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

Percentage Accurate: 76.1% → 99.8%

Time: 10.0s

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: 76.1% 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

\[\begin{array}{l} \\ \frac{\tan \left(\frac{x}{2}\right)}{0.75} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (tan (/ x 2.0)) 0.75))

C

double code(double x) {
	return tan((x / 2.0)) / 0.75;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x / 2.0d0)) / 0.75d0
end function

Java

public static double code(double x) {
	return Math.tan((x / 2.0)) / 0.75;
}

Python

def code(x):
	return math.tan((x / 2.0)) / 0.75

Julia

function code(x)
	return Float64(tan(Float64(x / 2.0)) / 0.75)
end

MATLAB

function tmp = code(x)
	tmp = tan((x / 2.0)) / 0.75;
end

Wolfram

code[x_] := N[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{8}{3}} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\color{blue}{8}}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}}{\sin \color{blue}{x}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{\frac{8}{3}}{\sin x}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{\frac{8}{3}}{\sin x}\right)}\right)\right) \]

   11. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{\color{blue}{8}}{3}}{\sin x}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{8}{3}\right), \color{blue}{\sin x}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \sin \color{blue}{x}\right)\right)\right) \]

   15. sin-lowering-sin.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{sin.f64}\left(x\right)\right)\right)\right) \]

3. Simplified 99.3%

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

6. Applied egg-rr 99.8%

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

Alternative 2: 99.4% accurate, 3.0× 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 78.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{8}{3}} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\color{blue}{8}}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}}{\sin \color{blue}{x}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{\frac{8}{3}}{\sin x}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{\frac{8}{3}}{\sin x}\right)}\right)\right) \]

   11. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{\color{blue}{8}}{3}}{\sin x}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{8}{3}\right), \color{blue}{\sin x}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \sin \color{blue}{x}\right)\right)\right) \]

   15. sin-lowering-sin.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{sin.f64}\left(x\right)\right)\right)\right) \]

3. Simplified 99.3%

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

6. Applied egg-rr 99.5%

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

Alternative 3: 52.0% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{-0.75 - \left(x \cdot x\right) \cdot -0.0625}}{-2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ x (- -0.75 (* (* x x) -0.0625))) -2.0))

C

double code(double x) {
	return (x / (-0.75 - ((x * x) * -0.0625))) / -2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x / ((-0.75d0) - ((x * x) * (-0.0625d0)))) / (-2.0d0)
end function

Java

public static double code(double x) {
	return (x / (-0.75 - ((x * x) * -0.0625))) / -2.0;
}

Python

def code(x):
	return (x / (-0.75 - ((x * x) * -0.0625))) / -2.0

Julia

function code(x)
	return Float64(Float64(x / Float64(-0.75 - Float64(Float64(x * x) * -0.0625))) / -2.0)
end

MATLAB

function tmp = code(x)
	tmp = (x / (-0.75 - ((x * x) * -0.0625))) / -2.0;
end

Wolfram

code[x_] := N[(N[(x / N[(-0.75 - N[(N[(x * x), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{8}{3}} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\color{blue}{8}}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}}{\sin \color{blue}{x}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{\frac{8}{3}}{\sin x}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{\frac{8}{3}}{\sin x}\right)}\right)\right) \]

   11. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{\color{blue}{8}}{3}}{\sin x}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{8}{3}\right), \color{blue}{\sin x}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \sin \color{blue}{x}\right)\right)\right) \]

   15. sin-lowering-sin.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{sin.f64}\left(x\right)\right)\right)\right) \]

3. Simplified 99.3%

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\frac{1}{\frac{0.375}{\tan \left(\frac{x}{2}\right)}}}{2}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{3}{4} + \frac{-1}{16} \cdot {x}^{2}}{x}\right)}\right), 2\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{3}{4} + \frac{-1}{16} \cdot {x}^{2}\right), x\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \left(\frac{-1}{16} \cdot {x}^{2}\right)\right), x\right)\right), 2\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \left({x}^{2} \cdot \frac{-1}{16}\right)\right), x\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{16}\right)\right), x\right)\right), 2\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{16}\right)\right), x\right)\right), 2\right) \]

