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

Percentage Accurate: 76.8% → 99.5%

Time: 24.0s

Alternatives: 18

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: 76.8% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t_1 := {t_0}^{2}\\ \mathbf{if}\;x \leq -0.0005:\\ \;\;\;\;\frac{\frac{1}{\sin x} \cdot t_1}{0.375}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{0.75 + -0.09375 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{0.375 \cdot \sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))) (t_1 (pow t_0 2.0)))
   (if (<= x -0.0005)
     (/ (* (/ 1.0 (sin x)) t_1) 0.375)
     (if (<= x 2e-5)
       (/ t_0 (+ 0.75 (* -0.09375 (pow x 2.0))))
       (/ t_1 (* 0.375 (sin x)))))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	double t_1 = pow(t_0, 2.0);
	double tmp;
	if (x <= -0.0005) {
		tmp = ((1.0 / sin(x)) * t_1) / 0.375;
	} else if (x <= 2e-5) {
		tmp = t_0 / (0.75 + (-0.09375 * pow(x, 2.0)));
	} else {
		tmp = t_1 / (0.375 * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((x * 0.5d0))
    t_1 = t_0 ** 2.0d0
    if (x <= (-0.0005d0)) then
        tmp = ((1.0d0 / sin(x)) * t_1) / 0.375d0
    else if (x <= 2d-5) then
        tmp = t_0 / (0.75d0 + ((-0.09375d0) * (x ** 2.0d0)))
    else
        tmp = t_1 / (0.375d0 * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	double t_1 = Math.pow(t_0, 2.0);
	double tmp;
	if (x <= -0.0005) {
		tmp = ((1.0 / Math.sin(x)) * t_1) / 0.375;
	} else if (x <= 2e-5) {
		tmp = t_0 / (0.75 + (-0.09375 * Math.pow(x, 2.0)));
	} else {
		tmp = t_1 / (0.375 * Math.sin(x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	t_1 = math.pow(t_0, 2.0)
	tmp = 0
	if x <= -0.0005:
		tmp = ((1.0 / math.sin(x)) * t_1) / 0.375
	elif x <= 2e-5:
		tmp = t_0 / (0.75 + (-0.09375 * math.pow(x, 2.0)))
	else:
		tmp = t_1 / (0.375 * math.sin(x))
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	t_1 = t_0 ^ 2.0
	tmp = 0.0
	if (x <= -0.0005)
		tmp = Float64(Float64(Float64(1.0 / sin(x)) * t_1) / 0.375);
	elseif (x <= 2e-5)
		tmp = Float64(t_0 / Float64(0.75 + Float64(-0.09375 * (x ^ 2.0))));
	else
		tmp = Float64(t_1 / Float64(0.375 * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5));
	t_1 = t_0 ^ 2.0;
	tmp = 0.0;
	if (x <= -0.0005)
		tmp = ((1.0 / sin(x)) * t_1) / 0.375;
	elseif (x <= 2e-5)
		tmp = t_0 / (0.75 + (-0.09375 * (x ^ 2.0)));
	else
		tmp = t_1 / (0.375 * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[LessEqual[x, -0.0005], N[(N[(N[(1.0 / N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / 0.375), $MachinePrecision], If[LessEqual[x, 2e-5], N[(t$95$0 / N[(0.75 + N[(-0.09375 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(0.375 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.0000000000000001e-4

   1. Initial program 99.0%

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

      1. associate-*l* 99.1%

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

      2. associate-/l* 99.2%

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

      3. sqr-neg 99.2%

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

      4. sin-neg 99.2%

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

      5. distribute-lft-neg-out 99.2%

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

      6. sin-neg 99.2%

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

      7. distribute-lft-neg-out 99.2%

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

      8. associate-/r* 99.1%

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

      9. associate-/l* 99.0%

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

      10. distribute-lft-neg-out 99.0%

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

      11. sin-neg 99.0%

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

      12. neg-mul-1 99.0%

          \[\leadsto \frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\frac{\sin x}{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}} \]

      13. associate-/r* 99.0%

          \[\leadsto \frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\color{blue}{\frac{\frac{\sin x}{-1}}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.1%

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

      1. clear-num 99.1%

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

      2. inv-pow 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. times-frac 99.1%

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

      5. metadata-eval 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. unpow-1 99.1%

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

      2. associate-*r/ 99.2%

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

      3. associate-/r/ 99.0%

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

      4. *-commutative 99.0%

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

   7. Simplified 99.0%

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

      1. associate-*l* 99.1%

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

      2. associate-/r* 99.2%

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

      3. associate-*l/ 99.2%

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

      4. pow2 99.2%

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

   9. Applied egg-rr 99.2%

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

   if -5.0000000000000001e-4 < x < 2.00000000000000016e-5

   1. Initial program 50.6%

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

      1. associate-/l* 99.5%

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

      2. associate-*r/ 99.5%

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

      3. associate-/r/ 99.4%

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

      4. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r/ 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]

   if 2.00000000000000016e-5 < x

   1. Initial program 98.9%

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

      1. associate-*l* 99.0%

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

      2. associate-/l* 99.0%

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

      3. sqr-neg 99.0%

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

      4. sin-neg 99.0%

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

      5. distribute-lft-neg-out 99.0%

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

      6. sin-neg 99.0%

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

      7. distribute-lft-neg-out 99.0%

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

      8. associate-/r* 99.1%

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

      9. associate-/l* 98.9%

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

      10. distribute-lft-neg-out 98.9%

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

      11. sin-neg 98.9%

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

      12. neg-mul-1 98.9%

          \[\leadsto \frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\frac{\sin x}{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}} \]

      13. associate-/r* 98.9%

          \[\leadsto \frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\color{blue}{\frac{\frac{\sin x}{-1}}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 98.8%

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

      1. clear-num 98.8%

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

      2. inv-pow 98.8%

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

      3. *-un-lft-identity 98.8%

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

      4. times-frac 99.0%

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

      5. metadata-eval 99.0%

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

   5. Applied egg-rr 99.0%

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

      1. unpow-1 99.0%

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

      2. associate-*r/ 98.9%

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

      3. associate-/r/ 99.0%

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

      4. *-commutative 99.0%

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

   7. Simplified 99.0%

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

      1. associate-*l/ 99.1%

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

      2. *-un-lft-identity 99.1%

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

      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. pow2 99.1%

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

      5. *-commutative 99.1%

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

   9. Applied egg-rr 99.1%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0005:\\ \;\;\;\;\frac{\frac{1}{\sin x} \cdot {\sin \left(x \cdot 0.5\right)}^{2}}{0.375}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375 \cdot \sin x}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.002 \lor \neg \left(x \leq 10^{-30}\right):\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.002) (not (<= x 1e-30)))
   (* 2.6666666666666665 (/ (pow (sin (* x 0.5)) 2.0) (sin x)))
   (/ (+ (* 0.020833333333333332 (pow x 3.0)) (* x 0.25)) 0.375)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.002) || !(x <= 1e-30)) {
		tmp = 2.6666666666666665 * (pow(sin((x * 0.5)), 2.0) / sin(x));
	} else {
		tmp = ((0.020833333333333332 * pow(x, 3.0)) + (x * 0.25)) / 0.375;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.002d0)) .or. (.not. (x <= 1d-30))) then
        tmp = 2.6666666666666665d0 * ((sin((x * 0.5d0)) ** 2.0d0) / sin(x))
    else
        tmp = ((0.020833333333333332d0 * (x ** 3.0d0)) + (x * 0.25d0)) / 0.375d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.002) || !(x <= 1e-30)) {
		tmp = 2.6666666666666665 * (Math.pow(Math.sin((x * 0.5)), 2.0) / Math.sin(x));
	} else {
		tmp = ((0.020833333333333332 * Math.pow(x, 3.0)) + (x * 0.25)) / 0.375;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.002) or not (x <= 1e-30):
		tmp = 2.6666666666666665 * (math.pow(math.sin((x * 0.5)), 2.0) / math.sin(x))
	else:
		tmp = ((0.020833333333333332 * math.pow(x, 3.0)) + (x * 0.25)) / 0.375
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.002) || !(x <= 1e-30))
		tmp = Float64(2.6666666666666665 * Float64((sin(Float64(x * 0.5)) ^ 2.0) / sin(x)));
	else
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x ^ 3.0)) + Float64(x * 0.25)) / 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.002) || ~((x <= 1e-30)))
		tmp = 2.6666666666666665 * ((sin((x * 0.5)) ^ 2.0) / sin(x));
	else
		tmp = ((0.020833333333333332 * (x ^ 3.0)) + (x * 0.25)) / 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.002], N[Not[LessEqual[x, 1e-30]], $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[(N[(0.020833333333333332 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e-3 or 1e-30 < x

