ((0.5d0 - (cos(x) / 2.0d0)) / 0.375d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.000145) {
		tmp = ((0.5 + (-0.5 * Math.cos(x))) / 0.375) / Math.sin(x);
	} else if (x <= 0.00012) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (1.0 / Math.sin(x)) * ((0.5 - (Math.cos(x) / 2.0)) / 0.375);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.000145:
		tmp = ((0.5 + (-0.5 * math.cos(x))) / 0.375) / math.sin(x)
	elif x <= 0.00012:
		tmp = (x * 0.25) / 0.375
	else:
		tmp = (1.0 / math.sin(x)) * ((0.5 - (math.cos(x) / 2.0)) / 0.375)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.000145)
		tmp = Float64(Float64(Float64(0.5 + Float64(-0.5 * cos(x))) / 0.375) / sin(x));
	elseif (x <= 0.00012)
		tmp = Float64(Float64(x * 0.25) / 0.375);
	else
		tmp = Float64(Float64(1.0 / sin(x)) * Float64(Float64(0.5 - Float64(cos(x) / 2.0)) / 0.375));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.000145)
		tmp = ((0.5 + (-0.5 * cos(x))) / 0.375) / sin(x);
	elseif (x <= 0.00012)
		tmp = (x * 0.25) / 0.375;
	else
		tmp = (1.0 / sin(x)) * ((0.5 - (cos(x) / 2.0)) / 0.375);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.000145], N[(N[(N[(0.5 + N[(-0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00012], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(1.0 / N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e-4

   1. Initial program 99.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 99.2%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{2.6666666666666665}{\sin x}} \]

      2. clear-num 99.0%

         \[\leadsto \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{2.6666666666666665}}} \]

      3. un-div-inv 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{2.6666666666666665}}} \]

      4. pow2 99.0%

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\frac{\sin x}{2.6666666666666665}} \]

      5. div-inv 99.2%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\color{blue}{\sin x \cdot \frac{1}{2.6666666666666665}}} \]

      6. metadata-eval 99.2%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot \color{blue}{0.375}} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]
   6. Step-by-step derivation

      1. unpow2 99.2%

         \[\leadsto \frac{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}{\sin x \cdot 0.375} \]

      2. sin-mult 98.1%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

   7. Applied egg-rr 98.1%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]
   8. Step-by-step derivation

      1. div-sub 98.1%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right)}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

      2. +-inverses 98.1%

         \[\leadsto \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      3. cos-0 98.1%

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

      4. metadata-eval 98.1%

         \[\leadsto \frac{\color{blue}{0.5} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      5. distribute-lft-out 98.1%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot \left(0.5 + 0.5\right)\right)}}{2}}{\sin x \cdot 0.375} \]

      6. metadata-eval 98.1%

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

      7. *-rgt-identity 98.1%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x \cdot 0.375} \]

   9. Simplified 98.1%

      \[\leadsto \frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{\sin x \cdot 0.375} \]
   10. Step-by-step derivation

       1. div-inv 98.1%

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

       2. div-inv 98.1%

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

       3. metadata-eval 98.1%

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

       4. *-commutative 98.1%

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

   11. Applied egg-rr 98.1%

       \[\leadsto \color{blue}{\left(0.5 - \cos x \cdot 0.5\right) \cdot \frac{1}{0.375 \cdot \sin x}} \]
   12. Step-by-step derivation

       1. un-div-inv 98.1%

          \[\leadsto \color{blue}{\frac{0.5 - \cos x \cdot 0.5}{0.375 \cdot \sin x}} \]

       2. associate-/r* 98.2%

          \[\leadsto \color{blue}{\frac{\frac{0.5 - \cos x \cdot 0.5}{0.375}}{\sin x}} \]

       3. sub-neg 98.2%

          \[\leadsto \frac{\frac{\color{blue}{0.5 + \left(-\cos x \cdot 0.5\right)}}{0.375}}{\sin x} \]

       4. distribute-rgt-neg-in 98.2%

          \[\leadsto \frac{\frac{0.5 + \color{blue}{\cos x \cdot \left(-0.5\right)}}{0.375}}{\sin x} \]

       5. metadata-eval 98.2%

          \[\leadsto \frac{\frac{0.5 + \cos x \cdot \color{blue}{-0.5}}{0.375}}{\sin x} \]

   13. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{\frac{\frac{0.5 + \cos x \cdot -0.5}{0.375}}{\sin x}} \]

   if -1.45e-4 < x < 1.20000000000000003e-4

   1. Initial program 63.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

      2. *-commutative 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

      3. *-commutative 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)} \]

      4. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      5. add-sqr-sqrt 53.5%

         \[\leadsto \color{blue}{\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}} \]

      6. pow2 53.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}\right)}^{2}} \]

   5. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 53.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      2. swap-sqr 36.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      3. unpow2 36.4%

         \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \]

      4. add-sqr-sqrt 63.2%

         \[\leadsto {\sin \left(x \cdot 0.5\right)}^{2} \cdot \color{blue}{\frac{2.6666666666666665}{\sin x}} \]

      5. metadata-eval 63.2%

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

      6. associate-/r* 63.2%

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

      7. *-commutative 63.2%

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

      8. div-inv 63.2%

         \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]

      9. associate-/r* 63.4%

         \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]

   if 1.20000000000000003e-4 < x

   1. Initial program 99.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.1%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

      2. *-commutative 99.1%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

      3. clear-num 99.1%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}} \]

      4. un-div-inv 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}} \]

      5. *-un-lft-identity 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\color{blue}{1 \cdot \sin x}}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}} \]

      6. times-frac 99.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{1}{2.6666666666666665} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      7. metadata-eval 99.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.375} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   5. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{0.375 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{0.375 \cdot \sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      2. *-commutative 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\color{blue}{\sin x \cdot 0.375}}{\sin \left(x \cdot 0.5\right)}} \]

      3. associate-/l* 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x \cdot 0.375}} \]

      4. unpow2 99.1%

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x \cdot 0.375} \]

      5. *-un-lft-identity 99.1%

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

      6. times-frac 99.3%

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

   7. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{1}{\sin x} \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375}} \]
   8. Step-by-step derivation

      1. unpow2 99.1%

         \[\leadsto \frac{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}{\sin x \cdot 0.375} \]

      2. sin-mult 98.2%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

   9. Applied egg-rr 98.4%

      \[\leadsto \frac{1}{\sin x} \cdot \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{0.375} \]
   10. Step-by-step derivation

       1. div-sub 98.2%

          \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right)}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

