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

Percentage Accurate: 76.2% → 99.8%

Time: 12.8s

Alternatives: 7

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: 76.2% 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 82.6%

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

   1. associate-/l* 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 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. associate-*r/ N/A

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

   6. sin-mult N/A

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

   7. associate-*l/ N/A

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

   8. associate-/l/ N/A

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

   9. /-lowering-/.f64 N/A

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

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 82.6%

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

   1. associate-/l* 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 99.2%

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

3. Simplified 99.2%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. associate-*r/ N/A

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

   4. sin-mult N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. times-frac N/A

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

   9. un-div-inv N/A

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

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

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

6. Applied egg-rr 99.4%

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

Alternative 3: 52.4% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{4 + \left(x \cdot x\right) \cdot -0.3333333333333333}}{0.375} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ x (+ 4.0 (* (* x x) -0.3333333333333333))) 0.375))

C

double code(double x) {
	return (x / (4.0 + ((x * x) * -0.3333333333333333))) / 0.375;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x / (4.0d0 + ((x * x) * (-0.3333333333333333d0)))) / 0.375d0
end function

Java

public static double code(double x) {
	return (x / (4.0 + ((x * x) * -0.3333333333333333))) / 0.375;
}

Python

def code(x):
	return (x / (4.0 + ((x * x) * -0.3333333333333333))) / 0.375

Julia

function code(x)
	return Float64(Float64(x / Float64(4.0 + Float64(Float64(x * x) * -0.3333333333333333))) / 0.375)
end

MATLAB

function tmp = code(x)
	tmp = (x / (4.0 + ((x * x) * -0.3333333333333333))) / 0.375;
end

Wolfram

code[x_] := N[(N[(x / N[(4.0 + N[(N[(x * x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]

Derivation

1. Initial program 82.6%

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

   1. associate-/l* 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 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

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

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

   8. clear-num N/A

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

   9. /-lowering-/.f64 N/A

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

   10. div-inv N/A

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

   11. sin-mult N/A

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

   12. associate-*l/ N/A

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

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

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

6. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{1}{\frac{\tan \left(\frac{x}{2}\right)}{2}}}} \]

7. Taylor expanded in x around 0

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

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

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

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

      \[\leadsto \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \left(\frac{-1}{3} \cdot {x}^{2}\right)\right), x\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \left({x}^{2} \cdot \frac{-1}{3}\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{3}\right)\right), x\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{3}\right)\right), x\right)\right) \]

   6. *-lowering-*.f64 49.1%

      \[\leadsto \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{3}\right)\right), x\right)\right) \]

9. Simplified 49.1%

   \[\leadsto \frac{2.6666666666666665}{\color{blue}{\frac{4 + \left(x \cdot x\right) \cdot -0.3333333333333333}{x}}} \]
10. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{4 + \left(x \cdot x\right) \cdot \frac{-1}{3}}{x}}{\frac{8}{3}}}} \]

    2. div-inv N/A

       \[\leadsto \frac{1}{\frac{4 + \left(x \cdot x\right) \cdot \frac{-1}{3}}{x} \cdot \color{blue}{\frac{1}{\frac{8}{3}}}} \]

    3. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{\frac{4 + \left(x \cdot x\right) \cdot \frac{-1}{3}}{x}}}{\color{blue}{\frac{1}{\frac{8}{3}}}} \]

    4. clear-num N/A

       \[\leadsto \frac{\frac{x}{4 + \left(x \cdot x\right) \cdot \frac{-1}{3}}}{\frac{\color{blue}{1}}{\frac{8}{3}}} \]

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

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

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

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

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(4, \left(\left(x \cdot x\right) \cdot \frac{-1}{3}\right)\right)\right), \left(\frac{1}{\frac{8}{3}}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{3}\right)\right)\right), \left(\frac{1}{\frac{8}{3}}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{3}\right)\right)\right), \left(\frac{1}{\frac{8}{3}}\right)\right) \]

    10. metadata-eval 49.4%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{3}\right)\right)\right), \frac{3}{8}\right) \]

11. Applied egg-rr 49.4%

    \[\leadsto \color{blue}{\frac{\frac{x}{4 + \left(x \cdot x\right) \cdot -0.3333333333333333}}{0.375}} \]
12. Add Preprocessing

Alternative 4: 52.1% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{2.6666666666666665}{4 + \left(x \cdot x\right) \cdot -0.3333333333333333} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (/ 2.6666666666666665 (+ 4.0 (* (* x x) -0.3333333333333333)))))

C

double code(double x) {
	return x * (2.6666666666666665 / (4.0 + ((x * x) * -0.3333333333333333)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (2.6666666666666665d0 / (4.0d0 + ((x * x) * (-0.3333333333333333d0))))
end function

Java

public static double code(double x) {
	return x * (2.6666666666666665 / (4.0 + ((x * x) * -0.3333333333333333)));
}

Python

def code(x):
	return x * (2.6666666666666665 / (4.0 + ((x * x) * -0.3333333333333333)))

