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

Percentage Accurate: 76.6% → 99.8%

Time: 8.5s

Alternatives: 5

Speedup: 3.1×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\left(\frac{8}{3} \cdot t\_0\right) \cdot t\_0}{\sin x} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\left(\frac{8}{3} \cdot t\_0\right) \cdot t\_0}{\sin x} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin x}{\frac{8}{3}}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}}} \]

   8. lift-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \frac{1}{2}\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}} \]

   9. lift-sin.f64 N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(x \cdot \frac{1}{2}\right)}}{\frac{\sin x}{\frac{8}{3}}} \]

   10. sin-mult N/A

       \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)}{2}}}{\frac{\sin x}{\frac{8}{3}}} \]

   11. div-inv N/A

       \[\leadsto \frac{\color{blue}{\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}}}{\frac{\sin x}{\frac{8}{3}}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{1}{2}}}{\frac{\sin x}{\frac{8}{3}}} \]

   13. associate-/l* N/A

       \[\leadsto \color{blue}{\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{1}{2}}{\frac{\sin x}{\frac{8}{3}}}} \]

4. Applied rewrites 45.9%

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

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot \frac{\frac{1}{2}}{\frac{3}{8} \cdot \sin x}} \]

   2. lift-/.f64 N/A

      \[\leadsto \left(1 - \cos x\right) \cdot \color{blue}{\frac{\frac{1}{2}}{\frac{3}{8} \cdot \sin x}} \]

   3. clear-num N/A

      \[\leadsto \left(1 - \cos x\right) \cdot \color{blue}{\frac{1}{\frac{\frac{3}{8} \cdot \sin x}{\frac{1}{2}}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{1 - \cos x}{\frac{\frac{3}{8} \cdot \sin x}{\frac{1}{2}}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1 - \cos x}{\frac{\color{blue}{\frac{3}{8} \cdot \sin x}}{\frac{1}{2}}} \]

   6. *-commutative N/A

      \[\leadsto \frac{1 - \cos x}{\frac{\color{blue}{\sin x \cdot \frac{3}{8}}}{\frac{1}{2}}} \]

   7. associate-/l* N/A

      \[\leadsto \frac{1 - \cos x}{\color{blue}{\sin x \cdot \frac{\frac{3}{8}}{\frac{1}{2}}}} \]

   8. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{\sin x}}{\frac{\frac{3}{8}}{\frac{1}{2}}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{\sin x}}{\frac{\frac{3}{8}}{\frac{1}{2}}}} \]

   10. lift--.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{1 - \cos x}}{\sin x}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   11. lift-cos.f64 N/A

       \[\leadsto \frac{\frac{1 - \color{blue}{\cos x}}{\sin x}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{\frac{1 - \cos x}{\color{blue}{\sin x}}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   13. hang-p0-tan N/A

       \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{2}\right)}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   14. lower-tan.f64 N/A

       \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{2}\right)}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   15. lower-/.f64 N/A

       \[\leadsto \frac{\tan \color{blue}{\left(\frac{x}{2}\right)}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   16. metadata-eval 99.8

       \[\leadsto \frac{\tan \left(\frac{x}{2}\right)}{\color{blue}{0.75}} \]

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{0.75}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\tan \color{blue}{\left(\frac{x}{2}\right)}}{\frac{3}{4}} \]

   2. div-inv N/A

      \[\leadsto \frac{\tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}}{\frac{3}{4}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\tan \left(x \cdot \color{blue}{\frac{1}{2}}\right)}{\frac{3}{4}} \]

   4. lift-*.f64 99.8

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \frac{\tan \left(0.5 \cdot x\right)}{0.75} \]
10. Add Preprocessing

Alternative 2: 99.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin x}{\frac{8}{3}}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}}} \]

   8. lift-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \frac{1}{2}\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}} \]

   9. lift-sin.f64 N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(x \cdot \frac{1}{2}\right)}}{\frac{\sin x}{\frac{8}{3}}} \]

   10. sin-mult N/A

       \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)}{2}}}{\frac{\sin x}{\frac{8}{3}}} \]

   11. div-inv N/A

       \[\leadsto \frac{\color{blue}{\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}}}{\frac{\sin x}{\frac{8}{3}}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{1}{2}}}{\frac{\sin x}{\frac{8}{3}}} \]