       2. +-inverses 98.2%

          \[\leadsto \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

       3. cos-0 98.2%

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

       4. metadata-eval 98.2%

          \[\leadsto \frac{\color{blue}{0.5} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

       5. distribute-lft-out 98.2%

          \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot \left(0.5 + 0.5\right)\right)}}{2}}{\sin x \cdot 0.375} \]

       6. metadata-eval 98.2%

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

       7. *-rgt-identity 98.2%

          \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x \cdot 0.375} \]

   11. Simplified 98.4%

       \[\leadsto \frac{1}{\sin x} \cdot \frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{0.375} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;\frac{\frac{0.5 + -0.5 \cdot \cos x}{0.375}}{\sin x}\\ \mathbf{elif}\;x \leq 0.00012:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} \cdot \frac{0.5 - \frac{\cos x}{2}}{0.375}\\ \end{array} \]

Alternative 10: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000145 \lor \neg \left(x \leq 0.00012\right):\\ \;\;\;\;\frac{2.6666666666666665}{\sin x} \cdot \left(0.5 - 0.5 \cdot \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.000145) (not (<= x 0.00012)))
   (* (/ 2.6666666666666665 (sin x)) (- 0.5 (* 0.5 (cos x))))
   (/ (* x 0.25) 0.375)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.000145) || !(x <= 0.00012)) {
		tmp = (2.6666666666666665 / sin(x)) * (0.5 - (0.5 * cos(x)));
	} else {
		tmp = (x * 0.25) / 0.375;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.000145d0)) .or. (.not. (x <= 0.00012d0))) then
        tmp = (2.6666666666666665d0 / sin(x)) * (0.5d0 - (0.5d0 * cos(x)))
    else
        tmp = (x * 0.25d0) / 0.375d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.000145) || !(x <= 0.00012)) {
		tmp = (2.6666666666666665 / Math.sin(x)) * (0.5 - (0.5 * Math.cos(x)));
	} else {
		tmp = (x * 0.25) / 0.375;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.000145) or not (x <= 0.00012):
		tmp = (2.6666666666666665 / math.sin(x)) * (0.5 - (0.5 * math.cos(x)))
	else:
		tmp = (x * 0.25) / 0.375
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.000145) || !(x <= 0.00012))
		tmp = Float64(Float64(2.6666666666666665 / sin(x)) * Float64(0.5 - Float64(0.5 * cos(x))));
	else
		tmp = Float64(Float64(x * 0.25) / 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.000145) || ~((x <= 0.00012)))
		tmp = (2.6666666666666665 / sin(x)) * (0.5 - (0.5 * cos(x)));
	else
		tmp = (x * 0.25) / 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.000145], N[Not[LessEqual[x, 0.00012]], $MachinePrecision]], N[(N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45e-4 or 1.20000000000000003e-4 < x

   1. Initial program 99.1%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{2.6666666666666665}{\sin x}} \]

      2. clear-num 99.0%

         \[\leadsto \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{2.6666666666666665}}} \]

      3. un-div-inv 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{2.6666666666666665}}} \]

      4. pow2 99.0%

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\frac{\sin x}{2.6666666666666665}} \]

      5. div-inv 99.1%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\color{blue}{\sin x \cdot \frac{1}{2.6666666666666665}}} \]

      6. metadata-eval 99.1%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot \color{blue}{0.375}} \]

   5. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]
   6. Step-by-step derivation

      1. unpow2 99.1%

         \[\leadsto \frac{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}{\sin x \cdot 0.375} \]

      2. sin-mult 98.2%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

   7. Applied egg-rr 98.2%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]
   8. Step-by-step derivation

      1. div-sub 98.2%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right)}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

      2. +-inverses 98.2%

         \[\leadsto \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      3. cos-0 98.2%

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

      4. metadata-eval 98.2%

         \[\leadsto \frac{\color{blue}{0.5} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      5. distribute-lft-out 98.2%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot \left(0.5 + 0.5\right)\right)}}{2}}{\sin x \cdot 0.375} \]

      6. metadata-eval 98.2%

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

      7. *-rgt-identity 98.2%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x \cdot 0.375} \]

   9. Simplified 98.2%

      \[\leadsto \frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{\sin x \cdot 0.375} \]
   10. Step-by-step derivation

       1. div-inv 98.2%

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

       2. div-inv 98.2%

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

       3. metadata-eval 98.2%

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

       4. *-commutative 98.2%

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

   11. Applied egg-rr 98.2%

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

   12. Taylor expanded in x around inf 98.2%

       \[\leadsto \left(0.5 - \cos x \cdot 0.5\right) \cdot \color{blue}{\frac{2.6666666666666665}{\sin x}} \]

   if -1.45e-4 < x < 1.20000000000000003e-4

   1. Initial program 63.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

      2. *-commutative 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

      3. *-commutative 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)} \]

      4. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      5. add-sqr-sqrt 53.5%

         \[\leadsto \color{blue}{\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}} \]

      6. pow2 53.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}\right)}^{2}} \]

   5. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 53.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      2. swap-sqr 36.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      3. unpow2 36.4%

         \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \]

      4. add-sqr-sqrt 63.2%

         \[\leadsto {\sin \left(x \cdot 0.5\right)}^{2} \cdot \color{blue}{\frac{2.6666666666666665}{\sin x}} \]

      5. metadata-eval 63.2%

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

      6. associate-/r* 63.2%

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

      7. *-commutative 63.2%

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

      8. div-inv 63.2%

         \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]

      9. associate-/r* 63.4%

         \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000145 \lor \neg \left(x \leq 0.00012\right):\\ \;\;\;\;\frac{2.6666666666666665}{\sin x} \cdot \left(0.5 - 0.5 \cdot \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \end{array} \]

