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

Percentage Accurate: 77.0% → 99.8%

Time: 8.7s

Alternatives: 8

Speedup: 3.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return Math.tan((0.5 * x)) / -(-0.75);
}

Python

def code(x):
	return math.tan((0.5 * x)) / -(-0.75)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. clear-num N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. sin-mult N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. neg-mul-1 N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 55.0%

   \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x} \cdot 1.3333333333333333} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. frac-2neg N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval N/A

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

   9. clear-num N/A

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

   10. lift-/.f64 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 55.0

       \[\leadsto \frac{-1.3333333333333333}{\color{blue}{\frac{-1}{\frac{1 - \cos x}{\sin x}}}} \]

   14. lift-/.f64 N/A

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

   15. lift--.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. lift-sin.f64 N/A

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

   18. hang-p0-tan N/A

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

   19. div-inv N/A

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

   20. metadata-eval N/A

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

   21. lift-*.f64 N/A

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

   22. lower-tan.f64 99.4

       \[\leadsto \frac{-1.3333333333333333}{\frac{-1}{\color{blue}{\tan \left(x \cdot 0.5\right)}}} \]

   23. lift-*.f64 N/A

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

   24. *-commutative N/A

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

   25. lift-*.f64 99.4

       \[\leadsto \frac{-1.3333333333333333}{\frac{-1}{\tan \color{blue}{\left(0.5 \cdot x\right)}}} \]

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{-1.3333333333333333}{\frac{-1}{\tan \left(0.5 \cdot x\right)}}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. frac-2neg N/A

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

   10. metadata-eval N/A

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

   11. remove-double-div N/A

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

   12. lower-neg.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval 99.8

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \frac{\tan \left(0.5 \cdot x\right)}{--0.75} \]
10. Add Preprocessing

Alternative 2: 99.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. clear-num N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. sin-mult N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. neg-mul-1 N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 55.0%

   \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x} \cdot 1.3333333333333333} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. hang-p0-tan N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. lift-*.f64 N/A

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

   9. lower-tan.f64 99.4

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 99.4

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

6. Applied rewrites 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 50.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \frac{-1.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(6.613756613756614 \cdot 10^{-5}, x \cdot x, 0.002777777777777778\right), x \cdot x, 0.16666666666666666\right), x, \frac{-2}{x}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.3333333333333333
  (fma
   (fma
    (fma 6.613756613756614e-5 (* x x) 0.002777777777777778)
    (* x x)
    0.16666666666666666)
   x
   (/ -2.0 x))))

C

double code(double x) {
	return -1.3333333333333333 / fma(fma(fma(6.613756613756614e-5, (x * x), 0.002777777777777778), (x * x), 0.16666666666666666), x, (-2.0 / x));
}

Julia

function code(x)
	return Float64(-1.3333333333333333 / fma(fma(fma(6.613756613756614e-5, Float64(x * x), 0.002777777777777778), Float64(x * x), 0.16666666666666666), x, Float64(-2.0 / x)))
end

Wolfram

code[x_] := N[(-1.3333333333333333 / N[(N[(N[(6.613756613756614e-5 * N[(x * x), $MachinePrecision] + 0.002777777777777778), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * x + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. clear-num N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. sin-mult N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. neg-mul-1 N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 55.0%

   \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x} \cdot 1.3333333333333333} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. frac-2neg N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval N/A

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

   9. clear-num N/A

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

   10. lift-/.f64 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 55.0

       \[\leadsto \frac{-1.3333333333333333}{\color{blue}{\frac{-1}{\frac{1 - \cos x}{\sin x}}}} \]

   14. lift-/.f64 N/A

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

   15. lift--.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. lift-sin.f64 N/A

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

   18. hang-p0-tan N/A

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

   19. div-inv N/A

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

   20. metadata-eval N/A

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

   21. lift-*.f64 N/A

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

   22. lower-tan.f64 99.4

       \[\leadsto \frac{-1.3333333333333333}{\frac{-1}{\color{blue}{\tan \left(x \cdot 0.5\right)}}} \]

   23. lift-*.f64 N/A

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

   24. *-commutative N/A

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

   25. lift-*.f64 99.4

       \[\leadsto \frac{-1.3333333333333333}{\frac{-1}{\tan \color{blue}{\left(0.5 \cdot x\right)}}} \]

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{-1.3333333333333333}{\frac{-1}{\tan \left(0.5 \cdot x\right)}}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{-4}{3}}{\color{blue}{\frac{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{15120} \cdot {x}^{2}\right)\right) - 2}{x}}} \]
8. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \frac{\frac{-4}{3}}{\color{blue}{\frac{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{15120} \cdot {x}^{2}\right)\right)}{x} - \frac{2}{x}}} \]