   13. associate-/l* N/A

       \[\leadsto \color{blue}{\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{1}{2}}{\frac{\sin x}{\frac{8}{3}}}} \]

4. Applied rewrites 45.9%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{4}{3} \cdot \frac{1 - \cos x}{\sin x}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x} \cdot \frac{4}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x} \cdot \frac{4}{3}} \]

   3. hang-p0-tan N/A

      \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right)} \cdot \frac{4}{3} \]

   4. *-rgt-identity N/A

      \[\leadsto \tan \left(\frac{\color{blue}{x \cdot 1}}{2}\right) \cdot \frac{4}{3} \]

   5. associate-/l* N/A

      \[\leadsto \tan \color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \frac{4}{3} \]

   6. metadata-eval N/A

      \[\leadsto \tan \left(x \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{4}{3} \]

   7. *-commutative N/A

      \[\leadsto \tan \color{blue}{\left(\frac{1}{2} \cdot x\right)} \cdot \frac{4}{3} \]

   8. lower-tan.f64 N/A

      \[\leadsto \color{blue}{\tan \left(\frac{1}{2} \cdot x\right)} \cdot \frac{4}{3} \]

   9. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

   \[\leadsto 1.3333333333333333 \cdot \tan \left(0.5 \cdot x\right) \]
9. Add Preprocessing

Alternative 3: 51.0% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(-0.125, x \cdot x, 1.5\right)}{x}} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin x}{\frac{8}{3}}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}}} \]

   8. lift-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \frac{1}{2}\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}} \]

   9. lift-sin.f64 N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(x \cdot \frac{1}{2}\right)}}{\frac{\sin x}{\frac{8}{3}}} \]

   10. sin-mult N/A

       \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)}{2}}}{\frac{\sin x}{\frac{8}{3}}} \]

   11. div-inv N/A

       \[\leadsto \frac{\color{blue}{\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}}}{\frac{\sin x}{\frac{8}{3}}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{1}{2}}}{\frac{\sin x}{\frac{8}{3}}} \]

   13. associate-/l* N/A

       \[\leadsto \color{blue}{\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{1}{2}}{\frac{\sin x}{\frac{8}{3}}}} \]

4. Applied rewrites 45.9%

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

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot \frac{\frac{1}{2}}{\frac{3}{8} \cdot \sin x}} \]

   2. lift-/.f64 N/A

      \[\leadsto \left(1 - \cos x\right) \cdot \color{blue}{\frac{\frac{1}{2}}{\frac{3}{8} \cdot \sin x}} \]

   3. clear-num N/A

      \[\leadsto \left(1 - \cos x\right) \cdot \color{blue}{\frac{1}{\frac{\frac{3}{8} \cdot \sin x}{\frac{1}{2}}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{1 - \cos x}{\frac{\frac{3}{8} \cdot \sin x}{\frac{1}{2}}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1 - \cos x}{\frac{\color{blue}{\frac{3}{8} \cdot \sin x}}{\frac{1}{2}}} \]

   6. *-commutative N/A

      \[\leadsto \frac{1 - \cos x}{\frac{\color{blue}{\sin x \cdot \frac{3}{8}}}{\frac{1}{2}}} \]

   7. associate-/l* N/A

      \[\leadsto \frac{1 - \cos x}{\color{blue}{\sin x \cdot \frac{\frac{3}{8}}{\frac{1}{2}}}} \]

   8. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{\sin x}}{\frac{\frac{3}{8}}{\frac{1}{2}}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{\sin x}}{\frac{\frac{3}{8}}{\frac{1}{2}}}} \]

   10. lift--.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{1 - \cos x}}{\sin x}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   11. lift-cos.f64 N/A

       \[\leadsto \frac{\frac{1 - \color{blue}{\cos x}}{\sin x}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{\frac{1 - \cos x}{\color{blue}{\sin x}}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   13. hang-p0-tan N/A

       \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{2}\right)}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   14. lower-tan.f64 N/A

       \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{2}\right)}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   15. lower-/.f64 N/A

       \[\leadsto \frac{\tan \color{blue}{\left(\frac{x}{2}\right)}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   16. metadata-eval 99.8

       \[\leadsto \frac{\tan \left(\frac{x}{2}\right)}{\color{blue}{0.75}} \]