   1. Initial program 99.0%

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

      1. associate-/l* 99.0%

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

      2. associate-*r/ 99.0%

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

      3. associate-/r/ 99.0%

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

      4. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in x around inf 99.1%

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

      1. *-commutative 99.1%

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

   6. Simplified 99.1%

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

   if -2e-3 < x < 1e-30

   1. Initial program 48.9%

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

      1. associate-/l* 99.4%

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

      2. *-commutative 99.4%

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

      3. *-lft-identity 99.4%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{1 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      4. metadata-eval 99.4%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      5. times-frac 99.4%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      6. associate-/l* 99.4%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      7. *-commutative 99.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      8. neg-mul-1 99.4%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      9. sin-neg 99.4%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      10. distribute-lft-neg-out 99.4%

          \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      11. associate-*r/ 99.4%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.4%

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

      1. associate-/l* 48.9%

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

      2. associate-*l* 49.0%

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

      3. *-commutative 49.0%

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

      4. associate-*l/ 49.0%

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

      5. sqr-sin-a 5.8%

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

      6. count-2 5.8%

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

      7. distribute-lft-out 5.8%

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

      8. metadata-eval 5.8%

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

      9. *-commutative 5.8%

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

      10. *-un-lft-identity 5.8%

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

   5. Applied egg-rr 5.8%

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

      1. associate-/r/ 5.8%

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

      2. div-inv 5.8%

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

      3. metadata-eval 5.8%

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

      4. associate-/r* 5.8%

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

      5. cancel-sign-sub-inv 5.8%

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

      6. *-commutative 5.8%

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

      7. metadata-eval 5.8%

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

   7. Applied egg-rr 5.8%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.002 \lor \neg \left(x \leq 10^{-30}\right):\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \mathbf{if}\;x \leq -0.0002:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x \cdot {t_0}^{-2}}\\ \mathbf{elif}\;x \leq 10^{-30}:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{t_0}^{2}}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))))
   (if (<= x -0.0002)
     (/ 2.6666666666666665 (* (sin x) (pow t_0 -2.0)))
     (if (<= x 1e-30)
       (/ (+ (* 0.020833333333333332 (pow x 3.0)) (* x 0.25)) 0.375)
       (* 2.6666666666666665 (/ (pow t_0 2.0) (sin x)))))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	double tmp;
	if (x <= -0.0002) {
		tmp = 2.6666666666666665 / (sin(x) * pow(t_0, -2.0));
	} else if (x <= 1e-30) {
		tmp = ((0.020833333333333332 * pow(x, 3.0)) + (x * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 * (pow(t_0, 2.0) / 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))
    if (x <= (-0.0002d0)) then
        tmp = 2.6666666666666665d0 / (sin(x) * (t_0 ** (-2.0d0)))
    else if (x <= 1d-30) then
        tmp = ((0.020833333333333332d0 * (x ** 3.0d0)) + (x * 0.25d0)) / 0.375d0
    else
        tmp = 2.6666666666666665d0 * ((t_0 ** 2.0d0) / sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	double tmp;
	if (x <= -0.0002) {
		tmp = 2.6666666666666665 / (Math.sin(x) * Math.pow(t_0, -2.0));
	} else if (x <= 1e-30) {
		tmp = ((0.020833333333333332 * Math.pow(x, 3.0)) + (x * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 * (Math.pow(t_0, 2.0) / Math.sin(x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	tmp = 0
	if x <= -0.0002:
		tmp = 2.6666666666666665 / (math.sin(x) * math.pow(t_0, -2.0))
	elif x <= 1e-30:
		tmp = ((0.020833333333333332 * math.pow(x, 3.0)) + (x * 0.25)) / 0.375
	else:
		tmp = 2.6666666666666665 * (math.pow(t_0, 2.0) / math.sin(x))
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	tmp = 0.0
	if (x <= -0.0002)
		tmp = Float64(2.6666666666666665 / Float64(sin(x) * (t_0 ^ -2.0)));
	elseif (x <= 1e-30)
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x ^ 3.0)) + Float64(x * 0.25)) / 0.375);
	else
		tmp = Float64(2.6666666666666665 * Float64((t_0 ^ 2.0) / sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5));
	tmp = 0.0;
	if (x <= -0.0002)
		tmp = 2.6666666666666665 / (sin(x) * (t_0 ^ -2.0));
	elseif (x <= 1e-30)
		tmp = ((0.020833333333333332 * (x ^ 3.0)) + (x * 0.25)) / 0.375;
	else
		tmp = 2.6666666666666665 * ((t_0 ^ 2.0) / sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -0.0002], N[(2.6666666666666665 / N[(N[Sin[x], $MachinePrecision] * N[Power[t$95$0, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-30], N[(N[(N[(0.020833333333333332 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0000000000000001e-4

   1. Initial program 99.0%

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

      1. associate-/l* 99.0%

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

      2. associate-*r/ 99.1%

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

      3. associate-/r/ 99.0%

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

      4. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*l* 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-/r/ 99.0%

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

      5. *-un-lft-identity 99.0%

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

      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/ 99.2%

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

      2. *-commutative 99.2%

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

      3. clear-num 99.1%

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

      4. *-un-lft-identity 99.1%

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

      5. associate-*l/ 99.0%

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

      6. *-commutative 99.0%

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

      7. associate-/l/ 99.1%

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

      8. clear-num 99.1%

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

      9. associate-/l/ 99.0%

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

      10. associate-/l/ 99.0%

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

      11. *-un-lft-identity 99.0%

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

      12. *-commutative 99.0%

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

      13. associate-/r* 98.9%

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

      14. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{1}{\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 99.0%

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

      2. associate-/l* 99.2%

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

      3. div-inv 99.1%

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

      4. metadata-eval 99.1%

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

   9. Applied egg-rr 99.1%

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

       1. clear-num 98.8%

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

       2. clear-num 99.0%

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

       3. associate-/r* 98.9%

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

       4. *-commutative 98.9%

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

       5. unpow2 98.9%

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

       6. associate-*r/ 98.9%

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

       7. inv-pow 98.9%

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

       8. metadata-eval 98.9%

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

       9. unpow-prod-down 99.1%

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

       10. metadata-eval 99.1%

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

       11. metadata-eval 99.1%

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

       12. div-inv 99.0%

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

       13. pow-flip 99.1%

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

       14. metadata-eval 99.1%

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

       15. metadata-eval 99.1%

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

   11. Applied egg-rr 99.1%

       \[\leadsto \color{blue}{2.6666666666666665 \cdot {\left(\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 99.1%