   2. sub-neg N/A

      \[\leadsto \frac{\frac{-4}{3}}{\color{blue}{\frac{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{15120} \cdot {x}^{2}\right)\right)}{x} + \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{-4}{3}}{\frac{\color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{15120} \cdot {x}^{2}\right)\right) \cdot {x}^{2}}}{x} + \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)} \]

   4. unpow2 N/A

      \[\leadsto \frac{\frac{-4}{3}}{\frac{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{15120} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}}{x} + \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\frac{-4}{3}}{\frac{\color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{15120} \cdot {x}^{2}\right)\right) \cdot x\right) \cdot x}}{x} + \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)} \]

   6. associate-/l* N/A

      \[\leadsto \frac{\frac{-4}{3}}{\color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{15120} \cdot {x}^{2}\right)\right) \cdot x\right) \cdot \frac{x}{x}} + \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)} \]

   7. *-inverses N/A

      \[\leadsto \frac{\frac{-4}{3}}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{15120} \cdot {x}^{2}\right)\right) \cdot x\right) \cdot \color{blue}{1} + \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)} \]

   8. *-rgt-identity N/A

      \[\leadsto \frac{\frac{-4}{3}}{\color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{15120} \cdot {x}^{2}\right)\right) \cdot x} + \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{-4}{3}}{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{15120} \cdot {x}^{2}\right), x, \mathsf{neg}\left(\frac{2}{x}\right)\right)}} \]

9. Applied rewrites 49.9%

   \[\leadsto \frac{-1.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(6.613756613756614 \cdot 10^{-5}, x \cdot x, 0.002777777777777778\right), x \cdot x, 0.16666666666666666\right), x, \frac{-2}{x}\right)}} \]
10. Add Preprocessing

Alternative 4: 50.9% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \frac{2.6666666666666665}{\mathsf{fma}\left(-0.3333333333333333, x, \frac{4}{x}\right)} \end{array} \]

FPCore

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

C

double code(double x) {
	return 2.6666666666666665 / fma(-0.3333333333333333, x, (4.0 / x));
}

Julia

function code(x)
	return Float64(2.6666666666666665 / fma(-0.3333333333333333, x, Float64(4.0 / x)))
end

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/l* N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. sqr-sin-a N/A

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

   15. sub-neg N/A

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

   16. +-commutative N/A

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

4. Applied rewrites 55.0%

   \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{\sin x}{\mathsf{fma}\left(-0.5, \cos x, 0.5\right)}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{8}{3}}{\color{blue}{\frac{4 + \frac{-1}{3} \cdot {x}^{2}}{x}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{8}{3}}{\color{blue}{\frac{4 + \frac{-1}{3} \cdot {x}^{2}}{x}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\color{blue}{\frac{-1}{3} \cdot {x}^{2} + 4}}{x}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {x}^{2}, 4\right)}}{x}} \]

   4. unpow2 N/A

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

   5. lower-*.f64 49.8

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

7. Applied rewrites 49.8%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{8}{3}}{\color{blue}{\frac{4 + \frac{-1}{3} \cdot {x}^{2}}{x}}} \]
9. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\color{blue}{4 \cdot 1} + \frac{-1}{3} \cdot {x}^{2}}{x}} \]

   2. lft-mult-inverse N/A

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

   3. associate-*l* N/A

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

   4. distribute-rgt-in N/A

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

   5. unpow2 N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\left(x \cdot x\right) \cdot \left(4 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)}{x}} \]

   7. sub-neg N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\left(x \cdot x\right) \cdot \color{blue}{\left(4 \cdot \frac{1}{{x}^{2}} - \frac{1}{3}\right)}}{x}} \]

   8. associate-*l* N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\color{blue}{x \cdot \left(x \cdot \left(4 \cdot \frac{1}{{x}^{2}} - \frac{1}{3}\right)\right)}}{x}} \]

   9. *-commutative N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\color{blue}{\left(x \cdot \left(4 \cdot \frac{1}{{x}^{2}} - \frac{1}{3}\right)\right) \cdot x}}{x}} \]

   10. associate-/l* N/A

       \[\leadsto \frac{\frac{8}{3}}{\color{blue}{\left(x \cdot \left(4 \cdot \frac{1}{{x}^{2}} - \frac{1}{3}\right)\right) \cdot \frac{x}{x}}} \]

   11. *-inverses N/A

       \[\leadsto \frac{\frac{8}{3}}{\left(x \cdot \left(4 \cdot \frac{1}{{x}^{2}} - \frac{1}{3}\right)\right) \cdot \color{blue}{1}} \]

   12. *-rgt-identity N/A

       \[\leadsto \frac{\frac{8}{3}}{\color{blue}{x \cdot \left(4 \cdot \frac{1}{{x}^{2}} - \frac{1}{3}\right)}} \]

   13. sub-neg N/A

       \[\leadsto \frac{\frac{8}{3}}{x \cdot \color{blue}{\left(4 \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)}} \]