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{0.75}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{\frac{3}{4}}} \]

   2. lift-tan.f64 N/A

      \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{2}\right)}}{\frac{3}{4}} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\tan \color{blue}{\left(\frac{x}{2}\right)}}{\frac{3}{4}} \]

   4. hang-p0-tan N/A

      \[\leadsto \frac{\color{blue}{\frac{1 - \cos x}{\sin x}}}{\frac{3}{4}} \]

   5. lift-cos.f64 N/A

      \[\leadsto \frac{\frac{1 - \color{blue}{\cos x}}{\sin x}}{\frac{3}{4}} \]

   6. lift--.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{1 - \cos x}}{\sin x}}{\frac{3}{4}} \]

   7. lift-sin.f64 N/A

      \[\leadsto \frac{\frac{1 - \cos x}{\color{blue}{\sin x}}}{\frac{3}{4}} \]

   8. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1 - \cos x}{\sin x}}}{\frac{3}{4}} \]

   9. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{3}{4}}{\frac{1 - \cos x}{\sin x}}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{3}{4}}{\frac{1 - \cos x}{\sin x}}}} \]

   11. frac-2neg N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(\frac{3}{4}\right)}{\mathsf{neg}\left(\frac{1 - \cos x}{\sin x}\right)}}} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(\frac{3}{4}\right)}{\mathsf{neg}\left(\frac{1 - \cos x}{\sin x}\right)}}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\frac{-3}{4}}}{\mathsf{neg}\left(\frac{1 - \cos x}{\sin x}\right)}} \]

   14. lift-/.f64 N/A

       \[\leadsto \frac{1}{\frac{\frac{-3}{4}}{\mathsf{neg}\left(\color{blue}{\frac{1 - \cos x}{\sin x}}\right)}} \]

   15. distribute-neg-frac2 N/A

       \[\leadsto \frac{1}{\frac{\frac{-3}{4}}{\color{blue}{\frac{1 - \cos x}{\mathsf{neg}\left(\sin x\right)}}}} \]

   16. lift--.f64 N/A

       \[\leadsto \frac{1}{\frac{\frac{-3}{4}}{\frac{\color{blue}{1 - \cos x}}{\mathsf{neg}\left(\sin x\right)}}} \]

   17. lift-cos.f64 N/A

       \[\leadsto \frac{1}{\frac{\frac{-3}{4}}{\frac{1 - \color{blue}{\cos x}}{\mathsf{neg}\left(\sin x\right)}}} \]

   18. lift-sin.f64 N/A

       \[\leadsto \frac{1}{\frac{\frac{-3}{4}}{\frac{1 - \cos x}{\mathsf{neg}\left(\color{blue}{\sin x}\right)}}} \]

   19. hang-m0-tan N/A

       \[\leadsto \frac{1}{\frac{\frac{-3}{4}}{\color{blue}{\tan \left(\frac{\mathsf{neg}\left(x\right)}{2}\right)}}} \]

   20. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{\frac{-3}{4}}{\tan \left(\frac{\mathsf{neg}\left(x\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)}} \]

   21. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{\frac{-3}{4}}{\tan \left(\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)}} \]

   22. frac-2neg N/A

       \[\leadsto \frac{1}{\frac{\frac{-3}{4}}{\tan \color{blue}{\left(\frac{x}{\mathsf{neg}\left(2\right)}\right)}}} \]

   23. lower-tan.f64 N/A

       \[\leadsto \frac{1}{\frac{\frac{-3}{4}}{\color{blue}{\tan \left(\frac{x}{\mathsf{neg}\left(2\right)}\right)}}} \]

   24. div-inv N/A

       \[\leadsto \frac{1}{\frac{\frac{-3}{4}}{\tan \color{blue}{\left(x \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)}}} \]

8. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{-0.75}{\tan \left(x \cdot -0.5\right)}}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{\frac{3}{2} + \frac{-1}{8} \cdot {x}^{2}}{x}}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{3}{2} + \frac{-1}{8} \cdot {x}^{2}}{x}}} \]