   6. *-lowering-*.f64 49.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right), x\right)\right), 2\right) \]

9. Simplified 49.4%

   \[\leadsto \frac{\frac{1}{\color{blue}{\frac{0.75 + \left(x \cdot x\right) \cdot -0.0625}{x}}}}{2} \]
10. Step-by-step derivation

    1. frac-2neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{\frac{\frac{3}{4} + \left(x \cdot x\right) \cdot \frac{-1}{16}}{x}}\right)}{\color{blue}{\mathsf{neg}\left(2\right)}} \]

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

       \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\frac{\frac{3}{4} + \left(x \cdot x\right) \cdot \frac{-1}{16}}{x}}\right)\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right) \]

    3. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{x}{\frac{3}{4} + \left(x \cdot x\right) \cdot \frac{-1}{16}}\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right) \]

    4. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\mathsf{neg}\left(\left(\frac{3}{4} + \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right)}\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\mathsf{neg}\left(\left(\frac{3}{4} + \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right) \]

    6. distribute-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{3}{4}\right)\right) + \left(\mathsf{neg}\left(\left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right)\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right) \]

    7. unsub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{3}{4}\right)\right) - \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right) \]

    8. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{3}{4}\right)\right), \left(\left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right) \]

    9. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(\frac{-3}{4}, \left(\left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(\frac{-3}{4}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{16}\right)\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(\frac{-3}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right) \]

    12. metadata-eval 49.6%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(\frac{-3}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right), -2\right) \]

11. Applied egg-rr 49.6%

    \[\leadsto \color{blue}{\frac{\frac{x}{-0.75 - \left(x \cdot x\right) \cdot -0.0625}}{-2}} \]
12. Add Preprocessing

Alternative 4: 51.7% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{0.5}{0.75 + \left(x \cdot x\right) \cdot -0.0625} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (/ 0.5 (+ 0.75 (* (* x x) -0.0625)))))

C

double code(double x) {
	return x * (0.5 / (0.75 + ((x * x) * -0.0625)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.5d0 / (0.75d0 + ((x * x) * (-0.0625d0))))
end function

Java

public static double code(double x) {
	return x * (0.5 / (0.75 + ((x * x) * -0.0625)));
}

Python

def code(x):
	return x * (0.5 / (0.75 + ((x * x) * -0.0625)))

Julia

function code(x)
	return Float64(x * Float64(0.5 / Float64(0.75 + Float64(Float64(x * x) * -0.0625))))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.5 / (0.75 + ((x * x) * -0.0625)));
end

Wolfram

code[x_] := N[(x * N[(0.5 / N[(0.75 + N[(N[(x * x), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{8}{3}} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\color{blue}{8}}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}}{\sin \color{blue}{x}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{\frac{8}{3}}{\sin x}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{\frac{8}{3}}{\sin x}\right)}\right)\right) \]

   11. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{\color{blue}{8}}{3}}{\sin x}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{8}{3}\right), \color{blue}{\sin x}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \sin \color{blue}{x}\right)\right)\right) \]

   15. sin-lowering-sin.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{sin.f64}\left(x\right)\right)\right)\right) \]

3. Simplified 99.3%

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\frac{1}{\frac{0.375}{\tan \left(\frac{x}{2}\right)}}}{2}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{3}{4} + \frac{-1}{16} \cdot {x}^{2}}{x}\right)}\right), 2\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{3}{4} + \frac{-1}{16} \cdot {x}^{2}\right), x\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \left(\frac{-1}{16} \cdot {x}^{2}\right)\right), x\right)\right), 2\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \left({x}^{2} \cdot \frac{-1}{16}\right)\right), x\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{16}\right)\right), x\right)\right), 2\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{16}\right)\right), x\right)\right), 2\right) \]