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

       2. associate-*r/ 99.2%

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

       3. metadata-eval 99.2%

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

   13. Simplified 99.2%

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

   if -2.0000000000000001e-4 < x < 1e-30

   1. Initial program 48.9%

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

      1. associate-/l* 99.4%

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

      2. *-commutative 99.4%

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

      3. *-lft-identity 99.4%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{1 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      4. metadata-eval 99.4%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      5. times-frac 99.4%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      6. associate-/l* 99.4%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      7. *-commutative 99.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      8. neg-mul-1 99.4%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      9. sin-neg 99.4%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      10. distribute-lft-neg-out 99.4%

          \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      11. associate-*r/ 99.4%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.4%

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

      1. associate-/l* 48.9%

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

      2. associate-*l* 49.0%

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

      3. *-commutative 49.0%

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

      4. associate-*l/ 49.0%

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

      5. sqr-sin-a 5.8%

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

      6. count-2 5.8%

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

      7. distribute-lft-out 5.8%

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

      8. metadata-eval 5.8%

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

      9. *-commutative 5.8%

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

      10. *-un-lft-identity 5.8%

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

   5. Applied egg-rr 5.8%

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

      1. associate-/r/ 5.8%

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

      2. div-inv 5.8%

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

      3. metadata-eval 5.8%

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

      4. associate-/r* 5.8%

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

      5. cancel-sign-sub-inv 5.8%

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

      6. *-commutative 5.8%

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

      7. metadata-eval 5.8%

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

   7. Applied egg-rr 5.8%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]

   if 1e-30 < x

   1. Initial program 99.0%

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

      1. associate-/l* 99.0%

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

      2. associate-*r/ 99.0%

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

      3. associate-/r/ 98.9%

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

      4. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in x around inf 99.1%

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

      1. *-commutative 99.1%

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

   6. Simplified 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0002:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{-2}}\\ \mathbf{elif}\;x \leq 10^{-30}:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \mathbf{if}\;x \leq -0.002:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x \cdot {t_0}^{-2}}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{0.75 + -0.09375 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t_0}^{2}}{0.375 \cdot \sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))))
   (if (<= x -0.002)
     (/ 2.6666666666666665 (* (sin x) (pow t_0 -2.0)))
     (if (<= x 2e-5)
       (/ t_0 (+ 0.75 (* -0.09375 (pow x 2.0))))
       (/ (pow t_0 2.0) (* 0.375 (sin x)))))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	double tmp;
	if (x <= -0.002) {
		tmp = 2.6666666666666665 / (sin(x) * pow(t_0, -2.0));
	} else if (x <= 2e-5) {
		tmp = t_0 / (0.75 + (-0.09375 * pow(x, 2.0)));
	} else {
		tmp = pow(t_0, 2.0) / (0.375 * 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))
    if (x <= (-0.002d0)) then
        tmp = 2.6666666666666665d0 / (sin(x) * (t_0 ** (-2.0d0)))
    else if (x <= 2d-5) then
        tmp = t_0 / (0.75d0 + ((-0.09375d0) * (x ** 2.0d0)))
    else
        tmp = (t_0 ** 2.0d0) / (0.375d0 * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	double tmp;
	if (x <= -0.002) {
		tmp = 2.6666666666666665 / (Math.sin(x) * Math.pow(t_0, -2.0));
	} else if (x <= 2e-5) {
		tmp = t_0 / (0.75 + (-0.09375 * Math.pow(x, 2.0)));
	} else {
		tmp = Math.pow(t_0, 2.0) / (0.375 * Math.sin(x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	tmp = 0
	if x <= -0.002:
		tmp = 2.6666666666666665 / (math.sin(x) * math.pow(t_0, -2.0))
	elif x <= 2e-5:
		tmp = t_0 / (0.75 + (-0.09375 * math.pow(x, 2.0)))
	else:
		tmp = math.pow(t_0, 2.0) / (0.375 * math.sin(x))
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	tmp = 0.0
	if (x <= -0.002)
		tmp = Float64(2.6666666666666665 / Float64(sin(x) * (t_0 ^ -2.0)));
	elseif (x <= 2e-5)
		tmp = Float64(t_0 / Float64(0.75 + Float64(-0.09375 * (x ^ 2.0))));
	else
		tmp = Float64((t_0 ^ 2.0) / Float64(0.375 * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5));
	tmp = 0.0;
	if (x <= -0.002)
		tmp = 2.6666666666666665 / (sin(x) * (t_0 ^ -2.0));
	elseif (x <= 2e-5)
		tmp = t_0 / (0.75 + (-0.09375 * (x ^ 2.0)));
	else
		tmp = (t_0 ^ 2.0) / (0.375 * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -0.002], N[(2.6666666666666665 / N[(N[Sin[x], $MachinePrecision] * N[Power[t$95$0, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-5], N[(t$95$0 / N[(0.75 + N[(-0.09375 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[(0.375 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e-3

   1. Initial program 99.0%

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

      1. associate-/l* 99.0%

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

      2. associate-*r/ 99.1%

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

      3. associate-/r/ 99.0%

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

      4. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*l* 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-/r/ 99.0%

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

      5. *-un-lft-identity 99.0%

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

      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/ 99.2%

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

      2. *-commutative 99.2%

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

      3. clear-num 99.1%

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

      4. *-un-lft-identity 99.1%

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

      5. associate-*l/ 99.0%

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

      6. *-commutative 99.0%

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

      7. associate-/l/ 99.1%

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

      8. clear-num 99.1%

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

      9. associate-/l/ 99.0%

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

      10. associate-/l/ 99.0%

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

      11. *-un-lft-identity 99.0%

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

      12. *-commutative 99.0%

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

      13. associate-/r* 98.9%

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

      14. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{1}{\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 99.0%

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

      2. associate-/l* 99.2%

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

      3. div-inv 99.1%

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

      4. metadata-eval 99.1%

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

   9. Applied egg-rr 99.1%

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

       1. clear-num 98.8%

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

       2. clear-num 99.0%

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

       3. associate-/r* 98.9%

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

       4. *-commutative 98.9%

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

       5. unpow2 98.9%

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

       6. associate-*r/ 98.9%

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

       7. inv-pow 98.9%

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

       8. metadata-eval 98.9%

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

       9. unpow-prod-down 99.1%

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

       10. metadata-eval 99.1%

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

       11. metadata-eval 99.1%

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

       12. div-inv 99.0%

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

       13. pow-flip 99.1%

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

       14. metadata-eval 99.1%

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

       15. metadata-eval 99.1%

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

   11. Applied egg-rr 99.1%

       \[\leadsto \color{blue}{2.6666666666666665 \cdot {\left(\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 99.1%

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

       2. associate-*r/ 99.2%

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

       3. metadata-eval 99.2%

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

   13. Simplified 99.2%

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

   if -2e-3 < x < 2.00000000000000016e-5

   1. Initial program 50.6%

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

      1. associate-/l* 99.5%

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

      2. associate-*r/ 99.5%

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

      3. associate-/r/ 99.4%

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

      4. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r/ 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]

   if 2.00000000000000016e-5 < x

   1. Initial program 98.9%

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

      1. associate-*l* 99.0%

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

      2. associate-/l* 99.0%

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

      3. sqr-neg 99.0%

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

      4. sin-neg 99.0%

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

      5. distribute-lft-neg-out 99.0%

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

      6. sin-neg 99.0%

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

      7. distribute-lft-neg-out 99.0%

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

      8. associate-/r* 99.1%

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

      9. associate-/l* 98.9%

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

      10. distribute-lft-neg-out 98.9%

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

      11. sin-neg 98.9%

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

      12. neg-mul-1 98.9%

          \[\leadsto \frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\frac{\sin x}{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}} \]

