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

Percentage Accurate: 76.4% → 99.5%

Time: 23.2s

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: 76.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t_1 := {t_0}^{2}\\ \mathbf{if}\;x \leq -0.002:\\ \;\;\;\;\frac{-\frac{t_1}{0.375}}{-\sin x}\\ \mathbf{elif}\;x \leq 10^{-7}:\\ \;\;\;\;\frac{t_0}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{0.375 \cdot \sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))) (t_1 (pow t_0 2.0)))
   (if (<= x -0.002)
     (/ (- (/ t_1 0.375)) (- (sin x)))
     (if (<= x 1e-7)
       (/ t_0 (+ 0.75 (* -0.09375 (* x x))))
       (/ 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 <= -0.002) {
		tmp = -(t_1 / 0.375) / -sin(x);
	} else if (x <= 1e-7) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} 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 <= (-0.002d0)) then
        tmp = -(t_1 / 0.375d0) / -sin(x)
    else if (x <= 1d-7) then
        tmp = t_0 / (0.75d0 + ((-0.09375d0) * (x * x)))
    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 <= -0.002) {
		tmp = -(t_1 / 0.375) / -Math.sin(x);
	} else if (x <= 1e-7) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} 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 <= -0.002:
		tmp = -(t_1 / 0.375) / -math.sin(x)
	elif x <= 1e-7:
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)))
	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 <= -0.002)
		tmp = Float64(Float64(-Float64(t_1 / 0.375)) / Float64(-sin(x)));
	elseif (x <= 1e-7)
		tmp = Float64(t_0 / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	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 <= -0.002)
		tmp = -(t_1 / 0.375) / -sin(x);
	elseif (x <= 1e-7)
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	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, -0.002], N[((-N[(t$95$1 / 0.375), $MachinePrecision]) / (-N[Sin[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 1e-7], N[(t$95$0 / N[(0.75 + N[(-0.09375 * N[(x * x), $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 < -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. *-lft-identity 99.0%

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

      3. metadata-eval 99.0%

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

      4. times-frac 99.0%

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

      5. neg-mul-1 99.0%

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

      6. sin-neg 99.0%

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

      7. neg-mul-1 99.0%

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

      8. sin-neg 99.0%

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

      9. distribute-lft-neg-out 99.0%

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

      10. distribute-lft-neg-out 99.0%

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

      11. sin-neg 99.0%

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

      12. neg-mul-1 99.0%

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

      13. associate-/r* 99.0%

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

      14. associate-/r/ 99.1%

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

   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. *-commutative 99.1%

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

      2. clear-num 98.9%

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

      3. un-div-inv 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. frac-2neg 98.9%

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

      2. div-inv 98.9%

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

      3. div-inv 98.9%

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

      4. distribute-lft-neg-in 98.9%

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

      5. associate-/r* 99.0%

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

      6. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. distribute-lft-neg-out 99.0%

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

      2. associate-*r/ 99.0%

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

      3. *-rgt-identity 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-/r* 99.0%

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

      6. associate-/l* 99.1%

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

      7. unpow2 99.1%

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

   9. Simplified 99.1%

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

   if -2e-3 < x < 9.9999999999999995e-8

   1. Initial program 59.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. *-lft-identity 99.5%

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

      3. metadata-eval 99.5%

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

      4. times-frac 99.5%

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

      5. neg-mul-1 99.5%

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

      6. sin-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. sin-neg 99.5%

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

      9. distribute-lft-neg-out 99.5%

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

      10. distribute-lft-neg-out 99.5%

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

      11. sin-neg 99.5%

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

      12. neg-mul-1 99.5%

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

      13. associate-/r* 99.5%

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

      14. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.2%

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

      3. un-div-inv 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. Applied egg-rr 99.4%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

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

   8. Simplified 100.0%

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

   if 9.9999999999999995e-8 < x

   1. Initial program 99.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. *-lft-identity 99.2%

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

      3. metadata-eval 99.2%

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

      4. times-frac 99.2%

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

      5. neg-mul-1 99.2%

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

      6. sin-neg 99.2%

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

      7. neg-mul-1 99.2%

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

      8. sin-neg 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. distribute-lft-neg-out 99.2%

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

      11. sin-neg 99.2%

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

      12. neg-mul-1 99.2%

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

      13. associate-/r* 99.2%

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

      14. associate-/r/ 99.1%

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

   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. *-commutative 99.1%

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

      2. associate-*r/ 99.2%

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

      3. *-commutative 99.2%

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

      4. associate-*r* 99.0%

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

      5. clear-num 99.0%

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

      6. un-div-inv 99.3%

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

      7. pow2 99.3%

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

      8. div-inv 99.4%

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

      9. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.002:\\ \;\;\;\;\frac{-\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375}}{-\sin x}\\ \mathbf{elif}\;x \leq 10^{-7}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375 \cdot \sin x}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{-7} \lor \neg \left(x \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;2.6666666666666665 \cdot \frac{{t_0}^{2}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{0.75}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))))
   (if (or (<= x -2e-7) (not (<= x 5e-8)))
     (* 2.6666666666666665 (/ (pow t_0 2.0) (sin x)))
     (/ t_0 0.75))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	double tmp;
	if ((x <= -2e-7) || !(x <= 5e-8)) {
		tmp = 2.6666666666666665 * (pow(t_0, 2.0) / sin(x));
	} else {
		tmp = t_0 / 0.75;
	}
	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 <= (-2d-7)) .or. (.not. (x <= 5d-8))) then
        tmp = 2.6666666666666665d0 * ((t_0 ** 2.0d0) / sin(x))
    else
        tmp = t_0 / 0.75d0
    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 <= -2e-7) || !(x <= 5e-8)) {
		tmp = 2.6666666666666665 * (Math.pow(t_0, 2.0) / Math.sin(x));
	} else {
		tmp = t_0 / 0.75;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	tmp = 0
	if (x <= -2e-7) or not (x <= 5e-8):
		tmp = 2.6666666666666665 * (math.pow(t_0, 2.0) / math.sin(x))
	else:
		tmp = t_0 / 0.75
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	tmp = 0.0
	if ((x <= -2e-7) || !(x <= 5e-8))
		tmp = Float64(2.6666666666666665 * Float64((t_0 ^ 2.0) / sin(x)));
	else
		tmp = Float64(t_0 / 0.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5));
	tmp = 0.0;
	if ((x <= -2e-7) || ~((x <= 5e-8)))
		tmp = 2.6666666666666665 * ((t_0 ^ 2.0) / sin(x));
	else
		tmp = t_0 / 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -2e-7], N[Not[LessEqual[x, 5e-8]], $MachinePrecision]], N[(2.6666666666666665 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / 0.75), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999999999999e-7 or 4.9999999999999998e-8 < 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* 99.1%

