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

Percentage Accurate: 77.5% → 99.5%

Time: 20.9s

Alternatives: 14

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: 77.5% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8} \lor \neg \left(x \leq 2 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{0.75}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5e-8) (not (<= x 2e-21)))
   (/ (pow (sin (* x 0.5)) 2.0) (* (sin x) 0.375))
   (/ (* x 0.5) 0.75)))

C

double code(double x) {
	double tmp;
	if ((x <= -5e-8) || !(x <= 2e-21)) {
		tmp = pow(sin((x * 0.5)), 2.0) / (sin(x) * 0.375);
	} else {
		tmp = (x * 0.5) / 0.75;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5d-8)) .or. (.not. (x <= 2d-21))) then
        tmp = (sin((x * 0.5d0)) ** 2.0d0) / (sin(x) * 0.375d0)
    else
        tmp = (x * 0.5d0) / 0.75d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -5e-8) || !(x <= 2e-21)) {
		tmp = Math.pow(Math.sin((x * 0.5)), 2.0) / (Math.sin(x) * 0.375);
	} else {
		tmp = (x * 0.5) / 0.75;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -5e-8) or not (x <= 2e-21):
		tmp = math.pow(math.sin((x * 0.5)), 2.0) / (math.sin(x) * 0.375)
	else:
		tmp = (x * 0.5) / 0.75
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5e-8) || !(x <= 2e-21))
		tmp = Float64((sin(Float64(x * 0.5)) ^ 2.0) / Float64(sin(x) * 0.375));
	else
		tmp = Float64(Float64(x * 0.5) / 0.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -5e-8) || ~((x <= 2e-21)))
		tmp = (sin((x * 0.5)) ^ 2.0) / (sin(x) * 0.375);
	else
		tmp = (x * 0.5) / 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5e-8], N[Not[LessEqual[x, 2e-21]], $MachinePrecision]], N[(N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] / 0.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e-8 or 1.99999999999999982e-21 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. frac-2neg 99.1%

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

      2. div-inv 99.1%

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

      3. *-commutative 99.1%

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

      4. distribute-rgt-neg-in 99.1%

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

      5. metadata-eval 99.1%

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

      6. distribute-neg-frac 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in x around inf 99.0%

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

      1. *-commutative 99.0%

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

      2. *-commutative 99.0%

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

      3. unpow2 99.0%

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

      4. associate-/l* 98.9%

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

      5. associate-*l/ 99.1%

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

      6. *-lft-identity 99.1%

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

      7. associate-*r/ 99.1%

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

      8. associate-*l/ 99.1%

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

      9. times-frac 99.0%

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

      10. associate-/l* 99.1%

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

      11. unpow2 99.1%

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

      12. /-rgt-identity 99.1%

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

   8. Simplified 99.1%

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

      1. clear-num 99.0%

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

      2. un-div-inv 99.1%

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

      3. div-inv 99.3%

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

      4. metadata-eval 99.3%

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

   10. Applied egg-rr 99.3%

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

   if -4.9999999999999998e-8 < x < 1.99999999999999982e-21

   1. Initial program 50.5%

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

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

      8. 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 99.4%

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

      3. un-div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in x around 0 100.0%

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

   7. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.5 \cdot x}}{0.75} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{x \cdot 0.5}}{0.75} \]

   9. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8} \lor \neg \left(x \leq 2 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{0.75}\\ \end{array} \]

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{1}{\frac{\sin x}{t_0} \cdot \frac{0.375}{t_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ 1.0 (* (/ (sin x) t_0) (/ 0.375 t_0)))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return 1.0 / ((sin(x) / t_0) * (0.375 / t_0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = 1.0d0 / ((sin(x) / t_0) * (0.375d0 / t_0))
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return 1.0 / ((Math.sin(x) / t_0) * (0.375 / t_0));
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return 1.0 / ((math.sin(x) / t_0) * (0.375 / t_0))

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(1.0 / Float64(Float64(sin(x) / t_0) * Float64(0.375 / t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = 1.0 / ((sin(x) / t_0) * (0.375 / t_0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(1.0 / N[(N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(0.375 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.1%

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

   1. associate-/l* 99.3%

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

   2. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. div-inv 99.3%

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

   2. clear-num 99.3%

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

   3. associate-*l* 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. *-commutative 99.2%

