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

Percentage Accurate: 77.0% → 99.8%

Time: 9.5s

Alternatives: 6

Speedup: 3.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\left(\frac{8}{3} \cdot t\_0\right) \cdot t\_0}{\sin x} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\left(\frac{8}{3} \cdot t\_0\right) \cdot t\_0}{\sin x} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 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 82.0%

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

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

   5. metadata-eval N/A

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

   6. clear-num N/A

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

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

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

   8. metadata-eval N/A

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

   9. associate-*l* N/A

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

   10. associate-/r* N/A

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

   11. sin-mult N/A

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

   12. associate-/r/ N/A

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

   13. associate-*l/ N/A

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

6. Applied egg-rr 60.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sin x \cdot 0.75}{1 - \cos x}}} \]
7. Step-by-step derivation

   1. clear-num N/A

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

   2. associate-/r* N/A

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

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

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

   4. hang-p0-tan N/A

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

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

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

   6. /-lowering-/.f64 99.7%

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

8. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{0.75}} \]
9. 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 82.0%

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

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

   5. metadata-eval N/A

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

   6. clear-num N/A

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

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

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

   8. metadata-eval N/A

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

   9. associate-*l* N/A

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

   10. associate-/r* N/A

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

   11. sin-mult N/A

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

   12. associate-/r/ N/A

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

   13. associate-*l/ N/A

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

6. Applied egg-rr 60.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sin x \cdot 0.75}{1 - \cos x}}} \]
7. Step-by-step derivation

   1. associate-/r/ N/A

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

   2. associate-*l/ N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. metadata-eval N/A

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

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

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

   7. hang-p0-tan N/A

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

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

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

   9. /-lowering-/.f64 99.4%

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

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

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

Alternative 3: 52.0% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0020833333333333333\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ x (+ 1.5 (* (* x x) (+ -0.125 (* (* x x) -0.0020833333333333333))))))

C

double code(double x) {
	return x / (1.5 + ((x * x) * (-0.125 + ((x * x) * -0.0020833333333333333))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (1.5d0 + ((x * x) * ((-0.125d0) + ((x * x) * (-0.0020833333333333333d0)))))
end function

Java

public static double code(double x) {
	return x / (1.5 + ((x * x) * (-0.125 + ((x * x) * -0.0020833333333333333))));
}

Python

def code(x):
	return x / (1.5 + ((x * x) * (-0.125 + ((x * x) * -0.0020833333333333333))))

Julia

function code(x)
	return Float64(x / Float64(1.5 + Float64(Float64(x * x) * Float64(-0.125 + Float64(Float64(x * x) * -0.0020833333333333333)))))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.5 + ((x * x) * (-0.125 + ((x * x) * -0.0020833333333333333))));
end

Wolfram

code[x_] := N[(x / N[(1.5 + N[(N[(x * x), $MachinePrecision] * N[(-0.125 + N[(N[(x * x), $MachinePrecision] * -0.0020833333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.0%

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

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

   5. metadata-eval N/A

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

   6. clear-num N/A

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

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

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

   8. metadata-eval N/A

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

   9. associate-*l* N/A

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

   10. associate-/r* N/A

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

   11. sin-mult N/A

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

   12. associate-/r/ N/A

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

   13. associate-*l/ N/A

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

6. Applied egg-rr 60.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sin x \cdot 0.75}{1 - \cos x}}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{3}{2} + {x}^{2} \cdot \left(\frac{-1}{480} \cdot {x}^{2} - \frac{1}{8}\right)}{x}\right)}\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{3}{2} + {x}^{2} \cdot \left(\frac{-1}{480} \cdot {x}^{2} - \frac{1}{8}\right)\right), \color{blue}{x}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \left({x}^{2} \cdot \left(\frac{-1}{480} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right), x\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{480} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right), x\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \left(x \cdot \left(x \cdot \left(\frac{-1}{480} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{-1}{480} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{480} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right), x\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{480} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)\right)\right)\right), x\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{480} \cdot {x}^{2} + \frac{-1}{8}\right)\right)\right)\right), x\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{8} + \frac{-1}{480} \cdot {x}^{2}\right)\right)\right)\right), x\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(\frac{-1}{480} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left({x}^{2} \cdot \frac{-1}{480}\right)\right)\right)\right)\right), x\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{480}\right)\right)\right)\right)\right), x\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{480}\right)\right)\right)\right)\right), x\right)\right) \]

   14. *-lowering-*.f64 43.6%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{480}\right)\right)\right)\right)\right), x\right)\right) \]

9. Simplified 43.6%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0020833333333333333\right)\right)}{x}}} \]
10. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \frac{x}{\color{blue}{\frac{3}{2} + x \cdot \left(x \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{-1}{480}\right)\right)}} \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{3}{2} + x \cdot \left(x \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{-1}{480}\right)\right)\right)}\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{2}, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{-1}{480}\right)\right)\right)}\right)\right) \]

    4. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{2}, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{-1}{480}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{-1}{480}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{-1}{8}} + \left(x \cdot x\right) \cdot \frac{-1}{480}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{-1}{480}\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(x \cdot x\right), \color{blue}{\frac{-1}{480}}\right)\right)\right)\right)\right) \]

    9. *-lowering-*.f64 43.8%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{480}\right)\right)\right)\right)\right) \]

11. Applied egg-rr 43.8%

    \[\leadsto \color{blue}{\frac{x}{1.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0020833333333333333\right)}} \]
12. Add Preprocessing

Alternative 4: 51.9% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

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

   5. metadata-eval N/A

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

   6. clear-num N/A

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

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

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

   8. metadata-eval N/A

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

   9. associate-*l* N/A

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

   10. associate-/r* N/A

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

   11. sin-mult N/A

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

   12. associate-/r/ N/A

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

   13. associate-*l/ N/A

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

6. Applied egg-rr 60.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sin x \cdot 0.75}{1 - \cos x}}} \]

7. Taylor expanded in x around 0

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

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

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

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

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

   3. *-commutative N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 43.7%

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

9. Simplified 43.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.5 + \left(x \cdot x\right) \cdot -0.125}{x}}} \]
10. Add Preprocessing

Alternative 5: 51.5% 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 82.0%

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

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

   5. metadata-eval N/A

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

   6. clear-num N/A

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

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

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

   8. metadata-eval N/A

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

   9. associate-*l* N/A

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

   10. associate-/r* N/A

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

   11. sin-mult N/A

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

   12. associate-/r/ N/A

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

   13. associate-*l/ N/A

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

6. Applied egg-rr 60.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sin x \cdot 0.75}{1 - \cos x}}} \]

7. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 42.9%

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

9. Simplified 42.9%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
10. Step-by-step derivation

    1. clear-num N/A

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

    2. /-lowering-/.f64 43.0%

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

11. Applied egg-rr 43.0%

    \[\leadsto \color{blue}{\frac{x}{1.5}} \]
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 82.0%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 42.8%

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

7. Simplified 42.8%

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

8. Final simplification 42.8%

   \[\leadsto x \cdot 0.6666666666666666 \]
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 2024161 
(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)))