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

      2. *-lft-identity 99.1%

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

      3. metadata-eval 99.1%

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

      4. times-frac 99.1%

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

      5. neg-mul-1 99.1%

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

      6. sin-neg 99.1%

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

      7. neg-mul-1 99.1%

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

      8. sin-neg 99.1%

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

      9. distribute-lft-neg-out 99.1%

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

      10. distribute-lft-neg-out 99.1%

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

      11. sin-neg 99.1%

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

      12. neg-mul-1 99.1%

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

      13. associate-/r* 99.1%

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

      14. associate-/r/ 99.1%

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

   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. 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. associate-*r/ 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*r/ 99.1%

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

   6. Simplified 99.1%

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

   if -1.9999999999999999e-7 < x < 4.9999999999999998e-8

   1. Initial program 57.7%

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

      1. associate-/l* 99.5%

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

      2. *-lft-identity 99.5%

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

      3. metadata-eval 99.5%

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

      4. times-frac 99.5%

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

      5. neg-mul-1 99.5%

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

      6. sin-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. sin-neg 99.5%

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

      9. distribute-lft-neg-out 99.5%

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

      10. distribute-lft-neg-out 99.5%

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

      11. sin-neg 99.5%

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

      12. neg-mul-1 99.5%

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

      13. associate-/r* 99.5%

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

      14. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.2%

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

      3. un-div-inv 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. Applied egg-rr 99.4%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-7} \lor \neg \left(x \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75}\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t_1 := {t_0}^{2}\\ \mathbf{if}\;x \leq -0.00028:\\ \;\;\;\;t_1 \cdot \frac{2.6666666666666665}{\sin x}\\ \mathbf{elif}\;x \leq 10^{-7}:\\ \;\;\;\;\frac{t_0}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{t_1}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))) (t_1 (pow t_0 2.0)))
   (if (<= x -0.00028)
     (* t_1 (/ 2.6666666666666665 (sin x)))
     (if (<= x 1e-7)
       (/ t_0 (+ 0.75 (* -0.09375 (* x x))))
       (* 2.6666666666666665 (/ t_1 (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 <= -0.00028) {
		tmp = t_1 * (2.6666666666666665 / sin(x));
	} else if (x <= 1e-7) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * (t_1 / 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 <= (-0.00028d0)) then
        tmp = t_1 * (2.6666666666666665d0 / sin(x))
    else if (x <= 1d-7) then
        tmp = t_0 / (0.75d0 + ((-0.09375d0) * (x * x)))
    else
        tmp = 2.6666666666666665d0 * (t_1 / 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 <= -0.00028) {
		tmp = t_1 * (2.6666666666666665 / Math.sin(x));
	} else if (x <= 1e-7) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * (t_1 / 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 <= -0.00028:
		tmp = t_1 * (2.6666666666666665 / math.sin(x))
	elif x <= 1e-7:
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)))
	else:
		tmp = 2.6666666666666665 * (t_1 / 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 <= -0.00028)
		tmp = Float64(t_1 * Float64(2.6666666666666665 / sin(x)));
	elseif (x <= 1e-7)
		tmp = Float64(t_0 / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	else
		tmp = Float64(2.6666666666666665 * Float64(t_1 / 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 <= -0.00028)
		tmp = t_1 * (2.6666666666666665 / sin(x));
	elseif (x <= 1e-7)
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	else
		tmp = 2.6666666666666665 * (t_1 / 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, -0.00028], N[(t$95$1 * N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-7], N[(t$95$0 / N[(0.75 + N[(-0.09375 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(t$95$1 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7999999999999998e-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. *-lft-identity 99.0%

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

      3. metadata-eval 99.0%

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

      4. times-frac 99.0%

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

      5. neg-mul-1 99.0%

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

      6. sin-neg 99.0%

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

      7. neg-mul-1 99.0%

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

      8. sin-neg 99.0%

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

      9. distribute-lft-neg-out 99.0%

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

      10. distribute-lft-neg-out 99.0%

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

      11. sin-neg 99.0%

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

      12. neg-mul-1 99.0%

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

      13. associate-/r* 99.0%

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

      14. associate-/r/ 99.1%

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

   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. Taylor expanded in x around inf 99.0%

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

      1. associate-*r/ 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l/ 99.1%

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

      4. *-commutative 99.1%

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

   6. Simplified 99.1%

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

   if -2.7999999999999998e-4 < x < 9.9999999999999995e-8

   1. Initial program 59.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. *-lft-identity 99.5%

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

      3. metadata-eval 99.5%

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

      4. times-frac 99.5%

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

      5. neg-mul-1 99.5%

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

      6. sin-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. sin-neg 99.5%

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

      9. distribute-lft-neg-out 99.5%

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

      10. distribute-lft-neg-out 99.5%

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

      11. sin-neg 99.5%

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

      12. neg-mul-1 99.5%

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

      13. associate-/r* 99.5%

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

      14. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.2%

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

      3. un-div-inv 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. Applied egg-rr 99.4%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

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

   8. Simplified 100.0%

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

   if 9.9999999999999995e-8 < x

   1. Initial program 99.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. *-lft-identity 99.2%

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

      3. metadata-eval 99.2%

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

      4. times-frac 99.2%

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

      5. neg-mul-1 99.2%

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

      6. sin-neg 99.2%

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

      7. neg-mul-1 99.2%

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

      8. sin-neg 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. distribute-lft-neg-out 99.2%

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

      11. sin-neg 99.2%

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

      12. neg-mul-1 99.2%

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

      13. associate-/r* 99.2%

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

      14. associate-/r/ 99.1%

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

   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. Taylor expanded in x around inf 99.2%