   6. *-lowering-*.f64 49.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right), x\right)\right), 2\right) \]

9. Simplified 49.4%

   \[\leadsto \frac{\frac{1}{\color{blue}{\frac{0.75 + \left(x \cdot x\right) \cdot -0.0625}{x}}}}{2} \]
10. Step-by-step derivation

    1. associate-/l/ N/A

       \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{\frac{3}{4} + \left(x \cdot x\right) \cdot \frac{-1}{16}}{x}}} \]

    2. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{3}{4} + \left(x \cdot x\right) \cdot \frac{-1}{16}}{x}}} \]

    3. associate-/r/ N/A

       \[\leadsto \frac{\frac{1}{2}}{\frac{3}{4} + \left(x \cdot x\right) \cdot \frac{-1}{16}} \cdot \color{blue}{x} \]

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

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{\frac{3}{4} + \left(x \cdot x\right) \cdot \frac{-1}{16}}\right), \color{blue}{x}\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), \left(\frac{3}{4} + \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right), x\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{3}{4} + \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right), x\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\frac{3}{4}, \left(\left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right)\right), x\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{16}\right)\right)\right), x\right) \]

    9. *-lowering-*.f64 49.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\frac{3}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right), x\right) \]

11. Applied egg-rr 49.4%

    \[\leadsto \color{blue}{\frac{0.5}{0.75 + \left(x \cdot x\right) \cdot -0.0625} \cdot x} \]

12. Final simplification 49.4%

    \[\leadsto x \cdot \frac{0.5}{0.75 + \left(x \cdot x\right) \cdot -0.0625} \]
13. Add Preprocessing

Alternative 5: 51.4% 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 78.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{8}{3}} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\color{blue}{8}}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}}{\sin \color{blue}{x}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{\frac{8}{3}}{\sin x}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{\frac{8}{3}}{\sin x}\right)}\right)\right) \]

   11. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{\color{blue}{8}}{3}}{\sin x}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{8}{3}\right), \color{blue}{\sin x}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \sin \color{blue}{x}\right)\right)\right) \]

   15. sin-lowering-sin.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{sin.f64}\left(x\right)\right)\right)\right) \]

3. Simplified 99.3%

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

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

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot x\right)}, \frac{3}{4}\right) \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{2}\right), \frac{3}{4}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{2}\right), \frac{3}{4}\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot 1}{2}\right), \frac{3}{4}\right) \]

   4. *-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{2}\right), \frac{3}{4}\right) \]

   5. /-lowering-/.f64 48.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, 2\right), \frac{3}{4}\right) \]

9. Simplified 48.9%

   \[\leadsto \frac{\color{blue}{\frac{x}{2}}}{0.75} \]
10. Step-by-step derivation

    1. associate-/l/ N/A

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

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

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{3}{4} \cdot 2\right)}\right) \]

    3. metadata-eval 48.9%

       \[\leadsto \mathsf{/.f64}\left(x, \frac{3}{2}\right) \]

11. Applied egg-rr 48.9%

    \[\leadsto \color{blue}{\frac{x}{1.5}} \]
12. Add Preprocessing

Alternative 6: 51.2% 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 78.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{8}{3}} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\color{blue}{8}}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}}{\sin \color{blue}{x}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{\frac{8}{3}}{\sin x}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{\frac{8}{3}}{\sin x}\right)}\right)\right) \]

   11. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{\color{blue}{8}}{3}}{\sin x}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{8}{3}\right), \color{blue}{\sin x}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \sin \color{blue}{x}\right)\right)\right) \]

   15. sin-lowering-sin.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{sin.f64}\left(x\right)\right)\right)\right) \]

3. Simplified 99.3%

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

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 48.6%

      \[\leadsto \mathsf{*.f64}\left(\frac{2}{3}, \color{blue}{x}\right) \]

7. Simplified 48.6%

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

8. Final simplification 48.6%

   \[\leadsto x \cdot 0.6666666666666666 \]
9. 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 2024138 
(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)))