      13. associate-/r* 98.9%

          \[\leadsto \frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\color{blue}{\frac{\frac{\sin x}{-1}}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 98.8%

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

      1. clear-num 98.8%

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

      2. inv-pow 98.8%

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

      3. *-un-lft-identity 98.8%

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

      4. times-frac 99.0%

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

      5. metadata-eval 99.0%

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

   5. Applied egg-rr 99.0%

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

      1. unpow-1 99.0%

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

      2. associate-*r/ 98.9%

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

      3. associate-/r/ 99.0%

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

      4. *-commutative 99.0%

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

   7. Simplified 99.0%

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

      1. associate-*l/ 99.1%

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

      2. *-un-lft-identity 99.1%

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

      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. pow2 99.1%

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

      5. *-commutative 99.1%

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

   9. Applied egg-rr 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.002:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{-2}}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375 \cdot \sin x}\\ \end{array} \]

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

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 75.9%

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

   1. associate-/l* 99.2%

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

   2. associate-*r/ 99.2%

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

   3. associate-/r/ 99.2%

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

   4. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-*l* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-/r/ 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 99.5%

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

   7. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 6: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return 2.6666666666666665 * (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 = 2.6666666666666665d0 * (t_0 * (t_0 / sin(x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. associate-/l* 99.2%

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

   2. associate-*r/ 99.2%

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

   3. associate-/r/ 99.2%

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

   4. metadata-eval 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 7: 99.2% accurate, 1.0× speedup

Math

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

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(Float64(t_0 * 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[(N[(t$95$0 * 2.6666666666666665), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.9%

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

   1. associate-*l* 76.0%

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

   2. associate-/l* 76.0%

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

   3. sqr-neg 76.0%

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

   4. sin-neg 76.0%

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

   5. distribute-lft-neg-out 76.0%

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

   6. sin-neg 76.0%

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

   7. distribute-lft-neg-out 76.0%

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

   8. associate-/r* 99.3%

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

   9. associate-/l* 99.2%

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

   10. distribute-lft-neg-out 99.2%

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

   11. sin-neg 99.2%

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

   12. neg-mul-1 99.2%

       \[\leadsto \frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\frac{\sin x}{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}} \]

   13. associate-/r* 99.2%

       \[\leadsto \frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\color{blue}{\frac{\frac{\sin x}{-1}}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 8: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return t_0 * ((t_0 / 0.375) / 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 / 0.375d0) / sin(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. associate-/l* 99.2%

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

   2. associate-*r/ 99.2%

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

   3. associate-/r/ 99.2%

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

   4. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-*l* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-/r/ 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 99.5%

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

   7. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

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

   1. associate-/r* 99.5%

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

   2. metadata-eval 99.5%

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

   3. associate-/l* 99.2%

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

   4. associate-/r/ 99.2%

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

   5. associate-/l* 99.4%

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

   6. metadata-eval 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 9: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \cos x \cdot -0.5\\ \mathbf{if}\;x \leq -0.0052:\\ \;\;\;\;\frac{t_0}{0.375 \cdot \sin x}\\ \mathbf{elif}\;x \leq 0.0058:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{0.375}}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* (cos x) -0.5))))
   (if (<= x -0.0052)
     (/ t_0 (* 0.375 (sin x)))
     (if (<= x 0.0058)
       (/ (sin (* x 0.5)) (+ 0.75 (* -0.09375 (pow x 2.0))))
       (/ (/ t_0 0.375) (sin x))))))

C

double code(double x) {
	double t_0 = 0.5 + (cos(x) * -0.5);
	double tmp;
	if (x <= -0.0052) {
		tmp = t_0 / (0.375 * sin(x));
	} else if (x <= 0.0058) {
		tmp = sin((x * 0.5)) / (0.75 + (-0.09375 * pow(x, 2.0)));
	} else {
		tmp = (t_0 / 0.375) / sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (cos(x) * (-0.5d0))
    if (x <= (-0.0052d0)) then
        tmp = t_0 / (0.375d0 * sin(x))
    else if (x <= 0.0058d0) then
        tmp = sin((x * 0.5d0)) / (0.75d0 + ((-0.09375d0) * (x ** 2.0d0)))
    else
        tmp = (t_0 / 0.375d0) / sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.5 + (Math.cos(x) * -0.5);
	double tmp;
	if (x <= -0.0052) {
		tmp = t_0 / (0.375 * Math.sin(x));
	} else if (x <= 0.0058) {
		tmp = Math.sin((x * 0.5)) / (0.75 + (-0.09375 * Math.pow(x, 2.0)));
	} else {
		tmp = (t_0 / 0.375) / Math.sin(x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (math.cos(x) * -0.5)
	tmp = 0
	if x <= -0.0052:
		tmp = t_0 / (0.375 * math.sin(x))
	elif x <= 0.0058:
		tmp = math.sin((x * 0.5)) / (0.75 + (-0.09375 * math.pow(x, 2.0)))
	else:
		tmp = (t_0 / 0.375) / math.sin(x)
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(cos(x) * -0.5))
	tmp = 0.0
	if (x <= -0.0052)
		tmp = Float64(t_0 / Float64(0.375 * sin(x)));
	elseif (x <= 0.0058)
		tmp = Float64(sin(Float64(x * 0.5)) / Float64(0.75 + Float64(-0.09375 * (x ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 / 0.375) / sin(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 + (cos(x) * -0.5);
	tmp = 0.0;
	if (x <= -0.0052)
		tmp = t_0 / (0.375 * sin(x));
	elseif (x <= 0.0058)
		tmp = sin((x * 0.5)) / (0.75 + (-0.09375 * (x ^ 2.0)));
	else
		tmp = (t_0 / 0.375) / sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(N[Cos[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0052], N[(t$95$0 / N[(0.375 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0058], N[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[(0.75 + N[(-0.09375 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / 0.375), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0051999999999999998

   1. Initial program 99.0%

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

      1. associate-/l* 99.0%

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

      2. *-commutative 99.0%

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

      3. *-lft-identity 99.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{1 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      4. metadata-eval 99.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      5. times-frac 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      6. associate-/l* 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      7. *-commutative 99.1%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      8. neg-mul-1 99.1%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      9. sin-neg 99.1%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      10. distribute-lft-neg-out 99.1%

          \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      11. associate-*r/ 99.0%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.0%

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

      1. associate-/l* 99.0%

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

      2. associate-*l* 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*l/ 99.1%

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

      5. sqr-sin-a 98.4%

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

      6. count-2 98.4%

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

      7. distribute-lft-out 98.4%

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

      8. metadata-eval 98.4%

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

      9. *-commutative 98.4%

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

      10. *-un-lft-identity 98.4%

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

   5. Applied egg-rr 98.4%

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

      1. *-commutative 98.4%

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

      2. metadata-eval 98.4%

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

      3. times-frac 98.7%

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

      4. *-un-lft-identity 98.7%

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

      5. cancel-sign-sub-inv 98.7%

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

      6. *-commutative 98.7%

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

      7. metadata-eval 98.7%

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

   7. Applied egg-rr 98.7%

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

   if -0.0051999999999999998 < x < 0.0058

   1. Initial program 50.6%

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

      1. associate-/l* 99.5%

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

      2. associate-*r/ 99.5%

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

      3. associate-/r/ 99.4%

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

      4. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r/ 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]

   if 0.0058 < x

   1. Initial program 98.9%

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

      1. associate-/l* 98.9%

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

      2. *-commutative 98.9%

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

      3. *-lft-identity 98.9%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{1 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      4. metadata-eval 98.9%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      5. times-frac 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      6. associate-/l* 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      7. *-commutative 99.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      8. neg-mul-1 99.0%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      9. sin-neg 99.0%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      10. distribute-lft-neg-out 99.0%