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

      1. associate-*r/ 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*r/ 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00028:\\ \;\;\;\;{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}\\ \mathbf{elif}\;x \leq 10^{-7}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t_1 := {t_0}^{2}\\ \mathbf{if}\;x \leq -0.0003:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{t_1}}\\ \mathbf{elif}\;x \leq 10^{-7}:\\ \;\;\;\;\frac{t_0}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{t_1}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))) (t_1 (pow t_0 2.0)))
   (if (<= x -0.0003)
     (/ 2.6666666666666665 (/ (sin x) t_1))
     (if (<= x 1e-7)
       (/ t_0 (+ 0.75 (* -0.09375 (* x x))))
       (* 2.6666666666666665 (/ t_1 (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 <= -0.0003) {
		tmp = 2.6666666666666665 / (sin(x) / t_1);
	} else if (x <= 1e-7) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * (t_1 / 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 <= (-0.0003d0)) then
        tmp = 2.6666666666666665d0 / (sin(x) / t_1)
    else if (x <= 1d-7) then
        tmp = t_0 / (0.75d0 + ((-0.09375d0) * (x * x)))
    else
        tmp = 2.6666666666666665d0 * (t_1 / 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 <= -0.0003) {
		tmp = 2.6666666666666665 / (Math.sin(x) / t_1);
	} else if (x <= 1e-7) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * (t_1 / 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 <= -0.0003:
		tmp = 2.6666666666666665 / (math.sin(x) / t_1)
	elif x <= 1e-7:
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)))
	else:
		tmp = 2.6666666666666665 * (t_1 / 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 <= -0.0003)
		tmp = Float64(2.6666666666666665 / Float64(sin(x) / t_1));
	elseif (x <= 1e-7)
		tmp = Float64(t_0 / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	else
		tmp = Float64(2.6666666666666665 * Float64(t_1 / 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 <= -0.0003)
		tmp = 2.6666666666666665 / (sin(x) / t_1);
	elseif (x <= 1e-7)
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	else
		tmp = 2.6666666666666665 * (t_1 / 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, -0.0003], N[(2.6666666666666665 / N[(N[Sin[x], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-7], N[(t$95$0 / N[(0.75 + N[(-0.09375 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(t$95$1 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.99999999999999974e-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. *-lft-identity 99.0%

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

      3. metadata-eval 99.0%

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

      4. times-frac 99.0%

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

      5. neg-mul-1 99.0%

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

      6. sin-neg 99.0%

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

      7. neg-mul-1 99.0%

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

      8. sin-neg 99.0%

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

      9. distribute-lft-neg-out 99.0%

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

      10. distribute-lft-neg-out 99.0%

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

      11. sin-neg 99.0%

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

      12. neg-mul-1 99.0%

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

      13. associate-/r* 99.0%

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

      14. associate-/r/ 99.1%

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

   3. Simplified 99.1%

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

   5. Applied egg-rr 99.1%

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

   if -2.99999999999999974e-4 < x < 9.9999999999999995e-8

   1. Initial program 59.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. *-lft-identity 99.5%

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

      3. metadata-eval 99.5%

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

      4. times-frac 99.5%

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

      5. neg-mul-1 99.5%

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

      6. sin-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. sin-neg 99.5%

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

      9. distribute-lft-neg-out 99.5%

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

      10. distribute-lft-neg-out 99.5%

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

      11. sin-neg 99.5%

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

      12. neg-mul-1 99.5%

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

      13. associate-/r* 99.5%

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

      14. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.2%

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

      3. un-div-inv 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. Applied egg-rr 99.4%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

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

   8. Simplified 100.0%

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

   if 9.9999999999999995e-8 < x

   1. Initial program 99.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. *-lft-identity 99.2%

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

      3. metadata-eval 99.2%

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

      4. times-frac 99.2%

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

      5. neg-mul-1 99.2%

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

      6. sin-neg 99.2%

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

      7. neg-mul-1 99.2%

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

      8. sin-neg 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. distribute-lft-neg-out 99.2%

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

      11. sin-neg 99.2%

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

      12. neg-mul-1 99.2%

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

      13. associate-/r* 99.2%

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

      14. associate-/r/ 99.1%

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

   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. Taylor expanded in x around inf 99.2%

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

      1. associate-*r/ 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*r/ 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0003:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}\\ \mathbf{elif}\;x \leq 10^{-7}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t_1 := {t_0}^{2}\\ \mathbf{if}\;x \leq -0.0003:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{t_1}}\\ \mathbf{elif}\;x \leq 10^{-7}:\\ \;\;\;\;\frac{t_0}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{0.375 \cdot \sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))) (t_1 (pow t_0 2.0)))
   (if (<= x -0.0003)
     (/ 2.6666666666666665 (/ (sin x) t_1))
     (if (<= x 1e-7)
       (/ t_0 (+ 0.75 (* -0.09375 (* x x))))
       (/ 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 <= -0.0003) {
		tmp = 2.6666666666666665 / (sin(x) / t_1);
	} else if (x <= 1e-7) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} 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 <= (-0.0003d0)) then
        tmp = 2.6666666666666665d0 / (sin(x) / t_1)
    else if (x <= 1d-7) then
        tmp = t_0 / (0.75d0 + ((-0.09375d0) * (x * x)))
    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 <= -0.0003) {
		tmp = 2.6666666666666665 / (Math.sin(x) / t_1);
	} else if (x <= 1e-7) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} 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 <= -0.0003:
		tmp = 2.6666666666666665 / (math.sin(x) / t_1)
	elif x <= 1e-7:
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)))
	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 <= -0.0003)
		tmp = Float64(2.6666666666666665 / Float64(sin(x) / t_1));
	elseif (x <= 1e-7)
		tmp = Float64(t_0 / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	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 <= -0.0003)
		tmp = 2.6666666666666665 / (sin(x) / t_1);
	elseif (x <= 1e-7)
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	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, -0.0003], N[(2.6666666666666665 / N[(N[Sin[x], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-7], N[(t$95$0 / N[(0.75 + N[(-0.09375 * N[(x * x), $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 < -2.99999999999999974e-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. *-lft-identity 99.0%

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

      3. metadata-eval 99.0%

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

      4. times-frac 99.0%

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

      5. neg-mul-1 99.0%

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

      6. sin-neg 99.0%

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

      7. neg-mul-1 99.0%

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

      8. sin-neg 99.0%

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

      9. distribute-lft-neg-out 99.0%

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

      10. distribute-lft-neg-out 99.0%

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

      11. sin-neg 99.0%

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

      12. neg-mul-1 99.0%

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

      13. associate-/r* 99.0%

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

      14. associate-/r/ 99.1%

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

   3. Simplified 99.1%

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

   5. Applied egg-rr 99.1%

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

   if -2.99999999999999974e-4 < x < 9.9999999999999995e-8

   1. Initial program 59.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. *-lft-identity 99.5%

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

      3. metadata-eval 99.5%

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

      4. times-frac 99.5%

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

      5. neg-mul-1 99.5%

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

      6. sin-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. sin-neg 99.5%

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

      9. distribute-lft-neg-out 99.5%

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

      10. distribute-lft-neg-out 99.5%

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

      11. sin-neg 99.5%

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

      12. neg-mul-1 99.5%

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

      13. associate-/r* 99.5%

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

      14. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.2%

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

      3. un-div-inv 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. Applied egg-rr 99.4%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