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

   5. associate-*r/ 74.0%

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

   6. pow2 74.0%

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

5. Applied egg-rr 74.0%

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

   1. associate-*l/ 74.0%

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

   2. unpow2 74.0%

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

   3. associate-*l* 74.1%

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

   4. metadata-eval 74.1%

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

   5. div-inv 74.2%

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

   6. associate-*l/ 99.6%

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

   7. clear-num 99.5%

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

   8. clear-num 99.4%

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

   9. frac-times 99.4%

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

   10. metadata-eval 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8} \lor \neg \left(x \leq 4 \cdot 10^{-32}\right):\\ \;\;\;\;{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{0.75}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5e-8) (not (<= x 4e-32)))
   (* (pow (sin (* x 0.5)) 2.0) (/ 2.6666666666666665 (sin x)))
   (/ (* x 0.5) 0.75)))

C

double code(double x) {
	double tmp;
	if ((x <= -5e-8) || !(x <= 4e-32)) {
		tmp = pow(sin((x * 0.5)), 2.0) * (2.6666666666666665 / sin(x));
	} else {
		tmp = (x * 0.5) / 0.75;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5d-8)) .or. (.not. (x <= 4d-32))) then
        tmp = (sin((x * 0.5d0)) ** 2.0d0) * (2.6666666666666665d0 / sin(x))
    else
        tmp = (x * 0.5d0) / 0.75d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -5e-8) || !(x <= 4e-32)) {
		tmp = Math.pow(Math.sin((x * 0.5)), 2.0) * (2.6666666666666665 / Math.sin(x));
	} else {
		tmp = (x * 0.5) / 0.75;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -5e-8) or not (x <= 4e-32):
		tmp = math.pow(math.sin((x * 0.5)), 2.0) * (2.6666666666666665 / math.sin(x))
	else:
		tmp = (x * 0.5) / 0.75
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5e-8) || !(x <= 4e-32))
		tmp = Float64((sin(Float64(x * 0.5)) ^ 2.0) * Float64(2.6666666666666665 / sin(x)));
	else
		tmp = Float64(Float64(x * 0.5) / 0.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -5e-8) || ~((x <= 4e-32)))
		tmp = (sin((x * 0.5)) ^ 2.0) * (2.6666666666666665 / sin(x));
	else
		tmp = (x * 0.5) / 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5e-8], N[Not[LessEqual[x, 4e-32]], $MachinePrecision]], N[(N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] / 0.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e-8 or 4.00000000000000022e-32 < 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. *-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. associate-/r* 99.1%

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

      8. 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. *-commutative 99.0%

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

      2. associate-*r/ 99.1%

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

      3. associate-*l/ 99.1%

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

   6. Simplified 99.1%

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

   if -4.9999999999999998e-8 < x < 4.00000000000000022e-32

   1. Initial program 50.1%

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

      1. associate-/l* 99.5%

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

      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. associate-/r* 99.5%

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

      8. 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 99.3%

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

      3. un-div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in x around 0 100.0%

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

   7. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.5 \cdot x}}{0.75} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{x \cdot 0.5}}{0.75} \]

   9. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8} \lor \neg \left(x \leq 4 \cdot 10^{-32}\right):\\ \;\;\;\;{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{0.75}\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8} \lor \neg \left(x \leq 4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{2.6666666666666665 \cdot {\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{0.75}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5e-8) (not (<= x 4e-32)))
   (/ (* 2.6666666666666665 (pow (sin (* x 0.5)) 2.0)) (sin x))
   (/ (* x 0.5) 0.75)))

C

double code(double x) {
	double tmp;
	if ((x <= -5e-8) || !(x <= 4e-32)) {
		tmp = (2.6666666666666665 * pow(sin((x * 0.5)), 2.0)) / sin(x);
	} else {
		tmp = (x * 0.5) / 0.75;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5d-8)) .or. (.not. (x <= 4d-32))) then
        tmp = (2.6666666666666665d0 * (sin((x * 0.5d0)) ** 2.0d0)) / sin(x)
    else
        tmp = (x * 0.5d0) / 0.75d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -5e-8) || !(x <= 4e-32)) {
		tmp = (2.6666666666666665 * Math.pow(Math.sin((x * 0.5)), 2.0)) / Math.sin(x);
	} else {
		tmp = (x * 0.5) / 0.75;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -5e-8) or not (x <= 4e-32):
		tmp = (2.6666666666666665 * math.pow(math.sin((x * 0.5)), 2.0)) / math.sin(x)
	else:
		tmp = (x * 0.5) / 0.75
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5e-8) || !(x <= 4e-32))
		tmp = Float64(Float64(2.6666666666666665 * (sin(Float64(x * 0.5)) ^ 2.0)) / sin(x));
	else
		tmp = Float64(Float64(x * 0.5) / 0.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -5e-8) || ~((x <= 4e-32)))
		tmp = (2.6666666666666665 * (sin((x * 0.5)) ^ 2.0)) / sin(x);
	else
		tmp = (x * 0.5) / 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5e-8], N[Not[LessEqual[x, 4e-32]], $MachinePrecision]], N[(N[(2.6666666666666665 * N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] / 0.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e-8 or 4.00000000000000022e-32 < 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. *-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. associate-/r* 99.1%

