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

Percentage Accurate: 76.4% → 99.8%

Time: 9.9s

Alternatives: 4

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. associate-*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. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r/ 76.6%

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

   2. associate-*r/ 76.5%

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

   3. expm1-log1p-u 60.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2.6666666666666665 \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}{\sin x}\right)\right)} \]

   4. associate-*l/ 60.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{2.6666666666666665}{\sin x} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}\right)\right) \]

   5. expm1-udef 35.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2.6666666666666665}{\sin x} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)\right)} - 1} \]

5. Applied egg-rr 34.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2.6666666666666665 \cdot \frac{0.5 + -0.5 \cdot \cos x}{\sin x}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 34.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2.6666666666666665 \cdot \frac{0.5 + -0.5 \cdot \cos x}{\sin x}\right)\right)} \]

   2. expm1-log1p 50.8%

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

   3. associate-*r/ 50.8%

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

   4. associate-*l/ 50.7%

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

   5. metadata-eval 50.7%

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

   6. metadata-eval 50.7%

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

   7. distribute-lft-neg-in 50.7%

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

   8. distribute-rgt-neg-in 50.7%

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

   9. distribute-lft-in 50.7%

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

   10. sub-neg 50.7%

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

   11. cos-0 50.7%

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

   12. metadata-eval 50.7%

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

   13. associate-/r/ 50.6%

       \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \color{blue}{\frac{1}{\frac{2}{\cos 0 - \cos x}}} \]

   14. associate-/l* 50.7%

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

   15. *-lft-identity 50.7%

       \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \frac{\color{blue}{\cos 0 - \cos x}}{2} \]

   16. times-frac 50.8%

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

   17. *-commutative 50.8%

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

   18. times-frac 50.8%

       \[\leadsto \color{blue}{\frac{2.6666666666666665}{2} \cdot \frac{\cos 0 - \cos x}{\sin x}} \]

   19. metadata-eval 50.8%

       \[\leadsto \color{blue}{1.3333333333333333} \cdot \frac{\cos 0 - \cos x}{\sin x} \]

7. Simplified 99.4%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \tan \left(\frac{x}{2}\right)} \]
8. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right) \cdot 1.3333333333333333} \]

   2. metadata-eval 99.4%

      \[\leadsto \tan \left(\frac{x}{2}\right) \cdot \color{blue}{\frac{1}{0.75}} \]

   3. div-inv 99.8%

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

   4. div-inv 99.8%

      \[\leadsto \frac{\tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}}{0.75} \]

   5. metadata-eval 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 1.3333333333333333 \cdot \tan \left(\frac{x}{2}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 1.3333333333333333 (tan (/ x 2.0))))

C

double code(double x) {
	return 1.3333333333333333 * tan((x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.3333333333333333d0 * tan((x / 2.0d0))
end function

Java

public static double code(double x) {
	return 1.3333333333333333 * Math.tan((x / 2.0));
}

Python

def code(x):
	return 1.3333333333333333 * math.tan((x / 2.0))

Julia

function code(x)
	return Float64(1.3333333333333333 * tan(Float64(x / 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.3333333333333333 * tan((x / 2.0));
end

Wolfram

code[x_] := N[(1.3333333333333333 * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

   1. associate-*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. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r/ 76.6%

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

   2. associate-*r/ 76.5%

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

   3. expm1-log1p-u 60.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2.6666666666666665 \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}{\sin x}\right)\right)} \]

   4. associate-*l/ 60.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{2.6666666666666665}{\sin x} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}\right)\right) \]

   5. expm1-udef 35.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2.6666666666666665}{\sin x} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)\right)} - 1} \]

5. Applied egg-rr 34.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2.6666666666666665 \cdot \frac{0.5 + -0.5 \cdot \cos x}{\sin x}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 34.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2.6666666666666665 \cdot \frac{0.5 + -0.5 \cdot \cos x}{\sin x}\right)\right)} \]

   2. expm1-log1p 50.8%

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

   3. associate-*r/ 50.8%

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

   4. associate-*l/ 50.7%

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

   5. metadata-eval 50.7%

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

   6. metadata-eval 50.7%

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

   7. distribute-lft-neg-in 50.7%

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

   8. distribute-rgt-neg-in 50.7%

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

   9. distribute-lft-in 50.7%

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

   10. sub-neg 50.7%

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

   11. cos-0 50.7%

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

   12. metadata-eval 50.7%

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

   13. associate-/r/ 50.6%

       \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \color{blue}{\frac{1}{\frac{2}{\cos 0 - \cos x}}} \]