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

   8. Simplified 100.0%

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

   if 9.9999999999999995e-8 < x

   1. Initial program 99.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. *-lft-identity 99.2%

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

      3. metadata-eval 99.2%

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

      4. times-frac 99.2%

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

      5. neg-mul-1 99.2%

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

      6. sin-neg 99.2%

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

      7. neg-mul-1 99.2%

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

      8. sin-neg 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. distribute-lft-neg-out 99.2%

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

      11. sin-neg 99.2%

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

      12. neg-mul-1 99.2%

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

      13. associate-/r* 99.2%

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

      14. associate-/r/ 99.1%

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

   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. *-commutative 99.1%

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

      2. associate-*r/ 99.2%

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

      3. *-commutative 99.2%

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

      4. associate-*r* 99.0%

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

      5. clear-num 99.0%

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

      6. un-div-inv 99.3%

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

      7. pow2 99.3%

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

      8. div-inv 99.4%

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

      9. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0003:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}\\ \mathbf{elif}\;x \leq 10^{-7}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375 \cdot \sin x}\\ \end{array} \]

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. associate-/l* 99.3%

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

   2. *-lft-identity 99.3%

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

   3. metadata-eval 99.3%

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

   4. times-frac 99.3%

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

   5. neg-mul-1 99.3%

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

   6. sin-neg 99.3%

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

   7. neg-mul-1 99.3%

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

   8. sin-neg 99.3%

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

   9. distribute-lft-neg-out 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. sin-neg 99.3%

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

   12. neg-mul-1 99.3%

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

   13. associate-/r* 99.3%

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

   14. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. clear-num 99.1%

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

   3. un-div-inv 99.2%

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

5. Applied egg-rr 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. times-frac 99.5%

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

   3. metadata-eval 99.5%

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

7. Applied egg-rr 99.5%

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

8. 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 7: 99.2% accurate, 1.0× speedup

Math

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

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 79.3%

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

   1. associate-/l* 99.3%

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

   2. associate-*r/ 99.3%

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

   3. metadata-eval 99.3%

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

   4. associate-/l* 79.3%

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

   5. sqr-neg 79.3%

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

   6. sin-neg 79.3%

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

   7. distribute-lft-neg-out 79.3%

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

   8. sin-neg 79.3%

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

   9. distribute-lft-neg-out 79.3%

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

   10. associate-*r/ 99.2%

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

   11. distribute-lft-neg-out 99.2%

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

   12. distribute-rgt-neg-in 99.2%

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

   13. metadata-eval 99.2%

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

   14. distribute-lft-neg-out 99.2%

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

   15. distribute-rgt-neg-in 99.2%

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

   16. metadata-eval 99.2%

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

3. Simplified 99.2%

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

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 8: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. associate-/l* 99.3%

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

   2. associate-*r/ 99.3%

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

   3. metadata-eval 99.3%

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

   4. remove-double-neg 99.3%

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

   5. sin-neg 99.3%

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

   6. distribute-lft-neg-out 99.3%

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

   7. neg-mul-1 99.3%

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

   8. *-commutative 99.3%

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

   9. associate-/l* 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. distribute-rgt-neg-in 99.3%

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

   12. metadata-eval 99.3%

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

   13. associate-/l/ 99.3%

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

   14. neg-mul-1 99.3%

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

   15. sin-neg 99.3%

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

   16. distribute-lft-neg-out 99.3%

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

   17. distribute-lft-neg-out 99.3%

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

   18. distribute-rgt-neg-in 99.3%

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

   19. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 9: 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 79.3%

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

   1. associate-/l* 99.3%

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

   2. *-lft-identity 99.3%

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

   3. metadata-eval 99.3%

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

   4. times-frac 99.3%

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

   5. neg-mul-1 99.3%

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

   6. sin-neg 99.3%

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

   7. neg-mul-1 99.3%

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

   8. sin-neg 99.3%

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

   9. distribute-lft-neg-out 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. sin-neg 99.3%

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

   12. neg-mul-1 99.3%

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

   13. associate-/r* 99.3%

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

   14. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 10: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0056:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}\\ \mathbf{elif}\;x \leq 0.0054:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \left(\left(0.5 - \frac{\cos x}{2}\right) \cdot \frac{1}{\sin x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0056)
   (* 2.6666666666666665 (/ (- 0.5 (* 0.5 (cos x))) (sin x)))
   (if (<= x 0.0054)
     (/ (sin (* x 0.5)) (+ 0.75 (* -0.09375 (* x x))))
     (* 2.6666666666666665 (* (- 0.5 (/ (cos x) 2.0)) (/ 1.0 (sin x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0056) {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x))) / sin(x));
	} else if (x <= 0.0054) {
		tmp = sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (cos(x) / 2.0)) * (1.0 / sin(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.0056d0)) then
        tmp = 2.6666666666666665d0 * ((0.5d0 - (0.5d0 * cos(x))) / sin(x))
    else if (x <= 0.0054d0) then
        tmp = sin((x * 0.5d0)) / (0.75d0 + ((-0.09375d0) * (x * x)))
    else
        tmp = 2.6666666666666665d0 * ((0.5d0 - (cos(x) / 2.0d0)) * (1.0d0 / sin(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0056) {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * Math.cos(x))) / Math.sin(x));
	} else if (x <= 0.0054) {
		tmp = Math.sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (Math.cos(x) / 2.0)) * (1.0 / Math.sin(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0056:
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * math.cos(x))) / math.sin(x))
	elif x <= 0.0054:
		tmp = math.sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)))
	else:
		tmp = 2.6666666666666665 * ((0.5 - (math.cos(x) / 2.0)) * (1.0 / math.sin(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0056)
		tmp = Float64(2.6666666666666665 * Float64(Float64(0.5 - Float64(0.5 * cos(x))) / sin(x)));
	elseif (x <= 0.0054)
		tmp = Float64(sin(Float64(x * 0.5)) / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	else
		tmp = Float64(2.6666666666666665 * Float64(Float64(0.5 - Float64(cos(x) / 2.0)) * Float64(1.0 / sin(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0056)
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x))) / sin(x));
	elseif (x <= 0.0054)
		tmp = sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	else
		tmp = 2.6666666666666665 * ((0.5 - (cos(x) / 2.0)) * (1.0 / sin(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0056], N[(2.6666666666666665 * N[(N[(0.5 - N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0054], N[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[(0.75 + N[(-0.09375 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[(0.5 - N[(N[Cos[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00559999999999999994