          \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      11. associate-*r/ 98.9%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 98.9%

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

      1. associate-/l* 98.9%

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

      2. associate-*l* 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l/ 99.0%

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

      5. sqr-sin-a 98.5%

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

      6. count-2 98.5%

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

      7. distribute-lft-out 98.5%

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

      8. metadata-eval 98.5%

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

      9. *-commutative 98.5%

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

      10. *-un-lft-identity 98.5%

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

   5. Applied egg-rr 98.5%

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

      1. associate-/r/ 98.5%

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

      2. div-inv 98.6%

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

      3. metadata-eval 98.6%

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

      4. associate-/l/ 98.6%

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

      5. cancel-sign-sub-inv 98.6%

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

      6. *-commutative 98.6%

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

      7. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0052:\\ \;\;\;\;\frac{0.5 + \cos x \cdot -0.5}{0.375 \cdot \sin x}\\ \mathbf{elif}\;x \leq 0.0058:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 + \cos x \cdot -0.5}{0.375}}{\sin x}\\ \end{array} \]

Alternative 10: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0039 \lor \neg \left(x \leq 0.0043\right):\\ \;\;\;\;\left(0.5 - 0.5 \cdot \cos x\right) \cdot \frac{2.6666666666666665}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.0039) (not (<= x 0.0043)))
   (* (- 0.5 (* 0.5 (cos x))) (/ 2.6666666666666665 (sin x)))
   (/ (+ (* 0.020833333333333332 (pow x 3.0)) (* x 0.25)) 0.375)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.0039) || !(x <= 0.0043)) {
		tmp = (0.5 - (0.5 * cos(x))) * (2.6666666666666665 / sin(x));
	} else {
		tmp = ((0.020833333333333332 * pow(x, 3.0)) + (x * 0.25)) / 0.375;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.0039d0)) .or. (.not. (x <= 0.0043d0))) then
        tmp = (0.5d0 - (0.5d0 * cos(x))) * (2.6666666666666665d0 / sin(x))
    else
        tmp = ((0.020833333333333332d0 * (x ** 3.0d0)) + (x * 0.25d0)) / 0.375d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.0039) || !(x <= 0.0043)) {
		tmp = (0.5 - (0.5 * Math.cos(x))) * (2.6666666666666665 / Math.sin(x));
	} else {
		tmp = ((0.020833333333333332 * Math.pow(x, 3.0)) + (x * 0.25)) / 0.375;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.0039) or not (x <= 0.0043):
		tmp = (0.5 - (0.5 * math.cos(x))) * (2.6666666666666665 / math.sin(x))
	else:
		tmp = ((0.020833333333333332 * math.pow(x, 3.0)) + (x * 0.25)) / 0.375
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.0039) || !(x <= 0.0043))
		tmp = Float64(Float64(0.5 - Float64(0.5 * cos(x))) * Float64(2.6666666666666665 / sin(x)));
	else
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x ^ 3.0)) + Float64(x * 0.25)) / 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.0039) || ~((x <= 0.0043)))
		tmp = (0.5 - (0.5 * cos(x))) * (2.6666666666666665 / sin(x));
	else
		tmp = ((0.020833333333333332 * (x ^ 3.0)) + (x * 0.25)) / 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.0039], N[Not[LessEqual[x, 0.0043]], $MachinePrecision]], N[(N[(0.5 - N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.020833333333333332 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0038999999999999998 or 0.0043 < x

   1. Initial program 99.0%

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

      1. associate-/l* 99.0%

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

      2. *-commutative 99.0%

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

      3. *-lft-identity 99.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{1 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      4. metadata-eval 99.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      5. times-frac 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      6. associate-/l* 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      7. *-commutative 99.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      8. neg-mul-1 99.0%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      9. sin-neg 99.0%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      10. distribute-lft-neg-out 99.0%

          \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      11. associate-*r/ 99.0%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.0%

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

      1. associate-/l* 99.1%

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

      2. associate-/l/ 99.1%

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

      3. associate-/r/ 99.0%

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

      4. sqr-sin-a 98.5%

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

      5. count-2 98.5%

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

      6. distribute-lft-out 98.5%

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

      7. metadata-eval 98.5%

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

      8. *-commutative 98.5%

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

      9. *-un-lft-identity 98.5%

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

   5. Applied egg-rr 98.5%

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

   if -0.0038999999999999998 < x < 0.0043

   1. Initial program 50.6%

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

      1. associate-/l* 99.5%

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

      2. *-commutative 99.5%

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

      3. *-lft-identity 99.5%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{1 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      4. metadata-eval 99.5%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      5. times-frac 99.4%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      6. associate-/l* 99.4%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      7. *-commutative 99.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      8. neg-mul-1 99.4%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      9. sin-neg 99.4%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      10. distribute-lft-neg-out 99.4%

          \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      11. associate-*r/ 99.5%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.5%

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

      1. associate-/l* 50.6%

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

      2. associate-*l* 50.6%

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

      3. *-commutative 50.6%

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

      4. associate-*l/ 50.7%

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

      5. sqr-sin-a 6.2%

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

      6. count-2 6.2%

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

      7. distribute-lft-out 6.2%

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

      8. metadata-eval 6.2%

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

      9. *-commutative 6.2%

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

      10. *-un-lft-identity 6.2%

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

   5. Applied egg-rr 6.2%

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

      1. associate-/r/ 6.2%

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

      2. div-inv 6.2%

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

      3. metadata-eval 6.2%

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

      4. associate-/r* 6.2%

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

      5. cancel-sign-sub-inv 6.2%

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

      6. *-commutative 6.2%

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

      7. metadata-eval 6.2%

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

   7. Applied egg-rr 6.2%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0039 \lor \neg \left(x \leq 0.0043\right):\\ \;\;\;\;\left(0.5 - 0.5 \cdot \cos x\right) \cdot \frac{2.6666666666666665}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \end{array} \]

Alternative 11: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos x\\ \mathbf{if}\;x \leq -0.0039:\\ \;\;\;\;t_0 \cdot \frac{2.6666666666666665}{\sin x}\\ \mathbf{elif}\;x \leq 0.0043:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos x)))))
   (if (<= x -0.0039)
     (* t_0 (/ 2.6666666666666665 (sin x)))
     (if (<= x 0.0043)
       (/ (+ (* 0.020833333333333332 (pow x 3.0)) (* x 0.25)) 0.375)
       (/ 2.6666666666666665 (/ (sin x) t_0))))))