Alternative 11: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000145 \lor \neg \left(x \leq 0.00012\right):\\ \;\;\;\;\frac{\frac{0.5 + -0.5 \cdot \cos x}{0.375}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.000145) (not (<= x 0.00012)))
   (/ (/ (+ 0.5 (* -0.5 (cos x))) 0.375) (sin x))
   (/ (* x 0.25) 0.375)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.000145) || !(x <= 0.00012)) {
		tmp = ((0.5 + (-0.5 * cos(x))) / 0.375) / sin(x);
	} else {
		tmp = (x * 0.25) / 0.375;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.000145d0)) .or. (.not. (x <= 0.00012d0))) then
        tmp = ((0.5d0 + ((-0.5d0) * cos(x))) / 0.375d0) / sin(x)
    else
        tmp = (x * 0.25d0) / 0.375d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.000145) || !(x <= 0.00012)) {
		tmp = ((0.5 + (-0.5 * Math.cos(x))) / 0.375) / Math.sin(x);
	} else {
		tmp = (x * 0.25) / 0.375;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.000145) or not (x <= 0.00012):
		tmp = ((0.5 + (-0.5 * math.cos(x))) / 0.375) / math.sin(x)
	else:
		tmp = (x * 0.25) / 0.375
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.000145) || !(x <= 0.00012))
		tmp = Float64(Float64(Float64(0.5 + Float64(-0.5 * cos(x))) / 0.375) / sin(x));
	else
		tmp = Float64(Float64(x * 0.25) / 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.000145) || ~((x <= 0.00012)))
		tmp = ((0.5 + (-0.5 * cos(x))) / 0.375) / sin(x);
	else
		tmp = (x * 0.25) / 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.000145], N[Not[LessEqual[x, 0.00012]], $MachinePrecision]], N[(N[(N[(0.5 + N[(-0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45e-4 or 1.20000000000000003e-4 < x

   1. Initial program 99.1%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{2.6666666666666665}{\sin x}} \]

      2. clear-num 99.0%

         \[\leadsto \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{2.6666666666666665}}} \]

      3. un-div-inv 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{2.6666666666666665}}} \]

      4. pow2 99.0%

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\frac{\sin x}{2.6666666666666665}} \]

      5. div-inv 99.1%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\color{blue}{\sin x \cdot \frac{1}{2.6666666666666665}}} \]

      6. metadata-eval 99.1%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot \color{blue}{0.375}} \]

   5. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]
   6. Step-by-step derivation

      1. unpow2 99.1%

         \[\leadsto \frac{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}{\sin x \cdot 0.375} \]

      2. sin-mult 98.2%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

   7. Applied egg-rr 98.2%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]
   8. Step-by-step derivation

      1. div-sub 98.2%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right)}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

      2. +-inverses 98.2%

         \[\leadsto \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      3. cos-0 98.2%

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

      4. metadata-eval 98.2%

         \[\leadsto \frac{\color{blue}{0.5} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      5. distribute-lft-out 98.2%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot \left(0.5 + 0.5\right)\right)}}{2}}{\sin x \cdot 0.375} \]

      6. metadata-eval 98.2%

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

      7. *-rgt-identity 98.2%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x \cdot 0.375} \]

   9. Simplified 98.2%

      \[\leadsto \frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{\sin x \cdot 0.375} \]
   10. Step-by-step derivation

       1. div-inv 98.2%

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

       2. div-inv 98.2%

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

       3. metadata-eval 98.2%

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

       4. *-commutative 98.2%

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

   11. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{\left(0.5 - \cos x \cdot 0.5\right) \cdot \frac{1}{0.375 \cdot \sin x}} \]
   12. Step-by-step derivation

       1. un-div-inv 98.2%

          \[\leadsto \color{blue}{\frac{0.5 - \cos x \cdot 0.5}{0.375 \cdot \sin x}} \]

       2. associate-/r* 98.2%

          \[\leadsto \color{blue}{\frac{\frac{0.5 - \cos x \cdot 0.5}{0.375}}{\sin x}} \]

       3. sub-neg 98.2%

          \[\leadsto \frac{\frac{\color{blue}{0.5 + \left(-\cos x \cdot 0.5\right)}}{0.375}}{\sin x} \]

       4. distribute-rgt-neg-in 98.2%

          \[\leadsto \frac{\frac{0.5 + \color{blue}{\cos x \cdot \left(-0.5\right)}}{0.375}}{\sin x} \]

       5. metadata-eval 98.2%

          \[\leadsto \frac{\frac{0.5 + \cos x \cdot \color{blue}{-0.5}}{0.375}}{\sin x} \]

   13. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{\frac{\frac{0.5 + \cos x \cdot -0.5}{0.375}}{\sin x}} \]

   if -1.45e-4 < x < 1.20000000000000003e-4

   1. Initial program 63.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

      2. *-commutative 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

      3. *-commutative 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)} \]

      4. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      5. add-sqr-sqrt 53.5%

         \[\leadsto \color{blue}{\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}} \]

      6. pow2 53.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}\right)}^{2}} \]

   5. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 53.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      2. swap-sqr 36.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      3. unpow2 36.4%

         \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \]

      4. add-sqr-sqrt 63.2%

         \[\leadsto {\sin \left(x \cdot 0.5\right)}^{2} \cdot \color{blue}{\frac{2.6666666666666665}{\sin x}} \]

      5. metadata-eval 63.2%

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

      6. associate-/r* 63.2%

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

      7. *-commutative 63.2%

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

      8. div-inv 63.2%

         \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]

      9. associate-/r* 63.4%

         \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000145 \lor \neg \left(x \leq 0.00012\right):\\ \;\;\;\;\frac{\frac{0.5 + -0.5 \cdot \cos x}{0.375}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \end{array} \]

Alternative 12: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x} \cdot \left(0.5 - 0.5 \cdot \cos x\right)\\ \mathbf{elif}\;x \leq 0.00012:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665 \cdot \left(0.5 + -0.5 \cdot \cos x\right)}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.000145)
   (* (/ 2.6666666666666665 (sin x)) (- 0.5 (* 0.5 (cos x))))
   (if (<= x 0.00012)
     (/ (* x 0.25) 0.375)
     (/ (* 2.6666666666666665 (+ 0.5 (* -0.5 (cos x)))) (sin x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.000145) {
		tmp = (2.6666666666666665 / sin(x)) * (0.5 - (0.5 * cos(x)));
	} else if (x <= 0.00012) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (2.6666666666666665 * (0.5 + (-0.5 * cos(x)))) / sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.000145d0)) then
        tmp = (2.6666666666666665d0 / sin(x)) * (0.5d0 - (0.5d0 * cos(x)))
    else if (x <= 0.00012d0) then
        tmp = (x * 0.25d0) / 0.375d0
    else
        tmp = (2.6666666666666665d0 * (0.5d0 + ((-0.5d0) * cos(x)))) / sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.000145) {
		tmp = (2.6666666666666665 / Math.sin(x)) * (0.5 - (0.5 * Math.cos(x)));
	} else if (x <= 0.00012) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (2.6666666666666665 * (0.5 + (-0.5 * Math.cos(x)))) / Math.sin(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.000145:
		tmp = (2.6666666666666665 / math.sin(x)) * (0.5 - (0.5 * math.cos(x)))
	elif x <= 0.00012:
		tmp = (x * 0.25) / 0.375
	else:
		tmp = (2.6666666666666665 * (0.5 + (-0.5 * math.cos(x)))) / math.sin(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.000145)
		tmp = Float64(Float64(2.6666666666666665 / sin(x)) * Float64(0.5 - Float64(0.5 * cos(x))));
	elseif (x <= 0.00012)
		tmp = Float64(Float64(x * 0.25) / 0.375);
	else
		tmp = Float64(Float64(2.6666666666666665 * Float64(0.5 + Float64(-0.5 * cos(x)))) / sin(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.000145)
		tmp = (2.6666666666666665 / sin(x)) * (0.5 - (0.5 * cos(x)));
	elseif (x <= 0.00012)
		tmp = (x * 0.25) / 0.375;
	else
		tmp = (2.6666666666666665 * (0.5 + (-0.5 * cos(x)))) / sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.000145], N[(N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00012], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(2.6666666666666665 * N[(0.5 + N[(-0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e-4

