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

Percentage Accurate: 77.8% → 99.7%

Time: 11.1s

Alternatives: 6

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: 77.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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/ 78.6%

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

   2. associate-*r/ 78.6%

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

   3. expm1-log1p-u 63.4%

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

   4. associate-*l/ 63.4%

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

   5. expm1-udef 41.2%

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

5. Applied egg-rr 40.3%

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

   1. expm1-def 40.3%

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

   2. expm1-log1p 55.4%

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

   3. associate-*r/ 55.5%

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

   4. associate-*l/ 55.5%

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

   5. metadata-eval 55.5%

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

   6. metadata-eval 55.5%

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

   7. distribute-lft-neg-in 55.5%

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

   8. distribute-rgt-neg-in 55.5%

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

   9. distribute-lft-in 55.5%

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

   10. sub-neg 55.5%

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

   11. cos-0 55.5%

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

   12. metadata-eval 55.5%

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

   13. associate-/r/ 55.5%

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

   14. associate-/l* 55.5%

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

   15. *-lft-identity 55.5%

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

   16. times-frac 55.5%

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

   17. *-commutative 55.5%

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

   18. times-frac 55.4%

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

   19. metadata-eval 55.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around inf 99.4%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{\sin \left(0.5 \cdot x\right)}{\cos \left(0.5 \cdot x\right)}} \]
9. Step-by-step derivation

   1. *-commutative 99.4%

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

   2. associate-*r/ 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-/l* 99.6%

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

   5. *-commutative 99.6%

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

10. Simplified 99.6%

    \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{\cos \left(x \cdot 0.5\right)}{1.3333333333333333}}} \]
11. Step-by-step derivation

    1. div-inv 99.6%

       \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\cos \left(x \cdot 0.5\right) \cdot \frac{1}{1.3333333333333333}}} \]

    2. metadata-eval 99.6%

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

12. Applied egg-rr 99.6%

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

13. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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/ 78.6%

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

   2. associate-*r/ 78.6%

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

   3. expm1-log1p-u 63.4%

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

   4. associate-*l/ 63.4%

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

   5. expm1-udef 41.2%

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

5. Applied egg-rr 40.3%

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

   1. expm1-def 40.3%

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

   2. expm1-log1p 55.4%

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

   3. associate-*r/ 55.5%

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

   4. associate-*l/ 55.5%

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

   5. metadata-eval 55.5%

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

   6. metadata-eval 55.5%

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

   7. distribute-lft-neg-in 55.5%

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

   8. distribute-rgt-neg-in 55.5%

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

   9. distribute-lft-in 55.5%

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

   10. sub-neg 55.5%

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

   11. cos-0 55.5%

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

   12. metadata-eval 55.5%

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

   13. associate-/r/ 55.5%

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

   14. associate-/l* 55.5%

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

   15. *-lft-identity 55.5%

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

   16. times-frac 55.5%

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

   17. *-commutative 55.5%

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

   18. times-frac 55.4%

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

   19. metadata-eval 55.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around inf 99.4%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{\sin \left(0.5 \cdot x\right)}{\cos \left(0.5 \cdot x\right)}} \]
9. Step-by-step derivation

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

    \[\leadsto 1.3333333333333333 \cdot \frac{\sin \left(x \cdot 0.5\right)}{\cos \left(x \cdot 0.5\right)} \]

Alternative 3: 99.5% 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 78.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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/ 78.6%

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

   2. associate-*r/ 78.6%

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

   3. expm1-log1p-u 63.4%

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

   4. associate-*l/ 63.4%

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

   5. expm1-udef 41.2%

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

5. Applied egg-rr 40.3%

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

   1. expm1-def 40.3%

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

   2. expm1-log1p 55.4%

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

   3. associate-*r/ 55.5%

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

   4. associate-*l/ 55.5%

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

   5. metadata-eval 55.5%

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

   6. metadata-eval 55.5%

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

   7. distribute-lft-neg-in 55.5%

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

   8. distribute-rgt-neg-in 55.5%

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

   9. distribute-lft-in 55.5%

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

   10. sub-neg 55.5%

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

   11. cos-0 55.5%

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

   12. metadata-eval 55.5%

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

   13. associate-/r/ 55.5%

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

   14. associate-/l* 55.5%

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

   15. *-lft-identity 55.5%

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

   16. times-frac 55.5%

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

   17. *-commutative 55.5%

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

   18. times-frac 55.4%

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

   19. metadata-eval 55.4%

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

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 4: 51.0% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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/ 78.6%

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

   2. associate-*r/ 78.6%

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

   3. clear-num 78.5%

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

   4. associate-/r* 78.5%

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

   5. div-inv 78.6%

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

   6. metadata-eval 78.6%

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

   7. sqr-sin-a 55.5%

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

   8. cancel-sign-sub-inv 55.5%

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

   9. metadata-eval 55.5%

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

   10. *-commutative 55.5%

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

   11. associate-*r* 55.5%

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

   12. metadata-eval 55.5%

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

   13. *-un-lft-identity 55.5%

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

5. Applied egg-rr 55.5%

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

6. Taylor expanded in x around 0 49.7%

   \[\leadsto \frac{1}{\color{blue}{-0.125 \cdot x + 1.5 \cdot \frac{1}{x}}} \]

7. Final simplification 49.7%

   \[\leadsto \frac{1}{x \cdot -0.125 + 1.5 \cdot \frac{1}{x}} \]

Alternative 5: 50.5% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1.5}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ 1.5 x)))

C

double code(double x) {
	return 1.0 / (1.5 / x);
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 / (1.5 / x);
}

Python

def code(x):
	return 1.0 / (1.5 / x)

Julia

function code(x)
	return Float64(1.0 / Float64(1.5 / x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.5 / x);
end

Wolfram

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

Derivation

1. Initial program 78.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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/ 78.6%

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

   2. associate-*r/ 78.6%

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

   3. clear-num 78.5%

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

   4. associate-/r* 78.5%

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

   5. div-inv 78.6%

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

   6. metadata-eval 78.6%

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

   7. sqr-sin-a 55.5%

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

   8. cancel-sign-sub-inv 55.5%

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

   9. metadata-eval 55.5%

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

   10. *-commutative 55.5%

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

   11. associate-*r* 55.5%

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

   12. metadata-eval 55.5%

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

   13. *-un-lft-identity 55.5%

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

5. Applied egg-rr 55.5%

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

6. Taylor expanded in x around 0 48.5%

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

7. Final simplification 48.5%

   \[\leadsto \frac{1}{\frac{1.5}{x}} \]

Alternative 6: 50.5% 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 78.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 48.5%

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

5. Final simplification 48.5%

   \[\leadsto x \cdot 0.6666666666666666 \]

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 2023200 
(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)))