   14. metadata-eval N/A

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

   15. +-commutative N/A

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

   16. distribute-rgt-in N/A

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

   17. cancel-sign-sub N/A

       \[\leadsto \frac{\frac{8}{3}}{\color{blue}{\frac{-1}{3} \cdot x - \left(\mathsf{neg}\left(4 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x}} \]

   18. *-commutative N/A

       \[\leadsto \frac{\frac{8}{3}}{\frac{-1}{3} \cdot x - \color{blue}{x \cdot \left(\mathsf{neg}\left(4 \cdot \frac{1}{{x}^{2}}\right)\right)}} \]

   19. cancel-sign-sub-inv N/A

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

10. Applied rewrites 49.8%

    \[\leadsto \frac{2.6666666666666665}{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, x, \frac{4}{x}\right)}} \]
11. Add Preprocessing

Alternative 5: 50.2% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.005555555555555556, x \cdot x, 0.05555555555555555\right), x \cdot x, 0.6666666666666666\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (fma 0.005555555555555556 (* x x) 0.05555555555555555)
   (* x x)
   0.6666666666666666)
  x))

C

double code(double x) {
	return fma(fma(0.005555555555555556, (x * x), 0.05555555555555555), (x * x), 0.6666666666666666) * x;
}

Julia

function code(x)
	return Float64(fma(fma(0.005555555555555556, Float64(x * x), 0.05555555555555555), Float64(x * x), 0.6666666666666666) * x)
end

Wolfram

code[x_] := N[(N[(N[(0.005555555555555556 * N[(x * x), $MachinePrecision] + 0.05555555555555555), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + \frac{1}{180} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + \frac{1}{180} \cdot {x}^{2}\right)\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + \frac{1}{180} \cdot {x}^{2}\right)\right) \cdot x} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{18} + \frac{1}{180} \cdot {x}^{2}\right) + \frac{2}{3}\right)} \cdot x \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{18} + \frac{1}{180} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{2}{3}\right) \cdot x \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{18} + \frac{1}{180} \cdot {x}^{2}, {x}^{2}, \frac{2}{3}\right)} \cdot x \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{180} \cdot {x}^{2} + \frac{1}{18}}, {x}^{2}, \frac{2}{3}\right) \cdot x \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{180}, {x}^{2}, \frac{1}{18}\right)}, {x}^{2}, \frac{2}{3}\right) \cdot x \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{180}, \color{blue}{x \cdot x}, \frac{1}{18}\right), {x}^{2}, \frac{2}{3}\right) \cdot x \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{180}, \color{blue}{x \cdot x}, \frac{1}{18}\right), {x}^{2}, \frac{2}{3}\right) \cdot x \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{180}, x \cdot x, \frac{1}{18}\right), \color{blue}{x \cdot x}, \frac{2}{3}\right) \cdot x \]

   11. lower-*.f64 49.1

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.005555555555555556, x \cdot x, 0.05555555555555555\right), \color{blue}{x \cdot x}, 0.6666666666666666\right) \cdot x \]

5. Applied rewrites 49.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.005555555555555556, x \cdot x, 0.05555555555555555\right), x \cdot x, 0.6666666666666666\right) \cdot x} \]
6. Add Preprocessing

Alternative 6: 50.6% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot x}{0.375} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* 0.25 x) 0.375))

C

double code(double x) {
	return (0.25 * x) / 0.375;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.25d0 * x) / 0.375d0
end function

Java

public static double code(double x) {
	return (0.25 * x) / 0.375;
}

Python

def code(x):
	return (0.25 * x) / 0.375

Julia

function code(x)
	return Float64(Float64(0.25 * x) / 0.375)
end

MATLAB

function tmp = code(x)
	tmp = (0.25 * x) / 0.375;
end

Wolfram

code[x_] := N[(N[(0.25 * x), $MachinePrecision] / 0.375), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. clear-num N/A

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

   8. frac-2neg N/A

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

   9. neg-mul-1 N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 55.0%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.5, \cos x, 0.5\right)}{\sin x}}{0.375}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot x}}{\frac{3}{8}} \]
6. Step-by-step derivation

   1. lower-*.f64 49.0

      \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]

7. Applied rewrites 49.0%

   \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]
8. Add Preprocessing

Alternative 7: 50.2% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, 0.05555555555555555, 0.6666666666666666\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma (* x x) 0.05555555555555555 0.6666666666666666) x))

C

double code(double x) {
	return fma((x * x), 0.05555555555555555, 0.6666666666666666) * x;
}

Julia

function code(x)
	return Float64(fma(Float64(x * x), 0.05555555555555555, 0.6666666666666666) * x)
end

Wolfram

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

Derivation

1. Initial program 75.8%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 48.9

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

5. Applied rewrites 48.9%

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

Alternative 8: 50.4% accurate, 57.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 0.6666666666666666 * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 48.7

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

5. Applied rewrites 48.7%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
6. Add Preprocessing

Developer Target 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024294 
(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)))