   1. Initial program 99.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 99.2%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{2.6666666666666665}{\sin x}} \]

      2. clear-num 99.0%

         \[\leadsto \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{2.6666666666666665}}} \]

      3. un-div-inv 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{2.6666666666666665}}} \]

      4. pow2 99.0%

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\frac{\sin x}{2.6666666666666665}} \]

      5. div-inv 99.2%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\color{blue}{\sin x \cdot \frac{1}{2.6666666666666665}}} \]

      6. metadata-eval 99.2%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot \color{blue}{0.375}} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]
   6. Step-by-step derivation

      1. unpow2 99.2%

         \[\leadsto \frac{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}{\sin x \cdot 0.375} \]

      2. sin-mult 98.1%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

   7. Applied egg-rr 98.1%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]
   8. Step-by-step derivation

      1. div-sub 98.1%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right)}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

      2. +-inverses 98.1%

         \[\leadsto \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      3. cos-0 98.1%

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

      4. metadata-eval 98.1%

         \[\leadsto \frac{\color{blue}{0.5} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      5. distribute-lft-out 98.1%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot \left(0.5 + 0.5\right)\right)}}{2}}{\sin x \cdot 0.375} \]

      6. metadata-eval 98.1%

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

      7. *-rgt-identity 98.1%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x \cdot 0.375} \]

   9. Simplified 98.1%

      \[\leadsto \frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{\sin x \cdot 0.375} \]
   10. Step-by-step derivation

       1. div-inv 98.1%

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

       2. div-inv 98.1%

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

       3. metadata-eval 98.1%

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

       4. *-commutative 98.1%

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

   11. Applied egg-rr 98.1%

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

   12. Taylor expanded in x around inf 98.2%

       \[\leadsto \left(0.5 - \cos x \cdot 0.5\right) \cdot \color{blue}{\frac{2.6666666666666665}{\sin x}} \]

   if -1.45e-4 < x < 1.20000000000000003e-4

   1. Initial program 63.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

      2. *-commutative 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

      3. *-commutative 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)} \]

      4. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      5. add-sqr-sqrt 53.5%

         \[\leadsto \color{blue}{\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}} \]

      6. pow2 53.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}\right)}^{2}} \]

   5. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 53.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      2. swap-sqr 36.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      3. unpow2 36.4%

         \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \]

      4. add-sqr-sqrt 63.2%

         \[\leadsto {\sin \left(x \cdot 0.5\right)}^{2} \cdot \color{blue}{\frac{2.6666666666666665}{\sin x}} \]

      5. metadata-eval 63.2%

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

      6. associate-/r* 63.2%

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

      7. *-commutative 63.2%

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

      8. div-inv 63.2%

         \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]

      9. associate-/r* 63.4%

         \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]

   if 1.20000000000000003e-4 < x

   1. Initial program 99.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{2.6666666666666665}{\sin x}} \]

      2. clear-num 99.1%

         \[\leadsto \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{2.6666666666666665}}} \]

      3. un-div-inv 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{2.6666666666666665}}} \]

      4. pow2 99.0%

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\frac{\sin x}{2.6666666666666665}} \]

      5. div-inv 99.1%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\color{blue}{\sin x \cdot \frac{1}{2.6666666666666665}}} \]

      6. metadata-eval 99.1%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot \color{blue}{0.375}} \]

   5. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]
   6. Step-by-step derivation

      1. unpow2 99.1%

         \[\leadsto \frac{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}{\sin x \cdot 0.375} \]

      2. sin-mult 98.2%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

   7. Applied egg-rr 98.2%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]
   8. Step-by-step derivation

      1. div-sub 98.2%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right)}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

      2. +-inverses 98.2%

         \[\leadsto \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      3. cos-0 98.2%

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

      4. metadata-eval 98.2%

         \[\leadsto \frac{\color{blue}{0.5} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      5. distribute-lft-out 98.2%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot \left(0.5 + 0.5\right)\right)}}{2}}{\sin x \cdot 0.375} \]

      6. metadata-eval 98.2%

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

      7. *-rgt-identity 98.2%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x \cdot 0.375} \]

   9. Simplified 98.2%

      \[\leadsto \frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{\sin x \cdot 0.375} \]
   10. Step-by-step derivation

       1. div-inv 98.2%

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

       2. div-inv 98.2%

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

       3. metadata-eval 98.2%

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

       4. *-commutative 98.2%

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

   11. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{\left(0.5 - \cos x \cdot 0.5\right) \cdot \frac{1}{0.375 \cdot \sin x}} \]
   12. Step-by-step derivation

       1. associate-/r* 98.2%

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

       2. metadata-eval 98.2%

          \[\leadsto \left(0.5 - \cos x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665}}{\sin x} \]

       3. associate-*r/ 98.3%

          \[\leadsto \color{blue}{\frac{\left(0.5 - \cos x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

       4. sub-neg 98.3%

          \[\leadsto \frac{\color{blue}{\left(0.5 + \left(-\cos x \cdot 0.5\right)\right)} \cdot 2.6666666666666665}{\sin x} \]

       5. distribute-rgt-neg-in 98.3%

          \[\leadsto \frac{\left(0.5 + \color{blue}{\cos x \cdot \left(-0.5\right)}\right) \cdot 2.6666666666666665}{\sin x} \]

       6. metadata-eval 98.3%

          \[\leadsto \frac{\left(0.5 + \cos x \cdot \color{blue}{-0.5}\right) \cdot 2.6666666666666665}{\sin x} \]