   14. associate-/l* 50.7%

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

   15. *-lft-identity 50.7%

       \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \frac{\color{blue}{\cos 0 - \cos x}}{2} \]

   16. times-frac 50.8%

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

   17. *-commutative 50.8%

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

   18. times-frac 50.8%

       \[\leadsto \color{blue}{\frac{2.6666666666666665}{2} \cdot \frac{\cos 0 - \cos x}{\sin x}} \]

   19. metadata-eval 50.8%

       \[\leadsto \color{blue}{1.3333333333333333} \cdot \frac{\cos 0 - \cos x}{\sin x} \]

7. Simplified 99.4%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \tan \left(\frac{x}{2}\right)} \]

8. Final simplification 99.4%

   \[\leadsto 1.3333333333333333 \cdot \tan \left(\frac{x}{2}\right) \]

Alternative 3: 50.6% 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 76.6%

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

   1. associate-*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. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 52.7%

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

5. Final simplification 52.7%

   \[\leadsto x \cdot 0.6666666666666666 \]

Alternative 4: 50.9% accurate, 104.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x 1.5))

C

double code(double x) {
	return x / 1.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / 1.5d0
end function

Java

public static double code(double x) {
	return x / 1.5;
}

Python

def code(x):
	return x / 1.5

Julia

function code(x)
	return Float64(x / 1.5)
end

MATLAB

function tmp = code(x)
	tmp = x / 1.5;
end

Wolfram

code[x_] := N[(x / 1.5), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

   1. associate-*l* 76.5%

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

   2. associate-*l/ 76.5%

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

   3. metadata-eval 76.5%

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

3. Simplified 76.5%

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

4. Taylor expanded in x around 0 30.4%

   \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \color{blue}{\left(0.25 \cdot {x}^{2}\right)} \]
5. Step-by-step derivation

   1. *-commutative 30.4%

      \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \color{blue}{\left({x}^{2} \cdot 0.25\right)} \]

   2. unpow2 30.4%

      \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.25\right) \]

   3. associate-*l* 30.4%

      \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.25\right)\right)} \]

6. Simplified 30.4%

   \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.25\right)\right)} \]

7. Taylor expanded in x around 0 29.8%

   \[\leadsto \color{blue}{\frac{2.6666666666666665}{x}} \cdot \left(x \cdot \left(x \cdot 0.25\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 29.8%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot 0.25\right)\right) \cdot \frac{2.6666666666666665}{x}} \]

   2. clear-num 29.7%

      \[\leadsto \left(x \cdot \left(x \cdot 0.25\right)\right) \cdot \color{blue}{\frac{1}{\frac{x}{2.6666666666666665}}} \]

   3. un-div-inv 29.9%

      \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot 0.25\right)}{\frac{x}{2.6666666666666665}}} \]

   4. div-inv 29.9%

      \[\leadsto \frac{x \cdot \left(x \cdot 0.25\right)}{\color{blue}{x \cdot \frac{1}{2.6666666666666665}}} \]

   5. metadata-eval 29.9%

      \[\leadsto \frac{x \cdot \left(x \cdot 0.25\right)}{x \cdot \color{blue}{0.375}} \]

9. Applied egg-rr 29.9%

   \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot 0.25\right)}{x \cdot 0.375}} \]
10. Step-by-step derivation

    1. associate-/l* 52.9%

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

    2. times-frac 52.9%

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

    3. *-inverses 52.9%

       \[\leadsto \frac{x}{\color{blue}{1} \cdot \frac{0.375}{0.25}} \]

    4. metadata-eval 52.9%

       \[\leadsto \frac{x}{1 \cdot \color{blue}{1.5}} \]

    5. metadata-eval 52.9%

       \[\leadsto \frac{x}{\color{blue}{1.5}} \]

11. Simplified 52.9%

    \[\leadsto \color{blue}{\frac{x}{1.5}} \]

12. Final simplification 52.9%

    \[\leadsto \frac{x}{1.5} \]

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 2023176 
(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)))