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

      8. 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/ 99.1%

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

      2. associate-*l* 99.1%

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

      3. pow2 99.1%

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

   5. Applied egg-rr 99.1%

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

   if -4.9999999999999998e-8 < x < 4.00000000000000022e-32

   1. Initial program 50.1%

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

      1. associate-/l* 99.5%

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

      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. associate-/r* 99.5%

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

      8. 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 99.3%

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

      3. un-div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in x around 0 100.0%

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

   7. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.5 \cdot x}}{0.75} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{x \cdot 0.5}}{0.75} \]

   9. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8} \lor \neg \left(x \leq 4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{2.6666666666666665 \cdot {\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{0.75}\\ \end{array} \]

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (sin (* x 0.5)) 2.0)))
   (if (<= x -5e-8)
     (* t_0 (/ 2.6666666666666665 (sin x)))
     (if (<= x 1e-8)
       (/ (* x 0.5) 0.75)
       (* 2.6666666666666665 (/ t_0 (sin x)))))))

C

double code(double x) {
	double t_0 = pow(sin((x * 0.5)), 2.0);
	double tmp;
	if (x <= -5e-8) {
		tmp = t_0 * (2.6666666666666665 / sin(x));
	} else if (x <= 1e-8) {
		tmp = (x * 0.5) / 0.75;
	} else {
		tmp = 2.6666666666666665 * (t_0 / sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x * 0.5d0)) ** 2.0d0
    if (x <= (-5d-8)) then
        tmp = t_0 * (2.6666666666666665d0 / sin(x))
    else if (x <= 1d-8) then
        tmp = (x * 0.5d0) / 0.75d0
    else
        tmp = 2.6666666666666665d0 * (t_0 / sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.sin((x * 0.5)), 2.0);
	double tmp;
	if (x <= -5e-8) {
		tmp = t_0 * (2.6666666666666665 / Math.sin(x));
	} else if (x <= 1e-8) {
		tmp = (x * 0.5) / 0.75;
	} else {
		tmp = 2.6666666666666665 * (t_0 / Math.sin(x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.pow(math.sin((x * 0.5)), 2.0)
	tmp = 0
	if x <= -5e-8:
		tmp = t_0 * (2.6666666666666665 / math.sin(x))
	elif x <= 1e-8:
		tmp = (x * 0.5) / 0.75
	else:
		tmp = 2.6666666666666665 * (t_0 / math.sin(x))
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5)) ^ 2.0
	tmp = 0.0
	if (x <= -5e-8)
		tmp = Float64(t_0 * Float64(2.6666666666666665 / sin(x)));
	elseif (x <= 1e-8)
		tmp = Float64(Float64(x * 0.5) / 0.75);
	else
		tmp = Float64(2.6666666666666665 * Float64(t_0 / sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5)) ^ 2.0;
	tmp = 0.0;
	if (x <= -5e-8)
		tmp = t_0 * (2.6666666666666665 / sin(x));
	elseif (x <= 1e-8)
		tmp = (x * 0.5) / 0.75;
	else
		tmp = 2.6666666666666665 * (t_0 / sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -5e-8], N[(t$95$0 * N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-8], N[(N[(x * 0.5), $MachinePrecision] / 0.75), $MachinePrecision], N[(2.6666666666666665 * N[(t$95$0 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999998e-8

   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* 98.9%

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

      2. *-lft-identity 98.9%

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

      3. metadata-eval 98.9%

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

      4. times-frac 98.9%

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

      5. neg-mul-1 98.9%

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

      6. sin-neg 98.9%

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

      7. associate-/r* 98.9%

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

      8. 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 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-*r/ 99.0%

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

      3. associate-*l/ 98.9%

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

   6. Simplified 98.9%

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

   if -4.9999999999999998e-8 < x < 1e-8

   1. Initial program 51.2%

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

      1. associate-/l* 99.5%

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

      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. associate-/r* 99.5%

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

      8. 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 99.3%

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

      3. un-div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in x around 0 100.0%

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

   7. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.5 \cdot x}}{0.75} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{x \cdot 0.5}}{0.75} \]