   13. Applied egg-rr 98.3%

       \[\leadsto \color{blue}{\frac{\left(0.5 + \cos x \cdot -0.5\right) \cdot 2.6666666666666665}{\sin x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x} \cdot \left(0.5 - 0.5 \cdot \cos x\right)\\ \mathbf{elif}\;x \leq 0.00012:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665 \cdot \left(0.5 + -0.5 \cdot \cos x\right)}{\sin x}\\ \end{array} \]

Alternative 13: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0002 \lor \neg \left(x \leq 0.0002\right):\\ \;\;\;\;\frac{1.3333333333333333 + \cos x \cdot -1.3333333333333333}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.0002) (not (<= x 0.0002)))
   (/ (+ 1.3333333333333333 (* (cos x) -1.3333333333333333)) (sin x))
   (/ (* x 0.25) 0.375)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.0002) || !(x <= 0.0002)) {
		tmp = (1.3333333333333333 + (cos(x) * -1.3333333333333333)) / sin(x);
	} else {
		tmp = (x * 0.25) / 0.375;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.0002d0)) .or. (.not. (x <= 0.0002d0))) then
        tmp = (1.3333333333333333d0 + (cos(x) * (-1.3333333333333333d0))) / sin(x)
    else
        tmp = (x * 0.25d0) / 0.375d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.0002) || !(x <= 0.0002)) {
		tmp = (1.3333333333333333 + (Math.cos(x) * -1.3333333333333333)) / Math.sin(x);
	} else {
		tmp = (x * 0.25) / 0.375;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.0002) or not (x <= 0.0002):
		tmp = (1.3333333333333333 + (math.cos(x) * -1.3333333333333333)) / math.sin(x)
	else:
		tmp = (x * 0.25) / 0.375
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.0002) || !(x <= 0.0002))
		tmp = Float64(Float64(1.3333333333333333 + Float64(cos(x) * -1.3333333333333333)) / sin(x));
	else
		tmp = Float64(Float64(x * 0.25) / 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.0002) || ~((x <= 0.0002)))
		tmp = (1.3333333333333333 + (cos(x) * -1.3333333333333333)) / sin(x);
	else
		tmp = (x * 0.25) / 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.0002], N[Not[LessEqual[x, 0.0002]], $MachinePrecision]], N[(N[(1.3333333333333333 + N[(N[Cos[x], $MachinePrecision] * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e-4 or 2.0000000000000001e-4 < x

   1. Initial program 99.1%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{2.6666666666666665}{\sin x}} \]

      2. clear-num 99.0%

         \[\leadsto \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{2.6666666666666665}}} \]

      3. un-div-inv 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{2.6666666666666665}}} \]

      4. pow2 99.0%

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\frac{\sin x}{2.6666666666666665}} \]

      5. div-inv 99.1%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\color{blue}{\sin x \cdot \frac{1}{2.6666666666666665}}} \]

      6. metadata-eval 99.1%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot \color{blue}{0.375}} \]

   5. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]
   6. Step-by-step derivation

      1. unpow2 99.1%

         \[\leadsto \frac{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}{\sin x \cdot 0.375} \]

      2. sin-mult 98.2%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

   7. Applied egg-rr 98.2%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]
   8. Step-by-step derivation

      1. div-sub 98.2%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right)}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

      2. +-inverses 98.2%

         \[\leadsto \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      3. cos-0 98.2%

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

      4. metadata-eval 98.2%

         \[\leadsto \frac{\color{blue}{0.5} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      5. distribute-lft-out 98.2%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot \left(0.5 + 0.5\right)\right)}}{2}}{\sin x \cdot 0.375} \]

      6. metadata-eval 98.2%

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

      7. *-rgt-identity 98.2%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x \cdot 0.375} \]

   9. Simplified 98.2%

      \[\leadsto \frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{\sin x \cdot 0.375} \]
   10. Step-by-step derivation

       1. div-inv 98.2%

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

       2. div-inv 98.2%

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

       3. metadata-eval 98.2%

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

       4. *-commutative 98.2%

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

   11. Applied egg-rr 98.2%

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

   12. Taylor expanded in x around inf 98.1%

       \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}} \]
   13. Step-by-step derivation

       1. associate-*r/ 98.2%

          \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \left(0.5 - 0.5 \cdot \cos x\right)}{\sin x}} \]

       2. cancel-sign-sub-inv 98.2%

          \[\leadsto \frac{2.6666666666666665 \cdot \color{blue}{\left(0.5 + \left(-0.5\right) \cdot \cos x\right)}}{\sin x} \]

       3. metadata-eval 98.2%

          \[\leadsto \frac{2.6666666666666665 \cdot \left(0.5 + \color{blue}{-0.5} \cdot \cos x\right)}{\sin x} \]

       4. *-commutative 98.2%

          \[\leadsto \frac{2.6666666666666665 \cdot \left(0.5 + \color{blue}{\cos x \cdot -0.5}\right)}{\sin x} \]

       5. distribute-lft-in 98.1%

          \[\leadsto \frac{\color{blue}{2.6666666666666665 \cdot 0.5 + 2.6666666666666665 \cdot \left(\cos x \cdot -0.5\right)}}{\sin x} \]

       6. metadata-eval 98.1%

          \[\leadsto \frac{\color{blue}{1.3333333333333333} + 2.6666666666666665 \cdot \left(\cos x \cdot -0.5\right)}{\sin x} \]

       7. *-commutative 98.1%

          \[\leadsto \frac{1.3333333333333333 + 2.6666666666666665 \cdot \color{blue}{\left(-0.5 \cdot \cos x\right)}}{\sin x} \]

       8. associate-*r* 98.1%

          \[\leadsto \frac{1.3333333333333333 + \color{blue}{\left(2.6666666666666665 \cdot -0.5\right) \cdot \cos x}}{\sin x} \]

       9. metadata-eval 98.1%

          \[\leadsto \frac{1.3333333333333333 + \color{blue}{-1.3333333333333333} \cdot \cos x}{\sin x} \]

   14. Simplified 98.1%

       \[\leadsto \color{blue}{\frac{1.3333333333333333 + -1.3333333333333333 \cdot \cos x}{\sin x}} \]

   if -2.0000000000000001e-4 < x < 2.0000000000000001e-4

   1. Initial program 63.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

      2. *-commutative 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

      3. *-commutative 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)} \]

      4. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      5. add-sqr-sqrt 53.5%

         \[\leadsto \color{blue}{\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}} \]

      6. pow2 53.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}\right)}^{2}} \]