   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. metadata-eval 99.0%

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

      4. remove-double-neg 99.0%

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

      5. sin-neg 99.0%

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

      6. distribute-lft-neg-out 99.0%

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

      7. neg-mul-1 99.0%

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

      8. *-commutative 99.0%

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

      9. associate-/l* 99.0%

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

      10. distribute-lft-neg-out 99.0%

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

      11. distribute-rgt-neg-in 99.0%

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

      12. metadata-eval 99.0%

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

      13. associate-/l/ 99.0%

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

      14. neg-mul-1 99.0%

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

      15. sin-neg 99.0%

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

      16. distribute-lft-neg-out 99.0%

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

      17. distribute-lft-neg-out 99.0%

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

      18. distribute-rgt-neg-in 99.0%

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

      19. metadata-eval 99.0%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{\sin \left(x \cdot -0.5\right)}{\frac{\sin x}{\sin \left(x \cdot -0.5\right)}}} \]
   4. 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. div-inv 98.9%

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

      3. sqr-sin-a 98.0%

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

      4. add-sqr-sqrt 58.9%

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

      5. sqrt-unprod 52.3%

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

      6. swap-sqr 52.3%

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

      7. metadata-eval 52.3%

         \[\leadsto 2.6666666666666665 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\left(x \cdot x\right) \cdot \color{blue}{0.25}}\right)\right) \cdot \frac{1}{\sin x}\right) \]

      8. metadata-eval 52.3%

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

      9. swap-sqr 52.3%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 98.0%

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

      12. sqr-sin-a 98.9%

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

      13. pow2 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. unpow2 98.9%

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

      2. sin-mult 98.0%

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

   7. Applied egg-rr 98.0%

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

      1. div-sub 98.0%

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

      2. +-inverses 98.0%

         \[\leadsto 2.6666666666666665 \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}\right) \cdot \frac{1}{\sin x}\right) \]

      3. cos-0 98.0%

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

      4. metadata-eval 98.0%

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

      5. distribute-lft-out 98.0%

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

      6. metadata-eval 98.0%

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

      7. *-rgt-identity 98.0%

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

   9. Simplified 98.0%

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

       1. un-div-inv 98.2%

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

       2. div-inv 98.2%

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

       3. metadata-eval 98.2%

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

   11. Applied egg-rr 98.2%

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

   if -0.00559999999999999994 < x < 0.0054000000000000003

   1. Initial program 59.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. *-lft-identity 99.5%

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

      3. metadata-eval 99.5%

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

      4. times-frac 99.5%

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

      5. neg-mul-1 99.5%

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

      6. sin-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. sin-neg 99.5%

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

      9. distribute-lft-neg-out 99.5%

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

      10. distribute-lft-neg-out 99.5%

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

      11. sin-neg 99.5%

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

      12. neg-mul-1 99.5%

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

      13. associate-/r* 99.5%

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

      14. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.2%

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

      3. un-div-inv 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. Applied egg-rr 99.4%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

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

   8. Simplified 100.0%

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

   if 0.0054000000000000003 < x

   1. Initial program 99.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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/ 99.1%

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

      3. metadata-eval 99.1%

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

      4. remove-double-neg 99.1%

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

      5. sin-neg 99.1%

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

      6. distribute-lft-neg-out 99.1%

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

      7. neg-mul-1 99.1%

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

      8. *-commutative 99.1%

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

      9. associate-/l* 99.1%

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

      10. distribute-lft-neg-out 99.1%

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

      11. distribute-rgt-neg-in 99.1%

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

      12. metadata-eval 99.1%

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

      13. associate-/l/ 99.1%

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

      14. neg-mul-1 99.1%

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

      15. sin-neg 99.1%

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

      16. distribute-lft-neg-out 99.1%

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

      17. distribute-lft-neg-out 99.1%

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

      18. distribute-rgt-neg-in 99.1%

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

      19. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. associate-/l* 99.2%

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

      2. div-inv 99.1%

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

      3. sqr-sin-a 98.2%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 50.6%

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

      6. swap-sqr 50.6%

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

      7. metadata-eval 50.6%

         \[\leadsto 2.6666666666666665 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\left(x \cdot x\right) \cdot \color{blue}{0.25}}\right)\right) \cdot \frac{1}{\sin x}\right) \]

      8. metadata-eval 50.6%

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

      9. swap-sqr 50.6%

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

      10. sqrt-unprod 57.5%

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

      11. add-sqr-sqrt 98.2%

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

      12. sqr-sin-a 99.1%

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

      13. pow2 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. unpow2 99.1%

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

      2. sin-mult 98.2%

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

   7. Applied egg-rr 98.2%

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

      1. div-sub 98.2%

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

      2. +-inverses 98.2%

         \[\leadsto 2.6666666666666665 \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}\right) \cdot \frac{1}{\sin x}\right) \]

      3. cos-0 98.2%

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

      4. metadata-eval 98.2%

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

      5. distribute-lft-out 98.2%

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

      6. metadata-eval 98.2%

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

      7. *-rgt-identity 98.2%

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

   9. Simplified 98.2%

      \[\leadsto 2.6666666666666665 \cdot \left(\color{blue}{\left(0.5 - \frac{\cos x}{2}\right)} \cdot \frac{1}{\sin x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0056:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}\\ \mathbf{elif}\;x \leq 0.0054:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \left(\left(0.5 - \frac{\cos x}{2}\right) \cdot \frac{1}{\sin x}\right)\\ \end{array} \]