C

double code(double x) {
	double t_0 = 0.5 - (0.5 * cos(x));
	double tmp;
	if (x <= -0.0039) {
		tmp = t_0 * (2.6666666666666665 / sin(x));
	} else if (x <= 0.0043) {
		tmp = ((0.020833333333333332 * pow(x, 3.0)) + (x * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 / (sin(x) / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 - (0.5d0 * cos(x))
    if (x <= (-0.0039d0)) then
        tmp = t_0 * (2.6666666666666665d0 / sin(x))
    else if (x <= 0.0043d0) then
        tmp = ((0.020833333333333332d0 * (x ** 3.0d0)) + (x * 0.25d0)) / 0.375d0
    else
        tmp = 2.6666666666666665d0 / (sin(x) / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.5 - (0.5 * Math.cos(x));
	double tmp;
	if (x <= -0.0039) {
		tmp = t_0 * (2.6666666666666665 / Math.sin(x));
	} else if (x <= 0.0043) {
		tmp = ((0.020833333333333332 * Math.pow(x, 3.0)) + (x * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 / (Math.sin(x) / t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 - (0.5 * math.cos(x))
	tmp = 0
	if x <= -0.0039:
		tmp = t_0 * (2.6666666666666665 / math.sin(x))
	elif x <= 0.0043:
		tmp = ((0.020833333333333332 * math.pow(x, 3.0)) + (x * 0.25)) / 0.375
	else:
		tmp = 2.6666666666666665 / (math.sin(x) / t_0)
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(0.5 * cos(x)))
	tmp = 0.0
	if (x <= -0.0039)
		tmp = Float64(t_0 * Float64(2.6666666666666665 / sin(x)));
	elseif (x <= 0.0043)
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x ^ 3.0)) + Float64(x * 0.25)) / 0.375);
	else
		tmp = Float64(2.6666666666666665 / Float64(sin(x) / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 - (0.5 * cos(x));
	tmp = 0.0;
	if (x <= -0.0039)
		tmp = t_0 * (2.6666666666666665 / sin(x));
	elseif (x <= 0.0043)
		tmp = ((0.020833333333333332 * (x ^ 3.0)) + (x * 0.25)) / 0.375;
	else
		tmp = 2.6666666666666665 / (sin(x) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0039], N[(t$95$0 * N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0043], N[(N[(N[(0.020833333333333332 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(2.6666666666666665 / N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0038999999999999998

   1. Initial program 99.0%

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

      1. associate-/l* 99.0%

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

      2. *-commutative 99.0%

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

      3. *-lft-identity 99.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{1 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      4. metadata-eval 99.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      5. times-frac 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      6. associate-/l* 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      7. *-commutative 99.1%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      8. neg-mul-1 99.1%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      9. sin-neg 99.1%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      10. distribute-lft-neg-out 99.1%

          \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      11. associate-*r/ 99.0%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.0%

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

      1. associate-/l* 99.1%

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

      2. associate-/l/ 99.2%

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

      3. associate-/r/ 98.9%

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

      4. sqr-sin-a 98.5%

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

      5. count-2 98.5%

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

      6. distribute-lft-out 98.5%

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

      7. metadata-eval 98.5%

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

      8. *-commutative 98.5%

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

      9. *-un-lft-identity 98.5%

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

   5. Applied egg-rr 98.5%

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

   if -0.0038999999999999998 < x < 0.0043

   1. Initial program 50.6%

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

      1. associate-/l* 99.5%

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

      2. *-commutative 99.5%

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

      3. *-lft-identity 99.5%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{1 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      4. metadata-eval 99.5%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      5. times-frac 99.4%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      6. associate-/l* 99.4%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      7. *-commutative 99.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      8. neg-mul-1 99.4%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      9. sin-neg 99.4%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      10. distribute-lft-neg-out 99.4%

          \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      11. associate-*r/ 99.5%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.5%

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

      1. associate-/l* 50.6%

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

      2. associate-*l* 50.6%

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

      3. *-commutative 50.6%

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

      4. associate-*l/ 50.7%

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

      5. sqr-sin-a 6.2%

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

      6. count-2 6.2%

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

      7. distribute-lft-out 6.2%

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

      8. metadata-eval 6.2%

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

      9. *-commutative 6.2%

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

      10. *-un-lft-identity 6.2%

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

   5. Applied egg-rr 6.2%

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

      1. associate-/r/ 6.2%

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

      2. div-inv 6.2%

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

      3. metadata-eval 6.2%

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

      4. associate-/r* 6.2%

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

      5. cancel-sign-sub-inv 6.2%

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

      6. *-commutative 6.2%

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

      7. metadata-eval 6.2%

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

   7. Applied egg-rr 6.2%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]

   if 0.0043 < x

   1. Initial program 98.9%

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

      1. associate-/l* 98.9%

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

      2. associate-*r/ 98.9%

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

      3. associate-/r/ 98.9%

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

      4. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

      1. associate-*l/ 99.0%

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

      2. associate-*r/ 99.0%

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

      3. associate-/l* 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. sqr-sin-a 98.6%

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

      5. count-2 98.6%

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

      6. distribute-lft-out 98.6%

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

      7. metadata-eval 98.6%

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

      8. *-commutative 98.6%

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

      9. *-un-lft-identity 98.6%

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

   5. Applied egg-rr 98.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0039:\\ \;\;\;\;\left(0.5 - 0.5 \cdot \cos x\right) \cdot \frac{2.6666666666666665}{\sin x}\\ \mathbf{elif}\;x \leq 0.0043:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{0.5 - 0.5 \cdot \cos x}}\\ \end{array} \]

Alternative 12: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0039:\\ \;\;\;\;\frac{0.5 + \cos x \cdot -0.5}{0.375 \cdot \sin x}\\ \mathbf{elif}\;x \leq 0.0043:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{0.5 - 0.5 \cdot \cos x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0039)
   (/ (+ 0.5 (* (cos x) -0.5)) (* 0.375 (sin x)))
   (if (<= x 0.0043)
     (/ (+ (* 0.020833333333333332 (pow x 3.0)) (* x 0.25)) 0.375)
     (/ 2.6666666666666665 (/ (sin x) (- 0.5 (* 0.5 (cos x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0039) {
		tmp = (0.5 + (cos(x) * -0.5)) / (0.375 * sin(x));
	} else if (x <= 0.0043) {
		tmp = ((0.020833333333333332 * pow(x, 3.0)) + (x * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 / (sin(x) / (0.5 - (0.5 * cos(x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.0039d0)) then
        tmp = (0.5d0 + (cos(x) * (-0.5d0))) / (0.375d0 * sin(x))
    else if (x <= 0.0043d0) then
        tmp = ((0.020833333333333332d0 * (x ** 3.0d0)) + (x * 0.25d0)) / 0.375d0
    else
        tmp = 2.6666666666666665d0 / (sin(x) / (0.5d0 - (0.5d0 * cos(x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0039) {
		tmp = (0.5 + (Math.cos(x) * -0.5)) / (0.375 * Math.sin(x));
	} else if (x <= 0.0043) {
		tmp = ((0.020833333333333332 * Math.pow(x, 3.0)) + (x * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 / (Math.sin(x) / (0.5 - (0.5 * Math.cos(x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0039:
		tmp = (0.5 + (math.cos(x) * -0.5)) / (0.375 * math.sin(x))
	elif x <= 0.0043:
		tmp = ((0.020833333333333332 * math.pow(x, 3.0)) + (x * 0.25)) / 0.375
	else:
		tmp = 2.6666666666666665 / (math.sin(x) / (0.5 - (0.5 * math.cos(x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0039)
		tmp = Float64(Float64(0.5 + Float64(cos(x) * -0.5)) / Float64(0.375 * sin(x)));
	elseif (x <= 0.0043)
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x ^ 3.0)) + Float64(x * 0.25)) / 0.375);
	else
		tmp = Float64(2.6666666666666665 / Float64(sin(x) / Float64(0.5 - Float64(0.5 * cos(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0039)
		tmp = (0.5 + (cos(x) * -0.5)) / (0.375 * sin(x));
	elseif (x <= 0.0043)
		tmp = ((0.020833333333333332 * (x ^ 3.0)) + (x * 0.25)) / 0.375;
	else
		tmp = 2.6666666666666665 / (sin(x) / (0.5 - (0.5 * cos(x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0039], N[(N[(0.5 + N[(N[Cos[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(0.375 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0043], N[(N[(N[(0.020833333333333332 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(2.6666666666666665 / N[(N[Sin[x], $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0038999999999999998

   1. Initial program 99.0%

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

      1. associate-/l* 99.0%

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

      2. *-commutative 99.0%

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

      3. *-lft-identity 99.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{1 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      4. metadata-eval 99.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      5. times-frac 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      6. associate-/l* 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      7. *-commutative 99.1%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      8. neg-mul-1 99.1%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      9. sin-neg 99.1%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      10. distribute-lft-neg-out 99.1%