   5. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 53.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      2. swap-sqr 36.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      3. unpow2 36.4%

         \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \]

      4. add-sqr-sqrt 63.2%

         \[\leadsto {\sin \left(x \cdot 0.5\right)}^{2} \cdot \color{blue}{\frac{2.6666666666666665}{\sin x}} \]

      5. metadata-eval 63.2%

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

      6. associate-/r* 63.2%

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

      7. *-commutative 63.2%

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

      8. div-inv 63.2%

         \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]

      9. associate-/r* 63.4%

         \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0002 \lor \neg \left(x \leq 0.0002\right):\\ \;\;\;\;\frac{1.3333333333333333 + \cos x \cdot -1.3333333333333333}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \end{array} \]

Alternative 14: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}\\ \mathbf{elif}\;x \leq 0.0002:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.3333333333333333 + \cos x \cdot -1.3333333333333333}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.000145)
   (* 2.6666666666666665 (/ (- 0.5 (* 0.5 (cos x))) (sin x)))
   (if (<= x 0.0002)
     (/ (* x 0.25) 0.375)
     (/ (+ 1.3333333333333333 (* (cos x) -1.3333333333333333)) (sin x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.000145) {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x))) / sin(x));
	} else if (x <= 0.0002) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (1.3333333333333333 + (cos(x) * -1.3333333333333333)) / sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.000145d0)) then
        tmp = 2.6666666666666665d0 * ((0.5d0 - (0.5d0 * cos(x))) / sin(x))
    else if (x <= 0.0002d0) then
        tmp = (x * 0.25d0) / 0.375d0
    else
        tmp = (1.3333333333333333d0 + (cos(x) * (-1.3333333333333333d0))) / sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.000145) {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * Math.cos(x))) / Math.sin(x));
	} else if (x <= 0.0002) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (1.3333333333333333 + (Math.cos(x) * -1.3333333333333333)) / Math.sin(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.000145:
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * math.cos(x))) / math.sin(x))
	elif x <= 0.0002:
		tmp = (x * 0.25) / 0.375
	else:
		tmp = (1.3333333333333333 + (math.cos(x) * -1.3333333333333333)) / math.sin(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.000145)
		tmp = Float64(2.6666666666666665 * Float64(Float64(0.5 - Float64(0.5 * cos(x))) / sin(x)));
	elseif (x <= 0.0002)
		tmp = Float64(Float64(x * 0.25) / 0.375);
	else
		tmp = Float64(Float64(1.3333333333333333 + Float64(cos(x) * -1.3333333333333333)) / sin(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.000145)
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x))) / sin(x));
	elseif (x <= 0.0002)
		tmp = (x * 0.25) / 0.375;
	else
		tmp = (1.3333333333333333 + (cos(x) * -1.3333333333333333)) / sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.000145], N[(2.6666666666666665 * N[(N[(0.5 - N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0002], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(1.3333333333333333 + N[(N[Cos[x], $MachinePrecision] * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e-4

   1. Initial program 99.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 99.2%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{2.6666666666666665}{\sin x}} \]

      2. clear-num 99.0%

         \[\leadsto \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{2.6666666666666665}}} \]

      3. un-div-inv 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{2.6666666666666665}}} \]

      4. pow2 99.0%

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\frac{\sin x}{2.6666666666666665}} \]

      5. div-inv 99.2%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\color{blue}{\sin x \cdot \frac{1}{2.6666666666666665}}} \]

      6. metadata-eval 99.2%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot \color{blue}{0.375}} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]
   6. Step-by-step derivation

      1. unpow2 99.2%

         \[\leadsto \frac{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}{\sin x \cdot 0.375} \]

      2. sin-mult 98.1%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

   7. Applied egg-rr 98.1%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]
   8. Step-by-step derivation

      1. div-sub 98.1%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right)}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

      2. +-inverses 98.1%

         \[\leadsto \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      3. cos-0 98.1%

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

      4. metadata-eval 98.1%

         \[\leadsto \frac{\color{blue}{0.5} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      5. distribute-lft-out 98.1%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot \left(0.5 + 0.5\right)\right)}}{2}}{\sin x \cdot 0.375} \]

      6. metadata-eval 98.1%

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

      7. *-rgt-identity 98.1%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x \cdot 0.375} \]

   9. Simplified 98.1%

      \[\leadsto \frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{\sin x \cdot 0.375} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 98.1%

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

       2. *-commutative 98.1%

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

       3. times-frac 98.1%

          \[\leadsto \color{blue}{\frac{1}{0.375} \cdot \frac{0.5 - \frac{\cos x}{2}}{\sin x}} \]

       4. metadata-eval 98.1%

          \[\leadsto \color{blue}{2.6666666666666665} \cdot \frac{0.5 - \frac{\cos x}{2}}{\sin x} \]

       5. div-inv 98.1%

          \[\leadsto 2.6666666666666665 \cdot \frac{0.5 - \color{blue}{\cos x \cdot \frac{1}{2}}}{\sin x} \]

       6. metadata-eval 98.1%

          \[\leadsto 2.6666666666666665 \cdot \frac{0.5 - \cos x \cdot \color{blue}{0.5}}{\sin x} \]

   11. Applied egg-rr 98.1%

       \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{0.5 - \cos x \cdot 0.5}{\sin x}} \]

   if -1.45e-4 < x < 2.0000000000000001e-4

   1. Initial program 63.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

      2. *-commutative 99.5%

         \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

      3. *-commutative 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)} \]

      4. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      5. add-sqr-sqrt 53.5%

         \[\leadsto \color{blue}{\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}} \]

      6. pow2 53.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}\right)}^{2}} \]

   5. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 53.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      2. swap-sqr 36.4%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

      3. unpow2 36.4%

         \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \]

      4. add-sqr-sqrt 63.2%

         \[\leadsto {\sin \left(x \cdot 0.5\right)}^{2} \cdot \color{blue}{\frac{2.6666666666666665}{\sin x}} \]

      5. metadata-eval 63.2%

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

      6. associate-/r* 63.2%

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

      7. *-commutative 63.2%

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

      8. div-inv 63.2%

         \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]

      9. associate-/r* 63.4%

         \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]

   if 2.0000000000000001e-4 < x

   1. Initial program 99.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{2.6666666666666665}{\sin x}} \]

      2. clear-num 99.1%

         \[\leadsto \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{2.6666666666666665}}} \]

      3. un-div-inv 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{2.6666666666666665}}} \]

      4. pow2 99.0%

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\frac{\sin x}{2.6666666666666665}} \]

