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

Percentage Accurate: 76.5% → 99.5%

Time: 11.3s

Alternatives: 9

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: 76.5% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot x\right)\\ \frac{t\_0}{\sin x} \cdot \frac{t\_0}{0.375} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 x)))) (* (/ t_0 (sin x)) (/ t_0 0.375))))

C

double code(double x) {
	double t_0 = sin((0.5 * x));
	return (t_0 / sin(x)) * (t_0 / 0.375);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((0.5d0 * x))
    code = (t_0 / sin(x)) * (t_0 / 0.375d0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((0.5 * x));
	return (t_0 / Math.sin(x)) * (t_0 / 0.375);
}

Python

def code(x):
	t_0 = math.sin((0.5 * x))
	return (t_0 / math.sin(x)) * (t_0 / 0.375)

Julia

function code(x)
	t_0 = sin(Float64(0.5 * x))
	return Float64(Float64(t_0 / sin(x)) * Float64(t_0 / 0.375))
end

MATLAB

function tmp = code(x)
	t_0 = sin((0.5 * x));
	tmp = (t_0 / sin(x)) * (t_0 / 0.375);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 / N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / 0.375), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.9%

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

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

   13. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \frac{\sin \left(0.5 \cdot x\right)}{0.375}} \]
5. 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 75.9%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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. div-inv N/A

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

   8. associate-/r* N/A

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

   9. clear-num N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-sin.f64 N/A

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

   15. lift-sin.f64 N/A

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

4. Applied rewrites 49.1%

   \[\leadsto \color{blue}{\frac{\frac{2.6666666666666665}{\sin x}}{\frac{2}{1 - \cos x}}} \]

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. hang-p0-tan N/A

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

   3. *-rgt-identity N/A

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

   4. associate-/l* N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-tan.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 51.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{2.6666666666666665}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00013227513227513228, -0.005555555555555556\right), -0.3333333333333333\right), 4\right)}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.6666666666666665
  (/
   (fma
    (* x x)
    (fma
     (* x x)
     (fma (* x x) -0.00013227513227513228 -0.005555555555555556)
     -0.3333333333333333)
    4.0)
   x)))

C

double code(double x) {
	return 2.6666666666666665 / (fma((x * x), fma((x * x), fma((x * x), -0.00013227513227513228, -0.005555555555555556), -0.3333333333333333), 4.0) / x);
}

Julia

function code(x)
	return Float64(2.6666666666666665 / Float64(fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.00013227513227513228, -0.005555555555555556), -0.3333333333333333), 4.0) / x))
end

Wolfram

code[x_] := N[(2.6666666666666665 / N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.00013227513227513228 + -0.005555555555555556), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + 4.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.9%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 49.1%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{8}{3}}{\color{blue}{\frac{4 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{7560} \cdot {x}^{2} - \frac{1}{180}\right) - \frac{1}{3}\right)}{x}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{8}{3}}{\color{blue}{\frac{4 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{7560} \cdot {x}^{2} - \frac{1}{180}\right) - \frac{1}{3}\right)}{x}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{7560} \cdot {x}^{2} - \frac{1}{180}\right) - \frac{1}{3}\right) + 4}}{x}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-1}{7560} \cdot {x}^{2} - \frac{1}{180}\right) - \frac{1}{3}, 4\right)}}{x}} \]

   4. unpow2 N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{7560} \cdot {x}^{2} - \frac{1}{180}\right) - \frac{1}{3}, 4\right)}{x}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{7560} \cdot {x}^{2} - \frac{1}{180}\right) - \frac{1}{3}, 4\right)}{x}} \]

   6. sub-neg N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{7560} \cdot {x}^{2} - \frac{1}{180}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 4\right)}{x}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{-1}{7560} \cdot {x}^{2} - \frac{1}{180}\right) + \color{blue}{\frac{-1}{3}}, 4\right)}{x}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{7560} \cdot {x}^{2} - \frac{1}{180}, \frac{-1}{3}\right)}, 4\right)}{x}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\frac{8}{3}}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{7560} \cdot {x}^{2} - \frac{1}{180}, \frac{-1}{3}\right), 4\right)}{x}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\frac{8}{3}}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{7560} \cdot {x}^{2} - \frac{1}{180}, \frac{-1}{3}\right), 4\right)}{x}} \]