Alternative 11: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0056:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}\\ \mathbf{elif}\;x \leq 0.0054:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin x}{2.6666666666666665 \cdot \left(0.5 - \frac{\cos x}{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0056)
   (* 2.6666666666666665 (/ (- 0.5 (* 0.5 (cos x))) (sin x)))
   (if (<= x 0.0054)
     (/ (sin (* x 0.5)) (+ 0.75 (* -0.09375 (* x x))))
     (/ 1.0 (/ (sin x) (* 2.6666666666666665 (- 0.5 (/ (cos x) 2.0))))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0056) {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x))) / sin(x));
	} else if (x <= 0.0054) {
		tmp = sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 1.0 / (sin(x) / (2.6666666666666665 * (0.5 - (cos(x) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.0056d0)) then
        tmp = 2.6666666666666665d0 * ((0.5d0 - (0.5d0 * cos(x))) / sin(x))
    else if (x <= 0.0054d0) then
        tmp = sin((x * 0.5d0)) / (0.75d0 + ((-0.09375d0) * (x * x)))
    else
        tmp = 1.0d0 / (sin(x) / (2.6666666666666665d0 * (0.5d0 - (cos(x) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0056) {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * Math.cos(x))) / Math.sin(x));
	} else if (x <= 0.0054) {
		tmp = Math.sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 1.0 / (Math.sin(x) / (2.6666666666666665 * (0.5 - (Math.cos(x) / 2.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0056:
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * math.cos(x))) / math.sin(x))
	elif x <= 0.0054:
		tmp = math.sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)))
	else:
		tmp = 1.0 / (math.sin(x) / (2.6666666666666665 * (0.5 - (math.cos(x) / 2.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0056)
		tmp = Float64(2.6666666666666665 * Float64(Float64(0.5 - Float64(0.5 * cos(x))) / sin(x)));
	elseif (x <= 0.0054)
		tmp = Float64(sin(Float64(x * 0.5)) / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	else
		tmp = Float64(1.0 / Float64(sin(x) / Float64(2.6666666666666665 * Float64(0.5 - Float64(cos(x) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0056)
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x))) / sin(x));
	elseif (x <= 0.0054)
		tmp = sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	else
		tmp = 1.0 / (sin(x) / (2.6666666666666665 * (0.5 - (cos(x) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0056], N[(2.6666666666666665 * N[(N[(0.5 - N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0054], N[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[(0.75 + N[(-0.09375 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sin[x], $MachinePrecision] / N[(2.6666666666666665 * N[(0.5 - N[(N[Cos[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00559999999999999994

   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. metadata-eval 99.0%

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

      4. remove-double-neg 99.0%

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

      5. sin-neg 99.0%

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

      6. distribute-lft-neg-out 99.0%

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

      7. neg-mul-1 99.0%

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

      8. *-commutative 99.0%

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

      9. associate-/l* 99.0%

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

      10. distribute-lft-neg-out 99.0%

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

      11. distribute-rgt-neg-in 99.0%

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

      12. metadata-eval 99.0%

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

      13. associate-/l/ 99.0%

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

      14. neg-mul-1 99.0%

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

      15. sin-neg 99.0%

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

      16. distribute-lft-neg-out 99.0%

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

      17. distribute-lft-neg-out 99.0%

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

      18. distribute-rgt-neg-in 99.0%

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

      19. metadata-eval 99.0%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{\sin \left(x \cdot -0.5\right)}{\frac{\sin x}{\sin \left(x \cdot -0.5\right)}}} \]
   4. 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. div-inv 98.9%

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

      3. sqr-sin-a 98.0%

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

      4. add-sqr-sqrt 58.9%

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

      5. sqrt-unprod 52.3%

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

      6. swap-sqr 52.3%

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

      7. metadata-eval 52.3%

         \[\leadsto 2.6666666666666665 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\left(x \cdot x\right) \cdot \color{blue}{0.25}}\right)\right) \cdot \frac{1}{\sin x}\right) \]

      8. metadata-eval 52.3%

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

      9. swap-sqr 52.3%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 98.0%

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

      12. sqr-sin-a 98.9%

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

      13. pow2 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. unpow2 98.9%

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

      2. sin-mult 98.0%

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

   7. Applied egg-rr 98.0%

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

      1. div-sub 98.0%

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

      2. +-inverses 98.0%

         \[\leadsto 2.6666666666666665 \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}\right) \cdot \frac{1}{\sin x}\right) \]

      3. cos-0 98.0%

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

      4. metadata-eval 98.0%

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

      5. distribute-lft-out 98.0%

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

      6. metadata-eval 98.0%

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

      7. *-rgt-identity 98.0%

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

   9. Simplified 98.0%

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

       1. un-div-inv 98.2%

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

       2. div-inv 98.2%

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

       3. metadata-eval 98.2%

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

   11. Applied egg-rr 98.2%

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

   if -0.00559999999999999994 < x < 0.0054000000000000003

   1. Initial program 59.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. *-lft-identity 99.5%

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

      3. metadata-eval 99.5%

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

      4. times-frac 99.5%

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

      5. neg-mul-1 99.5%

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

      6. sin-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. sin-neg 99.5%

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

      9. distribute-lft-neg-out 99.5%

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

      10. distribute-lft-neg-out 99.5%

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

      11. sin-neg 99.5%

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

      12. neg-mul-1 99.5%

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

      13. associate-/r* 99.5%

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

      14. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.2%

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

      3. un-div-inv 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. Applied egg-rr 99.4%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

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

   8. Simplified 100.0%

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

   if 0.0054000000000000003 < x

   1. Initial program 99.2%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. *-lft-identity 99.2%

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

      3. metadata-eval 99.2%

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

      4. times-frac 99.2%

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

      5. neg-mul-1 99.2%

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

      6. sin-neg 99.2%

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

      7. neg-mul-1 99.2%

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

      8. sin-neg 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. distribute-lft-neg-out 99.2%

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

      11. sin-neg 99.2%

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

      12. neg-mul-1 99.2%

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

      13. associate-/r* 99.2%

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

      14. associate-/r/ 99.1%

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

   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. associate-*l/ 99.2%

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

      2. clear-num 99.0%

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

      3. associate-*l* 99.1%

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

      4. pow2 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. unpow2 99.1%

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

      2. sin-mult 98.2%

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

   7. Applied egg-rr 98.2%

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

      1. div-sub 98.2%

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

      2. +-inverses 98.2%

         \[\leadsto 2.6666666666666665 \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}\right) \cdot \frac{1}{\sin x}\right) \]

      3. cos-0 98.2%

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

      4. metadata-eval 98.2%

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

      5. distribute-lft-out 98.2%

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

      6. metadata-eval 98.2%

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

      7. *-rgt-identity 98.2%

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

   9. Simplified 98.2%

      \[\leadsto \frac{1}{\frac{\sin x}{2.6666666666666665 \cdot \color{blue}{\left(0.5 - \frac{\cos x}{2}\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0056:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}\\ \mathbf{elif}\;x \leq 0.0054:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin x}{2.6666666666666665 \cdot \left(0.5 - \frac{\cos x}{2}\right)}}\\ \end{array} \]