      5. div-inv 99.1%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\color{blue}{\sin x \cdot \frac{1}{2.6666666666666665}}} \]

      6. metadata-eval 99.1%

         \[\leadsto \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot \color{blue}{0.375}} \]

   5. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]
   6. Step-by-step derivation

      1. unpow2 99.1%

         \[\leadsto \frac{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}{\sin x \cdot 0.375} \]

      2. sin-mult 98.2%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

   7. Applied egg-rr 98.2%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]
   8. Step-by-step derivation

      1. div-sub 98.2%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right)}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x \cdot 0.375} \]

      2. +-inverses 98.2%

         \[\leadsto \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      3. cos-0 98.2%

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

      4. metadata-eval 98.2%

         \[\leadsto \frac{\color{blue}{0.5} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x \cdot 0.375} \]

      5. distribute-lft-out 98.2%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot \left(0.5 + 0.5\right)\right)}}{2}}{\sin x \cdot 0.375} \]

      6. metadata-eval 98.2%

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

      7. *-rgt-identity 98.2%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x \cdot 0.375} \]

   9. Simplified 98.2%

      \[\leadsto \frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{\sin x \cdot 0.375} \]
   10. Step-by-step derivation

       1. div-inv 98.2%

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

       2. div-inv 98.2%

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

       3. metadata-eval 98.2%

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

       4. *-commutative 98.2%

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

   11. Applied egg-rr 98.2%

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

   12. Taylor expanded in x around inf 98.1%

       \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}} \]
   13. Step-by-step derivation

       1. associate-*r/ 98.3%

          \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \left(0.5 - 0.5 \cdot \cos x\right)}{\sin x}} \]

       2. cancel-sign-sub-inv 98.3%

          \[\leadsto \frac{2.6666666666666665 \cdot \color{blue}{\left(0.5 + \left(-0.5\right) \cdot \cos x\right)}}{\sin x} \]

       3. metadata-eval 98.3%

          \[\leadsto \frac{2.6666666666666665 \cdot \left(0.5 + \color{blue}{-0.5} \cdot \cos x\right)}{\sin x} \]

       4. *-commutative 98.3%

          \[\leadsto \frac{2.6666666666666665 \cdot \left(0.5 + \color{blue}{\cos x \cdot -0.5}\right)}{\sin x} \]

       5. distribute-lft-in 98.2%

          \[\leadsto \frac{\color{blue}{2.6666666666666665 \cdot 0.5 + 2.6666666666666665 \cdot \left(\cos x \cdot -0.5\right)}}{\sin x} \]

       6. metadata-eval 98.2%

          \[\leadsto \frac{\color{blue}{1.3333333333333333} + 2.6666666666666665 \cdot \left(\cos x \cdot -0.5\right)}{\sin x} \]

       7. *-commutative 98.2%

          \[\leadsto \frac{1.3333333333333333 + 2.6666666666666665 \cdot \color{blue}{\left(-0.5 \cdot \cos x\right)}}{\sin x} \]

       8. associate-*r* 98.2%

          \[\leadsto \frac{1.3333333333333333 + \color{blue}{\left(2.6666666666666665 \cdot -0.5\right) \cdot \cos x}}{\sin x} \]

       9. metadata-eval 98.2%

          \[\leadsto \frac{1.3333333333333333 + \color{blue}{-1.3333333333333333} \cdot \cos x}{\sin x} \]

   14. Simplified 98.2%

       \[\leadsto \color{blue}{\frac{1.3333333333333333 + -1.3333333333333333 \cdot \cos x}{\sin x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}\\ \mathbf{elif}\;x \leq 0.0002:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.3333333333333333 + \cos x \cdot -1.3333333333333333}{\sin x}\\ \end{array} \]

Alternative 15: 54.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(x \cdot 0.5\right) \cdot 1.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sin (* x 0.5)) 1.3333333333333333))

C

double code(double x) {
	return sin((x * 0.5)) * 1.3333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sin((x * 0.5d0)) * 1.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.sin((x * 0.5)) * 1.3333333333333333;
}

Python

def code(x):
	return math.sin((x * 0.5)) * 1.3333333333333333

Julia

function code(x)
	return Float64(sin(Float64(x * 0.5)) * 1.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = sin((x * 0.5)) * 1.3333333333333333;
end

Wolfram

code[x_] := N[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * 1.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 81.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]

4. Taylor expanded in x around 0 54.8%

   \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{1.3333333333333333} \]

5. Final simplification 54.8%

   \[\leadsto \sin \left(x \cdot 0.5\right) \cdot 1.3333333333333333 \]

Alternative 16: 55.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(x \cdot 0.5\right)}{0.75} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (sin (* x 0.5)) 0.75))

C

double code(double x) {
	return sin((x * 0.5)) / 0.75;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sin((x * 0.5d0)) / 0.75d0
end function

Java

public static double code(double x) {
	return Math.sin((x * 0.5)) / 0.75;
}

Python

def code(x):
	return math.sin((x * 0.5)) / 0.75

Julia

function code(x)
	return Float64(sin(Float64(x * 0.5)) / 0.75)
end

MATLAB

function tmp = code(x)
	tmp = sin((x * 0.5)) / 0.75;
end

Wolfram

code[x_] := N[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision]

Derivation

1. Initial program 81.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
4. Step-by-step derivation

   1. associate-*r/ 99.3%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

   2. *-commutative 99.3%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

   3. clear-num 99.1%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}} \]

   4. un-div-inv 99.3%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}} \]

   5. *-un-lft-identity 99.3%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\color{blue}{1 \cdot \sin x}}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}} \]

   6. times-frac 99.6%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{1}{2.6666666666666665} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   7. metadata-eval 99.6%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.375} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{0.375 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

6. Taylor expanded in x around 0 55.0%

   \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75}} \]

7. Final simplification 55.0%

   \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{0.75} \]

Alternative 17: 50.9% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 0.25}{0.375} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* x 0.25) 0.375))

C

double code(double x) {
	return (x * 0.25) / 0.375;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 0.25d0) / 0.375d0
end function

Java

public static double code(double x) {
	return (x * 0.25) / 0.375;
}

Python

def code(x):
	return (x * 0.25) / 0.375

Julia

function code(x)
	return Float64(Float64(x * 0.25) / 0.375)
end

MATLAB

function tmp = code(x)
	tmp = (x * 0.25) / 0.375;
end

Wolfram

code[x_] := N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision]

Derivation

1. Initial program 81.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
4. Step-by-step derivation

   1. associate-*r/ 99.3%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}} \]

