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

Percentage Accurate: 77.0% → 99.8%

Time: 11.3s

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: 77.0% 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

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

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 84.3%

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

   1. clear-num N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. sin-mult N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. associate-/l* N/A

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

   9. *-lowering-*.f64 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 egg-rr 59.2%

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

6. Applied egg-rr 99.8%

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

   1. /-lowering-/.f64 N/A

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

   2. tan-lowering-tan.f64 N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. *-lowering-*.f64 99.8

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

8. Applied egg-rr 99.8%

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

Alternative 2: 99.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. clear-num N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. sin-mult N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. associate-/l* N/A

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

   9. *-lowering-*.f64 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 egg-rr 59.2%

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

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. *-lowering-*.f64 N/A

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

   2. hang-p0-tan N/A

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

   3. *-rgt-identity N/A

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

   4. associate-/l* N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. tan-lowering-tan.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 99.5

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 50.0% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 0.2962962962962963}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.0030864197530864196 - 0.037037037037037035, 0.4444444444444444\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* x 0.2962962962962963)
  (fma
   (* x x)
   (- (* (* x x) 0.0030864197530864196) 0.037037037037037035)
   0.4444444444444444)))

C

double code(double x) {
	return (x * 0.2962962962962963) / fma((x * x), (((x * x) * 0.0030864197530864196) - 0.037037037037037035), 0.4444444444444444);
}

Julia

function code(x)
	return Float64(Float64(x * 0.2962962962962963) / fma(Float64(x * x), Float64(Float64(Float64(x * x) * 0.0030864197530864196) - 0.037037037037037035), 0.4444444444444444))
end

Wolfram

code[x_] := N[(N[(x * 0.2962962962962963), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.0030864197530864196), $MachinePrecision] - 0.037037037037037035), $MachinePrecision] + 0.4444444444444444), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.3%

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

   1. clear-num N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. frac-2neg N/A

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

   6. neg-mul-1 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. times-frac N/A

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

   10. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + \frac{1}{18} \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + \frac{1}{18} \cdot {x}^{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{18} \cdot {x}^{2} + \frac{2}{3}\right)} \]

   3. unpow2 N/A

      \[\leadsto x \cdot \left(\frac{1}{18} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{2}{3}\right) \]

   4. associate-*r* N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(\frac{1}{18} \cdot x\right) \cdot x} + \frac{2}{3}\right) \]

   5. *-commutative N/A

      \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(\frac{1}{18} \cdot x\right)} + \frac{2}{3}\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{18} \cdot x, \frac{2}{3}\right)} \]

   7. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{18}}, \frac{2}{3}\right) \]

   8. *-lowering-*.f64 43.3

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.05555555555555555}, 0.6666666666666666\right) \]

7. Simplified 43.3%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot 0.05555555555555555, 0.6666666666666666\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{18}\right) + \frac{2}{3}\right) \cdot x} \]

   2. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{\left(x \cdot \left(x \cdot \frac{1}{18}\right)\right)}^{3} + {\frac{2}{3}}^{3}}{\left(x \cdot \left(x \cdot \frac{1}{18}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{18}\right)\right) + \left(\frac{2}{3} \cdot \frac{2}{3} - \left(x \cdot \left(x \cdot \frac{1}{18}\right)\right) \cdot \frac{2}{3}\right)}} \cdot x \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\left({\left(x \cdot \left(x \cdot \frac{1}{18}\right)\right)}^{3} + {\frac{2}{3}}^{3}\right) \cdot x}{\left(x \cdot \left(x \cdot \frac{1}{18}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{18}\right)\right) + \left(\frac{2}{3} \cdot \frac{2}{3} - \left(x \cdot \left(x \cdot \frac{1}{18}\right)\right) \cdot \frac{2}{3}\right)}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left({\left(x \cdot \left(x \cdot \frac{1}{18}\right)\right)}^{3} + {\frac{2}{3}}^{3}\right) \cdot x}{\left(x \cdot \left(x \cdot \frac{1}{18}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{18}\right)\right) + \left(\frac{2}{3} \cdot \frac{2}{3} - \left(x \cdot \left(x \cdot \frac{1}{18}\right)\right) \cdot \frac{2}{3}\right)}} \]

9. Applied egg-rr 42.5%

   \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot 0.00017146776406035664, 0.2962962962962963\right)}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.0030864197530864196 - 0.037037037037037035, 0.4444444444444444\right)}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{\color{blue}{\frac{8}{27} \cdot x}}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{324} - \frac{1}{27}, \frac{4}{9}\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\color{blue}{x \cdot \frac{8}{27}}}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{324} - \frac{1}{27}, \frac{4}{9}\right)} \]

    2. *-lowering-*.f64 44.0

       \[\leadsto \frac{\color{blue}{x \cdot 0.2962962962962963}}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.0030864197530864196 - 0.037037037037037035, 0.4444444444444444\right)} \]

12. Simplified 44.0%

    \[\leadsto \frac{\color{blue}{x \cdot 0.2962962962962963}}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.0030864197530864196 - 0.037037037037037035, 0.4444444444444444\right)} \]
13. Add Preprocessing

Alternative 4: 49.9% accurate, 20.2× 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 84.3%

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

   1. clear-num N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. sin-mult N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. associate-/l* N/A

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

   9. *-lowering-*.f64 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 egg-rr 59.2%

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

6. Applied egg-rr 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. *-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{1}{2}}}{\frac{3}{4}} \]

   2. *-lowering-*.f64 43.8

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

9. Simplified 43.8%

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

Alternative 5: 49.7% accurate, 57.2× 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 84.3%

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

3. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 43.7

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

5. Simplified 43.7%

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

6. Final simplification 43.7%

   \[\leadsto x \cdot 0.6666666666666666 \]
7. 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 2024199 
(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)))