Alternative 12: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0056 \lor \neg \left(x \leq 0.0054\right):\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.0056) (not (<= x 0.0054)))
   (* 2.6666666666666665 (/ (- 0.5 (* 0.5 (cos x))) (sin x)))
   (/ (sin (* x 0.5)) (+ 0.75 (* -0.09375 (* x x))))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.0056) || !(x <= 0.0054)) {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x))) / sin(x));
	} else {
		tmp = sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.0056d0)) .or. (.not. (x <= 0.0054d0))) then
        tmp = 2.6666666666666665d0 * ((0.5d0 - (0.5d0 * cos(x))) / sin(x))
    else
        tmp = sin((x * 0.5d0)) / (0.75d0 + ((-0.09375d0) * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.0056) || !(x <= 0.0054)) {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * Math.cos(x))) / Math.sin(x));
	} else {
		tmp = Math.sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.0056) or not (x <= 0.0054):
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * math.cos(x))) / math.sin(x))
	else:
		tmp = math.sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.0056) || !(x <= 0.0054))
		tmp = Float64(2.6666666666666665 * Float64(Float64(0.5 - Float64(0.5 * cos(x))) / sin(x)));
	else
		tmp = Float64(sin(Float64(x * 0.5)) / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.0056) || ~((x <= 0.0054)))
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x))) / sin(x));
	else
		tmp = sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.0056], N[Not[LessEqual[x, 0.0054]], $MachinePrecision]], N[(2.6666666666666665 * N[(N[(0.5 - N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[(0.75 + N[(-0.09375 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00559999999999999994 or 0.0054000000000000003 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. 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. metadata-eval 99.1%

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

      4. remove-double-neg 99.1%

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

      5. sin-neg 99.1%

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

      6. distribute-lft-neg-out 99.1%

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

      7. neg-mul-1 99.1%

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

      8. *-commutative 99.1%

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

      9. associate-/l* 99.1%

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

      10. distribute-lft-neg-out 99.1%

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

      11. distribute-rgt-neg-in 99.1%

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

      12. metadata-eval 99.1%

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

      13. associate-/l/ 99.1%

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

      14. neg-mul-1 99.1%

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

      15. sin-neg 99.1%

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

      16. distribute-lft-neg-out 99.1%

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

      17. distribute-lft-neg-out 99.1%

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

      18. distribute-rgt-neg-in 99.1%

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

      19. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. associate-/l* 99.1%

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

      2. div-inv 99.0%

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

      3. sqr-sin-a 98.1%

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

      4. add-sqr-sqrt 32.4%

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

      5. sqrt-unprod 51.6%

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

      6. swap-sqr 51.6%

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

      7. metadata-eval 51.6%

         \[\leadsto 2.6666666666666665 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\left(x \cdot x\right) \cdot \color{blue}{0.25}}\right)\right) \cdot \frac{1}{\sin x}\right) \]

      8. metadata-eval 51.6%

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

      9. swap-sqr 51.6%

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

      10. sqrt-unprod 25.9%

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

      11. add-sqr-sqrt 98.1%

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

      12. sqr-sin-a 99.0%

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

      13. pow2 99.0%

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

   5. Applied egg-rr 99.0%

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

      1. unpow2 99.0%

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

      2. sin-mult 98.1%

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

   7. Applied egg-rr 98.1%

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

      1. div-sub 98.1%

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

      2. +-inverses 98.1%

         \[\leadsto 2.6666666666666665 \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}\right) \cdot \frac{1}{\sin x}\right) \]

      3. cos-0 98.1%

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

      4. metadata-eval 98.1%

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

      5. distribute-lft-out 98.1%

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

      6. metadata-eval 98.1%

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

      7. *-rgt-identity 98.1%

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

   9. Simplified 98.1%

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

       1. un-div-inv 98.2%

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

       2. div-inv 98.2%

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

       3. metadata-eval 98.2%

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

   11. Applied egg-rr 98.2%

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

   if -0.00559999999999999994 < x < 0.0054000000000000003

   1. Initial program 59.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. *-lft-identity 99.5%

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

      3. metadata-eval 99.5%

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

      4. times-frac 99.5%

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

      5. neg-mul-1 99.5%

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

      6. sin-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. sin-neg 99.5%

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

      9. distribute-lft-neg-out 99.5%

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

      10. distribute-lft-neg-out 99.5%

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

      11. sin-neg 99.5%

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

      12. neg-mul-1 99.5%

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

      13. associate-/r* 99.5%

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

      14. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.2%

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

      3. un-div-inv 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. Applied egg-rr 99.4%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0056 \lor \neg \left(x \leq 0.0054\right):\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 13: 55.2% 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 79.3%

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

   1. associate-/l* 99.3%

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

   2. *-lft-identity 99.3%

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

   3. metadata-eval 99.3%

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

   4. times-frac 99.3%

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

   5. neg-mul-1 99.3%

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

   6. sin-neg 99.3%

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

   7. neg-mul-1 99.3%

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

   8. sin-neg 99.3%

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

   9. distribute-lft-neg-out 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. sin-neg 99.3%

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

   12. neg-mul-1 99.3%

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

   13. associate-/r* 99.3%

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

   14. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in x around 0 55.6%

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

5. Final simplification 55.6%

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

Alternative 14: 55.4% 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 79.3%

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

   1. associate-/l* 99.3%

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

   2. *-lft-identity 99.3%

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

   3. metadata-eval 99.3%

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

   4. times-frac 99.3%

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

   5. neg-mul-1 99.3%

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

   6. sin-neg 99.3%

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

   7. neg-mul-1 99.3%

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

   8. sin-neg 99.3%

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

   9. distribute-lft-neg-out 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. sin-neg 99.3%

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

   12. neg-mul-1 99.3%

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

   13. associate-/r* 99.3%

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

   14. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. clear-num 99.1%

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

   3. un-div-inv 99.2%

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

5. Applied egg-rr 99.2%

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

6. Taylor expanded in x around 0 55.9%

   \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75}} \]

7. Final simplification 55.9%

   \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{0.75} \]

Alternative 15: 51.7% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot -0.125 + 1.5 \cdot \frac{1}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ (* x -0.125) (* 1.5 (/ 1.0 x)))))

C

double code(double x) {
	return 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((x * (-0.125d0)) + (1.5d0 * (1.0d0 / x)))
end function

Java

public static double code(double x) {
	return 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
}

Python

def code(x):
	return 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(x * -0.125) + Float64(1.5 * Float64(1.0 / x))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(x * -0.125), $MachinePrecision] + N[(1.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. associate-/l* 99.3%

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

   2. *-lft-identity 99.3%

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

   3. metadata-eval 99.3%

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

   4. times-frac 99.3%

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

   5. neg-mul-1 99.3%

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

   6. sin-neg 99.3%

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

   7. neg-mul-1 99.3%

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

   8. sin-neg 99.3%

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

   9. distribute-lft-neg-out 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. sin-neg 99.3%