   2. *-commutative 99.3%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x} \]

   3. *-commutative 99.3%

      \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)} \]

   4. associate-/r/ 99.3%

      \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   5. add-sqr-sqrt 51.8%

      \[\leadsto \color{blue}{\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \cdot \sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}} \]

   6. pow2 51.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}\right)}^{2}} \]

5. Applied egg-rr 51.7%

   \[\leadsto \color{blue}{{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)}^{2}} \]
6. Step-by-step derivation

   1. unpow2 51.7%

      \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

   2. swap-sqr 43.5%

      \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right)} \]

   3. unpow2 43.5%

      \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \left(\sqrt{\frac{2.6666666666666665}{\sin x}} \cdot \sqrt{\frac{2.6666666666666665}{\sin x}}\right) \]

   4. add-sqr-sqrt 81.7%

      \[\leadsto {\sin \left(x \cdot 0.5\right)}^{2} \cdot \color{blue}{\frac{2.6666666666666665}{\sin x}} \]

   5. metadata-eval 81.7%

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

   6. associate-/r* 81.7%

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

   7. *-commutative 81.7%

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

   8. div-inv 81.7%

      \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}} \]

   9. associate-/r* 81.8%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

7. Applied egg-rr 81.8%

   \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

8. Taylor expanded in x around 0 50.4%

   \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]

9. Final simplification 50.4%

   \[\leadsto \frac{x \cdot 0.25}{0.375} \]

Alternative 18: 3.5% accurate, 104.3× speedup

Math

\[\begin{array}{l} \\ x \cdot -0.6666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x -0.6666666666666666))

C

double code(double x) {
	return x * -0.6666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (-0.6666666666666666d0)
end function

Java

public static double code(double x) {
	return x * -0.6666666666666666;
}

Python

def code(x):
	return x * -0.6666666666666666

Julia

function code(x)
	return Float64(x * -0.6666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = x * -0.6666666666666666;
end

Wolfram

code[x_] := N[(x * -0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 81.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]

4. Taylor expanded in x around 0 50.1%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
5. Step-by-step derivation

   1. *-commutative 50.1%

      \[\leadsto \color{blue}{x \cdot 0.6666666666666666} \]

6. Simplified 50.1%

   \[\leadsto \color{blue}{x \cdot 0.6666666666666666} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 27.0%

      \[\leadsto \color{blue}{\sqrt{x \cdot 0.6666666666666666} \cdot \sqrt{x \cdot 0.6666666666666666}} \]

   2. sqrt-unprod 20.4%

      \[\leadsto \color{blue}{\sqrt{\left(x \cdot 0.6666666666666666\right) \cdot \left(x \cdot 0.6666666666666666\right)}} \]

   3. *-commutative 20.4%

      \[\leadsto \sqrt{\color{blue}{\left(0.6666666666666666 \cdot x\right)} \cdot \left(x \cdot 0.6666666666666666\right)} \]

   4. *-commutative 20.4%

      \[\leadsto \sqrt{\left(0.6666666666666666 \cdot x\right) \cdot \color{blue}{\left(0.6666666666666666 \cdot x\right)}} \]

   5. swap-sqr 20.5%

      \[\leadsto \sqrt{\color{blue}{\left(0.6666666666666666 \cdot 0.6666666666666666\right) \cdot \left(x \cdot x\right)}} \]

   6. metadata-eval 20.5%

      \[\leadsto \sqrt{\color{blue}{0.4444444444444444} \cdot \left(x \cdot x\right)} \]

8. Applied egg-rr 20.5%

   \[\leadsto \color{blue}{\sqrt{0.4444444444444444 \cdot \left(x \cdot x\right)}} \]
9. Step-by-step derivation

   1. associate-*r* 20.5%

      \[\leadsto \sqrt{\color{blue}{\left(0.4444444444444444 \cdot x\right) \cdot x}} \]

   2. *-commutative 20.5%

      \[\leadsto \sqrt{\color{blue}{x \cdot \left(0.4444444444444444 \cdot x\right)}} \]

   3. *-commutative 20.5%

      \[\leadsto \sqrt{x \cdot \color{blue}{\left(x \cdot 0.4444444444444444\right)}} \]

10. Simplified 20.5%

    \[\leadsto \color{blue}{\sqrt{x \cdot \left(x \cdot 0.4444444444444444\right)}} \]

11. Taylor expanded in x around -inf 3.3%

    \[\leadsto \color{blue}{-0.6666666666666666 \cdot x} \]
12. Step-by-step derivation

    1. *-commutative 3.3%

       \[\leadsto \color{blue}{x \cdot -0.6666666666666666} \]

13. Simplified 3.3%

    \[\leadsto \color{blue}{x \cdot -0.6666666666666666} \]

14. Final simplification 3.3%

    \[\leadsto x \cdot -0.6666666666666666 \]

Alternative 19: 50.7% accurate, 104.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.6666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.6666666666666666))

C

double code(double x) {
	return x * 0.6666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.6666666666666666d0
end function

Java

public static double code(double x) {
	return x * 0.6666666666666666;
}

Python

def code(x):
	return x * 0.6666666666666666

Julia

function code(x)
	return Float64(x * 0.6666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.6666666666666666;
end

Wolfram

code[x_] := N[(x * 0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 81.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]

4. Taylor expanded in x around 0 50.1%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
5. Step-by-step derivation

   1. *-commutative 50.1%

      \[\leadsto \color{blue}{x \cdot 0.6666666666666666} \]

6. Simplified 50.1%

   \[\leadsto \color{blue}{x \cdot 0.6666666666666666} \]

7. Final simplification 50.1%

   \[\leadsto x \cdot 0.6666666666666666 \]

Developer target: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\frac{8 \cdot t_0}{3}}{\frac{\sin x}{t_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ (/ (* 8.0 t_0) 3.0) (/ (sin x) t_0))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return ((8.0 * t_0) / 3.0) / (sin(x) / t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = ((8.0d0 * t_0) / 3.0d0) / (sin(x) / t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return ((8.0 * t_0) / 3.0) / (Math.sin(x) / t_0);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return ((8.0 * t_0) / 3.0) / (math.sin(x) / t_0)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(Float64(8.0 * t_0) / 3.0) / Float64(sin(x) / t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = ((8.0 * t_0) / 3.0) / (sin(x) / t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(8.0 * t$95$0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023290 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :herbie-target
  (/ (/ (* 8.0 (sin (* x 0.5))) 3.0) (/ (sin x) (sin (* x 0.5))))

  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))