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

Percentage Accurate: 75.9% → 99.4%

Time: 8.3s

Alternatives: 4

Speedup: 2.9×

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: 75.9% 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: 99.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. 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. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 72.6

       \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x} \cdot \frac{8}{3} \]

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 72.6

       \[\leadsto \frac{{\sin \color{blue}{\left(0.5 \cdot x\right)}}^{2}}{\sin x} \cdot \frac{8}{3} \]

   15. lift-/.f64 N/A

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

   16. metadata-eval 72.6

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

4. Applied rewrites 72.6%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*l/ N/A

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

   5. lift-/.f64 N/A

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

   6. lower-*.f64 99.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.2

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.2

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

6. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. sin-mult N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

8. Applied rewrites 43.5%

   \[\leadsto \color{blue}{\frac{1 - \cos \left(1 \cdot x\right)}{2 \cdot \sin x}} \cdot 2.6666666666666665 \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-lft-identity N/A

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

   12. lift-sin.f64 N/A

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

   13. hang-p0-tan N/A

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

   14. lower-tan.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. metadata-eval 99.4

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

10. Applied rewrites 99.4%

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

Alternative 2: 55.4% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 0.5 (* (sin (* 0.5 x)) 2.6666666666666665)))

C

double code(double x) {
	return 0.5 * (sin((0.5 * x)) * 2.6666666666666665);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (sin((0.5d0 * x)) * 2.6666666666666665d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.5 * N[(N[Sin[N[(0.5 * x), $MachinePrecision]], $MachinePrecision] * 2.6666666666666665), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. 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. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.2

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.2

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.2

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

   16. lift-/.f64 N/A

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

   17. metadata-eval 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}\right) \]
6. Step-by-step derivation

7. Applied rewrites 64.1%

   \[\leadsto \color{blue}{0.5} \cdot \left(\sin \left(0.5 \cdot x\right) \cdot 2.6666666666666665\right) \]
8. Add Preprocessing

Alternative 3: 52.1% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ -0.4444444444444444 \cdot \frac{x}{0.05555555555555555 \cdot \left(x \cdot x\right) - 0.6666666666666666} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  -0.4444444444444444
  (/ x (- (* 0.05555555555555555 (* x x)) 0.6666666666666666))))

C

double code(double x) {
	return -0.4444444444444444 * (x / ((0.05555555555555555 * (x * x)) - 0.6666666666666666));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-0.4444444444444444d0) * (x / ((0.05555555555555555d0 * (x * x)) - 0.6666666666666666d0))
end function

Java

public static double code(double x) {
	return -0.4444444444444444 * (x / ((0.05555555555555555 * (x * x)) - 0.6666666666666666));
}

Python

def code(x):
	return -0.4444444444444444 * (x / ((0.05555555555555555 * (x * x)) - 0.6666666666666666))

Julia

function code(x)
	return Float64(-0.4444444444444444 * Float64(x / Float64(Float64(0.05555555555555555 * Float64(x * x)) - 0.6666666666666666)))
end

MATLAB

function tmp = code(x)
	tmp = -0.4444444444444444 * (x / ((0.05555555555555555 * (x * x)) - 0.6666666666666666));
end

Wolfram

code[x_] := N[(-0.4444444444444444 * N[(x / N[(N[(0.05555555555555555 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 60.4

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

5. Applied rewrites 60.4%

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

7. Applied rewrites 60.2%

   \[\leadsto \left(0.0030864197530864196 \cdot {x}^{4} - 0.4444444444444444\right) \cdot \color{blue}{\frac{x}{0.05555555555555555 \cdot \left(x \cdot x\right) - 0.6666666666666666}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{-4}{9} \cdot \frac{\color{blue}{x}}{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}} \]
9. Step-by-step derivation

10. Applied rewrites 61.0%

    \[\leadsto -0.4444444444444444 \cdot \frac{\color{blue}{x}}{0.05555555555555555 \cdot \left(x \cdot x\right) - 0.6666666666666666} \]
11. Add Preprocessing

Alternative 4: 51.5% 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 72.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. 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 60.6

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

5. Applied rewrites 60.6%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
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 2024329 
(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)))