    2. +-commutative N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\frac{-1}{8} \cdot {x}^{2} + \frac{3}{2}}}{x}} \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, {x}^{2}, \frac{3}{2}\right)}}{x}} \]

    4. unpow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\frac{-1}{8}, \color{blue}{x \cdot x}, \frac{3}{2}\right)}{x}} \]

    5. lower-*.f64 58.3

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-0.125, \color{blue}{x \cdot x}, 1.5\right)}{x}} \]

11. Applied rewrites 58.3%

    \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-0.125, x \cdot x, 1.5\right)}{x}}} \]
12. Add Preprocessing

Alternative 4: 50.6% accurate, 20.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin x}{\frac{8}{3}}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}}} \]

   8. lift-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \frac{1}{2}\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}} \]

   9. lift-sin.f64 N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(x \cdot \frac{1}{2}\right)}}{\frac{\sin x}{\frac{8}{3}}} \]

   10. sin-mult N/A

       \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)}{2}}}{\frac{\sin x}{\frac{8}{3}}} \]

   11. div-inv N/A

       \[\leadsto \frac{\color{blue}{\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}}}{\frac{\sin x}{\frac{8}{3}}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{1}{2}}}{\frac{\sin x}{\frac{8}{3}}} \]

   13. associate-/l* N/A

       \[\leadsto \color{blue}{\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{1}{2}}{\frac{\sin x}{\frac{8}{3}}}} \]

4. Applied rewrites 45.9%

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

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot \frac{\frac{1}{2}}{\frac{3}{8} \cdot \sin x}} \]

   2. lift-/.f64 N/A

      \[\leadsto \left(1 - \cos x\right) \cdot \color{blue}{\frac{\frac{1}{2}}{\frac{3}{8} \cdot \sin x}} \]

   3. clear-num N/A

      \[\leadsto \left(1 - \cos x\right) \cdot \color{blue}{\frac{1}{\frac{\frac{3}{8} \cdot \sin x}{\frac{1}{2}}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{1 - \cos x}{\frac{\frac{3}{8} \cdot \sin x}{\frac{1}{2}}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1 - \cos x}{\frac{\color{blue}{\frac{3}{8} \cdot \sin x}}{\frac{1}{2}}} \]

   6. *-commutative N/A

      \[\leadsto \frac{1 - \cos x}{\frac{\color{blue}{\sin x \cdot \frac{3}{8}}}{\frac{1}{2}}} \]

   7. associate-/l* N/A

      \[\leadsto \frac{1 - \cos x}{\color{blue}{\sin x \cdot \frac{\frac{3}{8}}{\frac{1}{2}}}} \]

   8. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{\sin x}}{\frac{\frac{3}{8}}{\frac{1}{2}}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{\sin x}}{\frac{\frac{3}{8}}{\frac{1}{2}}}} \]

   10. lift--.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{1 - \cos x}}{\sin x}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   11. lift-cos.f64 N/A

       \[\leadsto \frac{\frac{1 - \color{blue}{\cos x}}{\sin x}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{\frac{1 - \cos x}{\color{blue}{\sin x}}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   13. hang-p0-tan N/A

       \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{2}\right)}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   14. lower-tan.f64 N/A

       \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{2}\right)}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   15. lower-/.f64 N/A

       \[\leadsto \frac{\tan \color{blue}{\left(\frac{x}{2}\right)}}{\frac{\frac{3}{8}}{\frac{1}{2}}} \]

   16. metadata-eval 99.8

       \[\leadsto \frac{\tan \left(\frac{x}{2}\right)}{\color{blue}{0.75}} \]

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{0.75}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot x}}{\frac{3}{4}} \]
8. Step-by-step derivation

   1. lower-*.f64 57.8

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

9. Applied rewrites 57.8%

   \[\leadsto \frac{\color{blue}{0.5 \cdot x}}{0.75} \]
10. Add Preprocessing

Alternative 5: 50.3% accurate, 57.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 0.6666666666666666 * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{2}{3} \cdot x} \]
4. Step-by-step derivation

   1. lower-*.f64 57.5

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

5. Applied rewrites 57.5%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
6. Add Preprocessing

Developer Target 1: 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 2024296 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (* 8 (sin (* x 1/2))) 3) (/ (sin x) (sin (* x 1/2)))))

  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))