Julia

function code(x)
	return Float64(x * Float64(2.6666666666666665 / Float64(4.0 + Float64(Float64(x * x) * -0.3333333333333333))))
end

MATLAB

function tmp = code(x)
	tmp = x * (2.6666666666666665 / (4.0 + ((x * x) * -0.3333333333333333)));
end

Wolfram

code[x_] := N[(x * N[(2.6666666666666665 / N[(4.0 + N[(N[(x * x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

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

   1. associate-/l* 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 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

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

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

   8. clear-num N/A

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

   9. /-lowering-/.f64 N/A

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

   10. div-inv N/A

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

   11. sin-mult N/A

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

   12. associate-*l/ N/A

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

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

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

6. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{1}{\frac{\tan \left(\frac{x}{2}\right)}{2}}}} \]

7. Taylor expanded in x around 0

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

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

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

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

      \[\leadsto \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \left(\frac{-1}{3} \cdot {x}^{2}\right)\right), x\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \left({x}^{2} \cdot \frac{-1}{3}\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{3}\right)\right), x\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{3}\right)\right), x\right)\right) \]

   6. *-lowering-*.f64 49.1%

      \[\leadsto \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{3}\right)\right), x\right)\right) \]

9. Simplified 49.1%

   \[\leadsto \frac{2.6666666666666665}{\color{blue}{\frac{4 + \left(x \cdot x\right) \cdot -0.3333333333333333}{x}}} \]
10. Step-by-step derivation

    1. associate-/r/ N/A

       \[\leadsto \frac{\frac{8}{3}}{4 + \left(x \cdot x\right) \cdot \frac{-1}{3}} \cdot \color{blue}{x} \]

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

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{8}{3}}{4 + \left(x \cdot x\right) \cdot \frac{-1}{3}}\right), \color{blue}{x}\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{8}{3}, \left(4 + \left(x \cdot x\right) \cdot \frac{-1}{3}\right)\right), x\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{8}{3}, \mathsf{+.f64}\left(4, \left(\left(x \cdot x\right) \cdot \frac{-1}{3}\right)\right)\right), x\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{8}{3}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{3}\right)\right)\right), x\right) \]

    6. *-lowering-*.f64 49.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{8}{3}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{3}\right)\right)\right), x\right) \]

11. Applied egg-rr 49.2%

    \[\leadsto \color{blue}{\frac{2.6666666666666665}{4 + \left(x \cdot x\right) \cdot -0.3333333333333333} \cdot x} \]

12. Final simplification 49.2%

    \[\leadsto x \cdot \frac{2.6666666666666665}{4 + \left(x \cdot x\right) \cdot -0.3333333333333333} \]
13. Add Preprocessing

Alternative 5: 51.8% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{2}}{0.75} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return (x / 2.0) / 0.75

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

   1. associate-/l* 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 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. associate-*r/ N/A

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

   6. sin-mult N/A

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

   7. associate-*l/ N/A

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

   8. associate-/l/ N/A

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

   9. /-lowering-/.f64 N/A

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

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 48.7%

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

9. Simplified 48.7%

   \[\leadsto \frac{\color{blue}{\frac{x}{2}}}{0.75} \]
10. Add Preprocessing

Alternative 6: 51.5% accurate, 62.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

   1. associate-/l* 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 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

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

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

   8. clear-num N/A

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

   9. /-lowering-/.f64 N/A

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

   10. div-inv N/A

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

   11. sin-mult N/A

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

   12. associate-*l/ N/A

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

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

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

6. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{1}{\frac{\tan \left(\frac{x}{2}\right)}{2}}}} \]

7. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 48.4%

      \[\leadsto \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{/.f64}\left(4, \color{blue}{x}\right)\right) \]

9. Simplified 48.4%

   \[\leadsto \frac{2.6666666666666665}{\color{blue}{\frac{4}{x}}} \]
10. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{4}{x}}{\frac{8}{3}}}} \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{4}{x}}{\frac{8}{3}}\right)}\right) \]

    3. associate-/l/ N/A

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

    4. associate-/r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{4}{\frac{8}{3}}}{\color{blue}{x}}\right)\right) \]

    5. metadata-eval N/A

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

    6. metadata-eval N/A

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

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

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

    8. metadata-eval 48.5%

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

11. Applied egg-rr 48.5%

    \[\leadsto \color{blue}{\frac{1}{\frac{1.5}{x}}} \]
12. Add Preprocessing

Alternative 7: 51.5% accurate, 104.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.6666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.6666666666666666))

C

double code(double x) {
	return x * 0.6666666666666666;
}

Fortran

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

Java

public static double code(double x) {
	return x * 0.6666666666666666;
}

Python

def code(x):
	return x * 0.6666666666666666

Julia

function code(x)
	return Float64(x * 0.6666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.6666666666666666;
end

Wolfram

code[x_] := N[(x * 0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 82.6%

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

   1. associate-/l* 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 99.2%

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

3. Simplified 99.2%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
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 48.5%

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

7. Simplified 48.5%

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

8. Final simplification 48.5%

   \[\leadsto x \cdot 0.6666666666666666 \]
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 2024191 
(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)))