   9. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{x \cdot 0.5}}{0.75} \]

   if 1e-8 < 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* 99.2%

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

      2. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. div-inv 99.1%

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

      2. clear-num 99.2%

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

      3. associate-*l* 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-*r/ 99.2%

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

      6. pow2 99.2%

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

   5. Applied egg-rr 99.2%

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

4. Final simplification 99.6%

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

Alternative 6: 99.2% accurate, 1.0× speedup

Math

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

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 74.1%

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

   1. associate-/l* 99.3%

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

   2. associate-*r/ 99.2%

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

   3. metadata-eval 99.2%

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

   4. remove-double-neg 99.2%

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

   5. sin-neg 99.2%

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

   6. distribute-lft-neg-out 99.2%

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

   7. neg-mul-1 99.2%

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

   8. *-commutative 99.2%

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

   9. associate-/l* 99.2%

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

   10. distribute-lft-neg-out 99.2%

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

   11. distribute-rgt-neg-in 99.2%

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

   12. metadata-eval 99.2%

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

   13. associate-/l/ 99.2%

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

   14. neg-mul-1 99.2%

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

   15. sin-neg 99.2%

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

   16. distribute-rgt-neg-in 99.2%

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

   17. metadata-eval 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 7: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. associate-*r/ 99.3%

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

   2. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 8: 99.2% accurate, 1.0× speedup

Math

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

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 74.1%

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

   1. associate-/l* 99.3%

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

   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. associate-/r* 99.3%

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

   8. 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 9: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00013 \lor \neg \left(x \leq 0.00014\right):\\ \;\;\;\;\frac{2.6666666666666665}{\sin x} \cdot \left(0.5 - \frac{\cos x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{0.75}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.00013) (not (<= x 0.00014)))
   (* (/ 2.6666666666666665 (sin x)) (- 0.5 (/ (cos x) 2.0)))
   (/ (* x 0.5) 0.75)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.00013) || !(x <= 0.00014)) {
		tmp = (2.6666666666666665 / sin(x)) * (0.5 - (cos(x) / 2.0));
	} else {
		tmp = (x * 0.5) / 0.75;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.00013d0)) .or. (.not. (x <= 0.00014d0))) then
        tmp = (2.6666666666666665d0 / sin(x)) * (0.5d0 - (cos(x) / 2.0d0))
    else
        tmp = (x * 0.5d0) / 0.75d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.00013) || !(x <= 0.00014)) {
		tmp = (2.6666666666666665 / Math.sin(x)) * (0.5 - (Math.cos(x) / 2.0));
	} else {
		tmp = (x * 0.5) / 0.75;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.00013) or not (x <= 0.00014):
		tmp = (2.6666666666666665 / math.sin(x)) * (0.5 - (math.cos(x) / 2.0))
	else:
		tmp = (x * 0.5) / 0.75
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.00013) || !(x <= 0.00014))
		tmp = Float64(Float64(2.6666666666666665 / sin(x)) * Float64(0.5 - Float64(cos(x) / 2.0)));
	else
		tmp = Float64(Float64(x * 0.5) / 0.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.00013) || ~((x <= 0.00014)))
		tmp = (2.6666666666666665 / sin(x)) * (0.5 - (cos(x) / 2.0));
	else
		tmp = (x * 0.5) / 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.00013], N[Not[LessEqual[x, 0.00014]], $MachinePrecision]], N[(N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] / 0.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.29999999999999989e-4 or 1.3999999999999999e-4 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. frac-2neg 99.1%

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

      2. div-inv 99.1%

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

      3. *-commutative 99.1%

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

      4. distribute-rgt-neg-in 99.1%

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

      5. metadata-eval 99.1%

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

      6. distribute-neg-frac 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in x around inf 99.0%

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

      1. *-commutative 99.0%

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

      2. *-commutative 99.0%

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

      3. unpow2 99.0%

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

      4. associate-/l* 98.9%

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

      5. associate-*l/ 99.1%

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

      6. *-lft-identity 99.1%

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

      7. associate-*r/ 99.1%

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

      8. associate-*l/ 99.1%

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

      9. times-frac 99.0%

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

      10. associate-/l* 99.1%

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

      11. unpow2 99.1%

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

      12. /-rgt-identity 99.1%

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

   8. Simplified 99.1%

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

      1. unpow2 99.1%

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

      2. sin-mult 98.7%

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

   10. Applied egg-rr 98.7%

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

       1. div-sub 98.7%

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

       2. +-inverses 98.7%

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

       3. cos-0 98.7%

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

       4. metadata-eval 98.7%

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

       5. distribute-lft-out 98.7%

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

       6. metadata-eval 98.7%

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

       7. *-rgt-identity 98.7%

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

   12. Simplified 98.7%

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

   if -1.29999999999999989e-4 < x < 1.3999999999999999e-4

   1. Initial program 51.6%

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

      1. associate-/l* 99.5%

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

      2. *-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. associate-/r* 99.5%

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

      8. 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 99.3%

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

      3. un-div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in x around 0 100.0%