          \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      11. associate-*r/ 99.0%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.0%

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

      1. associate-/l* 99.0%

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

      2. associate-*l* 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*l/ 99.1%

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

      5. sqr-sin-a 98.4%

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

      6. count-2 98.4%

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

      7. distribute-lft-out 98.4%

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

      8. metadata-eval 98.4%

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

      9. *-commutative 98.4%

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

      10. *-un-lft-identity 98.4%

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

   5. Applied egg-rr 98.4%

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

      1. *-commutative 98.4%

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

      2. metadata-eval 98.4%

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

      3. times-frac 98.7%

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

      4. *-un-lft-identity 98.7%

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

      5. cancel-sign-sub-inv 98.7%

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

      6. *-commutative 98.7%

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

      7. metadata-eval 98.7%

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

   7. Applied egg-rr 98.7%

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

   if -0.0038999999999999998 < x < 0.0043

   1. Initial program 50.6%

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

      1. associate-/l* 99.5%

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

      2. *-commutative 99.5%

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

      3. *-lft-identity 99.5%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{1 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      4. metadata-eval 99.5%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      5. times-frac 99.4%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      6. associate-/l* 99.4%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      7. *-commutative 99.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      8. neg-mul-1 99.4%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      9. sin-neg 99.4%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      10. distribute-lft-neg-out 99.4%

          \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      11. associate-*r/ 99.5%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.5%

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

      1. associate-/l* 50.6%

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

      2. associate-*l* 50.6%

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

      3. *-commutative 50.6%

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

      4. associate-*l/ 50.7%

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

      5. sqr-sin-a 6.2%

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

      6. count-2 6.2%

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

      7. distribute-lft-out 6.2%

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

      8. metadata-eval 6.2%

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

      9. *-commutative 6.2%

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

      10. *-un-lft-identity 6.2%

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

   5. Applied egg-rr 6.2%

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

      1. associate-/r/ 6.2%

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

      2. div-inv 6.2%

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

      3. metadata-eval 6.2%

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

      4. associate-/r* 6.2%

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

      5. cancel-sign-sub-inv 6.2%

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

      6. *-commutative 6.2%

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

      7. metadata-eval 6.2%

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

   7. Applied egg-rr 6.2%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]

   if 0.0043 < x

   1. Initial program 98.9%

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

      1. associate-/l* 98.9%

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

      2. associate-*r/ 98.9%

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

      3. associate-/r/ 98.9%

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

      4. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

      1. associate-*l/ 99.0%

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

      2. associate-*r/ 99.0%

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

      3. associate-/l* 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. sqr-sin-a 98.6%

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

      5. count-2 98.6%

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

      6. distribute-lft-out 98.6%

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

      7. metadata-eval 98.6%

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

      8. *-commutative 98.6%

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

      9. *-un-lft-identity 98.6%

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

   5. Applied egg-rr 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0039:\\ \;\;\;\;\frac{0.5 + \cos x \cdot -0.5}{0.375 \cdot \sin x}\\ \mathbf{elif}\;x \leq 0.0043:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{0.5 - 0.5 \cdot \cos x}}\\ \end{array} \]

Alternative 13: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \cos x \cdot -0.5\\ \mathbf{if}\;x \leq -0.0039:\\ \;\;\;\;\frac{t_0}{0.375 \cdot \sin x}\\ \mathbf{elif}\;x \leq 0.0043:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{0.375}}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* (cos x) -0.5))))
   (if (<= x -0.0039)
     (/ t_0 (* 0.375 (sin x)))
     (if (<= x 0.0043)
       (/ (+ (* 0.020833333333333332 (pow x 3.0)) (* x 0.25)) 0.375)
       (/ (/ t_0 0.375) (sin x))))))

C

double code(double x) {
	double t_0 = 0.5 + (cos(x) * -0.5);
	double tmp;
	if (x <= -0.0039) {
		tmp = t_0 / (0.375 * sin(x));
	} else if (x <= 0.0043) {
		tmp = ((0.020833333333333332 * pow(x, 3.0)) + (x * 0.25)) / 0.375;
	} else {
		tmp = (t_0 / 0.375) / sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (cos(x) * (-0.5d0))
    if (x <= (-0.0039d0)) then
        tmp = t_0 / (0.375d0 * sin(x))
    else if (x <= 0.0043d0) then
        tmp = ((0.020833333333333332d0 * (x ** 3.0d0)) + (x * 0.25d0)) / 0.375d0
    else
        tmp = (t_0 / 0.375d0) / sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.5 + (Math.cos(x) * -0.5);
	double tmp;
	if (x <= -0.0039) {
		tmp = t_0 / (0.375 * Math.sin(x));
	} else if (x <= 0.0043) {
		tmp = ((0.020833333333333332 * Math.pow(x, 3.0)) + (x * 0.25)) / 0.375;
	} else {
		tmp = (t_0 / 0.375) / Math.sin(x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (math.cos(x) * -0.5)
	tmp = 0
	if x <= -0.0039:
		tmp = t_0 / (0.375 * math.sin(x))
	elif x <= 0.0043:
		tmp = ((0.020833333333333332 * math.pow(x, 3.0)) + (x * 0.25)) / 0.375
	else:
		tmp = (t_0 / 0.375) / math.sin(x)
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(cos(x) * -0.5))
	tmp = 0.0
	if (x <= -0.0039)
		tmp = Float64(t_0 / Float64(0.375 * sin(x)));
	elseif (x <= 0.0043)
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x ^ 3.0)) + Float64(x * 0.25)) / 0.375);
	else
		tmp = Float64(Float64(t_0 / 0.375) / sin(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 + (cos(x) * -0.5);
	tmp = 0.0;
	if (x <= -0.0039)
		tmp = t_0 / (0.375 * sin(x));
	elseif (x <= 0.0043)
		tmp = ((0.020833333333333332 * (x ^ 3.0)) + (x * 0.25)) / 0.375;
	else
		tmp = (t_0 / 0.375) / sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(N[Cos[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0039], N[(t$95$0 / N[(0.375 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0043], N[(N[(N[(0.020833333333333332 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(t$95$0 / 0.375), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0038999999999999998

   1. Initial program 99.0%

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

      1. associate-/l* 99.0%

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

      2. *-commutative 99.0%

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

      3. *-lft-identity 99.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{1 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      4. metadata-eval 99.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      5. times-frac 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      6. associate-/l* 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      7. *-commutative 99.1%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      8. neg-mul-1 99.1%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      9. sin-neg 99.1%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      10. distribute-lft-neg-out 99.1%

          \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      11. associate-*r/ 99.0%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.0%

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

      1. associate-/l* 99.0%

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

      2. associate-*l* 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*l/ 99.1%

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

      5. sqr-sin-a 98.4%

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

      6. count-2 98.4%

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

      7. distribute-lft-out 98.4%

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

      8. metadata-eval 98.4%

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

      9. *-commutative 98.4%

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

      10. *-un-lft-identity 98.4%

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

   5. Applied egg-rr 98.4%

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

      1. *-commutative 98.4%

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

      2. metadata-eval 98.4%

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

      3. times-frac 98.7%

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

      4. *-un-lft-identity 98.7%

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

      5. cancel-sign-sub-inv 98.7%

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

      6. *-commutative 98.7%

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

      7. metadata-eval 98.7%

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

   7. Applied egg-rr 98.7%

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

   if -0.0038999999999999998 < x < 0.0043

   1. Initial program 50.6%

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

      1. associate-/l* 99.5%

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

      2. *-commutative 99.5%

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

      3. *-lft-identity 99.5%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{1 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      4. metadata-eval 99.5%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      5. times-frac 99.4%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      6. associate-/l* 99.4%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      7. *-commutative 99.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      8. neg-mul-1 99.4%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      9. sin-neg 99.4%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      10. distribute-lft-neg-out 99.4%