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

   12. neg-mul-1 99.3%

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

   13. associate-/r* 99.3%

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

   14. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

   1. associate-*l/ 79.3%

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

   2. clear-num 79.2%

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

   3. associate-*l* 79.3%

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

   4. pow2 79.3%

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

5. Applied egg-rr 79.3%

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

6. Taylor expanded in x around 0 51.2%

   \[\leadsto \frac{1}{\color{blue}{-0.125 \cdot x + 1.5 \cdot \frac{1}{x}}} \]

7. Final simplification 51.2%

   \[\leadsto \frac{1}{x \cdot -0.125 + 1.5 \cdot \frac{1}{x}} \]

Alternative 16: 51.2% 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 79.3%

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

   1. associate-/l* 99.3%

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

   2. *-lft-identity 99.3%

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

   3. metadata-eval 99.3%

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

   4. times-frac 99.3%

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

   5. neg-mul-1 99.3%

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

   6. sin-neg 99.3%

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

   7. neg-mul-1 99.3%

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

   8. sin-neg 99.3%

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

   9. distribute-lft-neg-out 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. sin-neg 99.3%

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

   12. neg-mul-1 99.3%

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

   13. associate-/r* 99.3%

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

   14. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

   1. associate-*l/ 79.3%

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

   2. clear-num 79.2%

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

   3. associate-*l* 79.3%

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

   4. pow2 79.3%

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

5. Applied egg-rr 79.3%

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

6. Taylor expanded in x around 0 51.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]

7. Final simplification 51.2%

   \[\leadsto \frac{1}{\frac{1.5}{x}} \]

Alternative 17: 3.5% accurate, 104.3× speedup

Math

\[\begin{array}{l} \\ x \cdot -0.6666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x -0.6666666666666666))

C

double code(double x) {
	return x * -0.6666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (-0.6666666666666666d0)
end function

Java

public static double code(double x) {
	return x * -0.6666666666666666;
}

Python

def code(x):
	return x * -0.6666666666666666

Julia

function code(x)
	return Float64(x * -0.6666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = x * -0.6666666666666666;
end

Wolfram

code[x_] := N[(x * -0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. associate-/l* 99.3%

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

   2. *-lft-identity 99.3%

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

   3. metadata-eval 99.3%

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

   4. times-frac 99.3%

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

   5. neg-mul-1 99.3%

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

   6. sin-neg 99.3%

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

   7. neg-mul-1 99.3%

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

   8. sin-neg 99.3%

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

   9. distribute-lft-neg-out 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. sin-neg 99.3%

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

   12. neg-mul-1 99.3%

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

   13. associate-/r* 99.3%

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

   14. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in x around 0 51.1%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
5. Step-by-step derivation

   1. *-commutative 51.1%

      \[\leadsto \color{blue}{x \cdot 0.6666666666666666} \]

6. Simplified 51.1%

   \[\leadsto \color{blue}{x \cdot 0.6666666666666666} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 25.9%

      \[\leadsto \color{blue}{\sqrt{x \cdot 0.6666666666666666} \cdot \sqrt{x \cdot 0.6666666666666666}} \]

   2. sqrt-unprod 18.3%

      \[\leadsto \color{blue}{\sqrt{\left(x \cdot 0.6666666666666666\right) \cdot \left(x \cdot 0.6666666666666666\right)}} \]

   3. *-commutative 18.3%

      \[\leadsto \sqrt{\color{blue}{\left(0.6666666666666666 \cdot x\right)} \cdot \left(x \cdot 0.6666666666666666\right)} \]

   4. *-commutative 18.3%

      \[\leadsto \sqrt{\left(0.6666666666666666 \cdot x\right) \cdot \color{blue}{\left(0.6666666666666666 \cdot x\right)}} \]

   5. swap-sqr 18.3%

      \[\leadsto \sqrt{\color{blue}{\left(0.6666666666666666 \cdot 0.6666666666666666\right) \cdot \left(x \cdot x\right)}} \]

   6. metadata-eval 18.3%

      \[\leadsto \sqrt{\color{blue}{0.4444444444444444} \cdot \left(x \cdot x\right)} \]

8. Applied egg-rr 18.3%

   \[\leadsto \color{blue}{\sqrt{0.4444444444444444 \cdot \left(x \cdot x\right)}} \]
9. Step-by-step derivation

   1. unpow2 18.3%

      \[\leadsto \sqrt{0.4444444444444444 \cdot \color{blue}{{x}^{2}}} \]

   2. *-commutative 18.3%

      \[\leadsto \sqrt{\color{blue}{{x}^{2} \cdot 0.4444444444444444}} \]

   3. unpow2 18.3%

      \[\leadsto \sqrt{\color{blue}{\left(x \cdot x\right)} \cdot 0.4444444444444444} \]

10. Simplified 18.3%

    \[\leadsto \color{blue}{\sqrt{\left(x \cdot x\right) \cdot 0.4444444444444444}} \]

11. Taylor expanded in x around -inf 3.2%

    \[\leadsto \color{blue}{-0.6666666666666666 \cdot x} \]
12. Step-by-step derivation

    1. *-commutative 3.2%

       \[\leadsto \color{blue}{x \cdot -0.6666666666666666} \]

13. Simplified 3.2%

    \[\leadsto \color{blue}{x \cdot -0.6666666666666666} \]

14. Final simplification 3.2%

    \[\leadsto x \cdot -0.6666666666666666 \]

Alternative 18: 51.1% 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 79.3%

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

   1. associate-/l* 99.3%

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

   2. *-lft-identity 99.3%

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

   3. metadata-eval 99.3%

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

   4. times-frac 99.3%

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

   5. neg-mul-1 99.3%

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

   6. sin-neg 99.3%

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

   7. neg-mul-1 99.3%

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

   8. sin-neg 99.3%

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

   9. distribute-lft-neg-out 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. sin-neg 99.3%

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

   12. neg-mul-1 99.3%

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

   13. associate-/r* 99.3%

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

   14. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in x around 0 51.1%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
5. Step-by-step derivation

   1. *-commutative 51.1%

      \[\leadsto \color{blue}{x \cdot 0.6666666666666666} \]

6. Simplified 51.1%

   \[\leadsto \color{blue}{x \cdot 0.6666666666666666} \]

7. Final simplification 51.1%

   \[\leadsto x \cdot 0.6666666666666666 \]

Developer target: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023275 
(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)))