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

   7. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.5 \cdot x}}{0.75} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{x \cdot 0.5}}{0.75} \]

   9. Simplified 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00013 \lor \neg \left(x \leq 0.00014\right):\\ \;\;\;\;\frac{2.6666666666666665}{\sin x} \cdot \left(0.5 - \frac{\cos x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{0.75}\\ \end{array} \]

Alternative 10: 54.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. associate-/l* 99.3%

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

   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. associate-/r* 99.3%

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

   8. 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 57.3%

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

5. Final simplification 57.3%

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

Alternative 11: 55.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. associate-/l* 99.3%

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

   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. associate-/r* 99.3%

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

   8. 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 99.3%

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

   3. un-div-inv 99.3%

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

5. Applied egg-rr 99.3%

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

6. Taylor expanded in x around 0 57.5%

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

7. Final simplification 57.5%

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

Alternative 12: 51.0% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 0.5}{0.75} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return (x * 0.5) / 0.75

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. associate-/l* 99.3%

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

   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. associate-/r* 99.3%

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

   8. 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 99.3%

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

   3. un-div-inv 99.3%

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

5. Applied egg-rr 99.3%

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

6. Taylor expanded in x around 0 57.5%

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

7. Taylor expanded in x around 0 54.1%

   \[\leadsto \frac{\color{blue}{0.5 \cdot x}}{0.75} \]
8. Step-by-step derivation

   1. *-commutative 54.1%

      \[\leadsto \frac{\color{blue}{x \cdot 0.5}}{0.75} \]

9. Simplified 54.1%

   \[\leadsto \frac{\color{blue}{x \cdot 0.5}}{0.75} \]

10. Final simplification 54.1%

    \[\leadsto \frac{x \cdot 0.5}{0.75} \]

Alternative 13: 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 74.1%

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

   1. associate-/l* 99.3%

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

   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. associate-/r* 99.3%

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

   8. 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 53.9%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
5. Step-by-step derivation

   1. *-commutative 53.9%

      \[\leadsto \color{blue}{x \cdot 0.6666666666666666} \]

6. Simplified 53.9%

   \[\leadsto \color{blue}{x \cdot 0.6666666666666666} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 28.6%

      \[\leadsto \color{blue}{\sqrt{x \cdot 0.6666666666666666} \cdot \sqrt{x \cdot 0.6666666666666666}} \]

   2. sqrt-unprod 15.0%

      \[\leadsto \color{blue}{\sqrt{\left(x \cdot 0.6666666666666666\right) \cdot \left(x \cdot 0.6666666666666666\right)}} \]

   3. swap-sqr 15.0%

      \[\leadsto \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}} \]

   4. pow2 15.0%

      \[\leadsto \sqrt{\color{blue}{{x}^{2}} \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)} \]

   5. metadata-eval 15.0%

      \[\leadsto \sqrt{{x}^{2} \cdot \color{blue}{0.4444444444444444}} \]

8. Applied egg-rr 15.0%

   \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot 0.4444444444444444}} \]

9. Taylor expanded in x around -inf 3.9%

   \[\leadsto \color{blue}{-0.6666666666666666 \cdot x} \]
10. Step-by-step derivation

    1. *-commutative 3.9%

       \[\leadsto \color{blue}{x \cdot -0.6666666666666666} \]

11. Simplified 3.9%

    \[\leadsto \color{blue}{x \cdot -0.6666666666666666} \]

12. Final simplification 3.9%

    \[\leadsto x \cdot -0.6666666666666666 \]

Alternative 14: 50.7% accurate, 104.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.6666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.6666666666666666))

C

double code(double x) {
	return x * 0.6666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.6666666666666666d0
end function

Java

public static double code(double x) {
	return x * 0.6666666666666666;
}

Python

def code(x):
	return x * 0.6666666666666666

Julia

function code(x)
	return Float64(x * 0.6666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.6666666666666666;
end

Wolfram

code[x_] := N[(x * 0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 74.1%

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

   1. associate-/l* 99.3%

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

   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. associate-/r* 99.3%

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

   8. 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 53.9%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
5. Step-by-step derivation

   1. *-commutative 53.9%

      \[\leadsto \color{blue}{x \cdot 0.6666666666666666} \]

6. Simplified 53.9%

   \[\leadsto \color{blue}{x \cdot 0.6666666666666666} \]

7. Final simplification 53.9%

   \[\leadsto x \cdot 0.6666666666666666 \]

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 2023318 
(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)))