          \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      11. associate-*r/ 99.5%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.5%

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

      1. associate-/l* 50.6%

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

      2. associate-*l* 50.6%

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

      3. *-commutative 50.6%

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

      4. associate-*l/ 50.7%

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

      5. sqr-sin-a 6.2%

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

      6. count-2 6.2%

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

      7. distribute-lft-out 6.2%

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

      8. metadata-eval 6.2%

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

      9. *-commutative 6.2%

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

      10. *-un-lft-identity 6.2%

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

   5. Applied egg-rr 6.2%

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

      1. associate-/r/ 6.2%

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

      2. div-inv 6.2%

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

      3. metadata-eval 6.2%

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

      4. associate-/r* 6.2%

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

      5. cancel-sign-sub-inv 6.2%

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

      6. *-commutative 6.2%

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

      7. metadata-eval 6.2%

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

   7. Applied egg-rr 6.2%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]

   if 0.0043 < x

   1. Initial program 98.9%

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

      1. associate-/l* 98.9%

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

      2. *-commutative 98.9%

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

      3. *-lft-identity 98.9%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{1 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      4. metadata-eval 98.9%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      5. times-frac 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      6. associate-/l* 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      7. *-commutative 99.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      8. neg-mul-1 99.0%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      9. sin-neg 99.0%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      10. distribute-lft-neg-out 99.0%

          \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

      11. associate-*r/ 98.9%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 98.9%

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

      1. associate-/l* 98.9%

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

      2. associate-*l* 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l/ 99.0%

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

      5. sqr-sin-a 98.5%

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

      6. count-2 98.5%

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

      7. distribute-lft-out 98.5%

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

      8. metadata-eval 98.5%

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

      9. *-commutative 98.5%

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

      10. *-un-lft-identity 98.5%

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

   5. Applied egg-rr 98.5%

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

      1. associate-/r/ 98.5%

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

      2. div-inv 98.6%

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

      3. metadata-eval 98.6%

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

      4. associate-/l/ 98.6%

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

      5. cancel-sign-sub-inv 98.6%

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

      6. *-commutative 98.6%

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

      7. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0039:\\ \;\;\;\;\frac{0.5 + \cos x \cdot -0.5}{0.375 \cdot \sin x}\\ \mathbf{elif}\;x \leq 0.0043:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 + \cos x \cdot -0.5}{0.375}}{\sin x}\\ \end{array} \]

Alternative 14: 55.0% 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 75.9%

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

   1. associate-*l* 76.0%

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

   2. associate-/l* 76.0%

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

   3. sqr-neg 76.0%

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

   4. sin-neg 76.0%

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

   5. distribute-lft-neg-out 76.0%

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

   6. sin-neg 76.0%

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

   7. distribute-lft-neg-out 76.0%

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

   8. associate-/r* 99.3%

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

   9. associate-/l* 99.2%

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

   10. distribute-lft-neg-out 99.2%

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

   11. sin-neg 99.2%

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

   12. neg-mul-1 99.2%

       \[\leadsto \frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\frac{\sin x}{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}} \]

   13. associate-/r* 99.2%

       \[\leadsto \frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\color{blue}{\frac{\frac{\sin x}{-1}}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 53.3%

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

5. Final simplification 53.3%

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

Alternative 15: 55.3% 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 75.9%

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

   1. associate-/l* 99.2%

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

   2. associate-*r/ 99.2%

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

   3. associate-/r/ 99.2%

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

   4. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-*l* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-/r/ 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 99.5%

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

   7. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in x around 0 53.6%

   \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75}} \]

7. Final simplification 53.6%

   \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{0.75} \]

Alternative 16: 50.9% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1.5}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ 1.5 x)))

C

double code(double x) {
	return 1.0 / (1.5 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.5d0 / x)
end function

Java

public static double code(double x) {
	return 1.0 / (1.5 / x);
}

Python

def code(x):
	return 1.0 / (1.5 / x)

Julia

function code(x)
	return Float64(1.0 / Float64(1.5 / x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.5 / x);
end

Wolfram

code[x_] := N[(1.0 / N[(1.5 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.9%

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

   1. associate-*l* 76.0%

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

   2. associate-/l* 76.0%

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

   3. sqr-neg 76.0%

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

   4. sin-neg 76.0%

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

   5. distribute-lft-neg-out 76.0%

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

   6. sin-neg 76.0%

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

   7. distribute-lft-neg-out 76.0%

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

   8. associate-/r* 99.3%

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

   9. associate-/l* 99.2%

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

   10. distribute-lft-neg-out 99.2%

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

   11. sin-neg 99.2%

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

   12. neg-mul-1 99.2%

       \[\leadsto \frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\frac{\sin x}{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}} \]

   13. associate-/r* 99.2%

       \[\leadsto \frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\color{blue}{\frac{\frac{\sin x}{-1}}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.2%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

   3. *-un-lft-identity 99.1%

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

   4. times-frac 99.2%

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

   5. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. unpow-1 99.2%

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

   2. associate-*r/ 99.2%

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

   3. associate-/r/ 99.1%

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

   4. *-commutative 99.1%

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

7. Simplified 99.1%

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

   1. associate-/r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*r/ 99.2%

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

   4. associate-/r/ 99.3%

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

   5. *-un-lft-identity 99.3%

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

   6. times-frac 99.2%

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

   7. metadata-eval 99.2%

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

   8. associate-/l/ 75.9%

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

   9. pow2 75.9%

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

9. Applied egg-rr 75.9%

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

10. Taylor expanded in x around 0 49.4%

    \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]

11. Final simplification 49.4%

    \[\leadsto \frac{1}{\frac{1.5}{x}} \]

Alternative 17: 51.1% 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 75.9%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. *-lft-identity 99.2%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{1 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   4. metadata-eval 99.2%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   5. times-frac 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   6. associate-/l* 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   7. *-commutative 99.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   8. neg-mul-1 99.2%

      \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   9. sin-neg 99.2%

      \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   10. distribute-lft-neg-out 99.2%

       \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

   11. associate-*r/ 99.2%

       \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.2%

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

   1. associate-/l* 75.9%

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

   2. associate-*l* 76.0%

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

   3. *-commutative 76.0%

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

   4. associate-*l/ 76.0%

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

   5. sqr-sin-a 54.5%

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

   6. count-2 54.5%

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

   7. distribute-lft-out 54.5%

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

   8. metadata-eval 54.5%

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

   9. *-commutative 54.5%

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

   10. *-un-lft-identity 54.5%

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

5. Applied egg-rr 54.5%

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

   1. associate-/r/ 54.5%

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

   2. div-inv 54.5%

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

   3. metadata-eval 54.5%

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

   4. associate-/r* 54.5%

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

   5. cancel-sign-sub-inv 54.5%

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

   6. *-commutative 54.5%

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

   7. metadata-eval 54.5%

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

7. Applied egg-rr 54.5%

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

8. Taylor expanded in x around 0 49.5%

   \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]
9. Step-by-step derivation

   1. *-commutative 49.5%

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

10. Simplified 49.5%

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

11. Final simplification 49.5%

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

Alternative 18: 50.8% 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 75.9%

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

   1. associate-/l* 99.2%

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

   2. associate-*r/ 99.2%

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

   3. associate-/r/ 99.2%

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

   4. metadata-eval 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 49.2%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]

5. Final simplification 49.2%

   \[\leadsto x \cdot 0.6666666666666666 \]

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 2023301 
(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)))