   11. sub-neg N/A

       \[\leadsto \frac{\frac{8}{3}}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{7560} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right)}, \frac{-1}{3}\right), 4\right)}{x}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\frac{8}{3}}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{7560}} + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right), \frac{-1}{3}\right), 4\right)}{x}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\frac{8}{3}}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{-1}{7560} + \color{blue}{\frac{-1}{180}}, \frac{-1}{3}\right), 4\right)}{x}} \]

   14. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{8}{3}}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{7560}, \frac{-1}{180}\right)}, \frac{-1}{3}\right), 4\right)}{x}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\frac{8}{3}}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{7560}, \frac{-1}{180}\right), \frac{-1}{3}\right), 4\right)}{x}} \]

   16. lower-*.f64 56.2

       \[\leadsto \frac{2.6666666666666665}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.00013227513227513228, -0.005555555555555556\right), -0.3333333333333333\right), 4\right)}{x}} \]

7. Applied rewrites 56.2%

   \[\leadsto \frac{2.6666666666666665}{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00013227513227513228, -0.005555555555555556\right), -0.3333333333333333\right), 4\right)}{x}}} \]
8. Add Preprocessing

Alternative 4: 51.3% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.9%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 49.1%

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

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. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 56.1

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

7. Applied rewrites 56.1%

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

Alternative 5: 50.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ 1.3333333333333333 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  1.3333333333333333
  (*
   x
   (fma (* x x) (fma x (* x 0.004166666666666667) 0.041666666666666664) 0.5))))

C

double code(double x) {
	return 1.3333333333333333 * (x * fma((x * x), fma(x, (x * 0.004166666666666667), 0.041666666666666664), 0.5));
}

Julia

function code(x)
	return Float64(1.3333333333333333 * Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.004166666666666667), 0.041666666666666664), 0.5)))
end

Wolfram

code[x_] := N[(1.3333333333333333 * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.004166666666666667), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.9%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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. div-inv N/A

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

   8. associate-/r* N/A

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

   9. clear-num N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-sin.f64 N/A

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

   15. lift-sin.f64 N/A

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

4. Applied rewrites 49.1%

   \[\leadsto \color{blue}{\frac{\frac{2.6666666666666665}{\sin x}}{\frac{2}{1 - \cos x}}} \]

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. hang-p0-tan N/A

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

   3. *-rgt-identity N/A

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

   4. associate-/l* N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-tan.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{4}{3} \cdot \left(x \cdot \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)}\right) \]
9. Step-by-step derivation

10. Applied rewrites 55.3%

    \[\leadsto 1.3333333333333333 \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), 0.5\right)}\right) \]
11. Add Preprocessing

Alternative 6: 50.6% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 55.3

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

5. Applied rewrites 55.3%

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

Alternative 7: 50.6% accurate, 15.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. sqr-sin-a N/A

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

   7. sub-neg N/A

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

   8. distribute-rgt-in N/A

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

   9. lift-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lift-/.f64 N/A

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

   17. lower-+.f64 N/A

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

4. Applied rewrites 49.0%

   \[\leadsto \frac{\color{blue}{1.3333333333333333 + \left(\cos x \cdot -0.5\right) \cdot 2.6666666666666665}}{\sin x} \]

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 55.1

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

7. Applied rewrites 55.1%

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

9. Applied rewrites 55.1%

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

Alternative 8: 50.6% accurate, 20.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 55.1

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

5. Applied rewrites 55.1%

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

Alternative 9: 50.8% accurate, 57.2× 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 75.9%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 55.0

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

5. Applied rewrites 55.0%

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

6. Final simplification 55.0%

   \[\leadsto x \cdot 0.6666666666666666 \]
7. 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 2024222 
(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)))