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

Percentage Accurate: 77.3% → 99.5%

Time: 11.2s

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: 77.3% 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 76.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 99.5%

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

Alternative 2: 99.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 50.4%

   \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x} \cdot 1.3333333333333333} \]
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. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-tan.f64 99.4

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

6. Applied rewrites 99.4%

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

Alternative 3: 50.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -4.96031746031746 \cdot 10^{-5}, -0.0020833333333333333\right), -0.125\right), 1.5\right)}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (/
   (fma
    (* x x)
    (fma
     x
     (* x (fma x (* x -4.96031746031746e-5) -0.0020833333333333333))
     -0.125)
    1.5)
   x)))

C

double code(double x) {
	return 1.0 / (fma((x * x), fma(x, (x * fma(x, (x * -4.96031746031746e-5), -0.0020833333333333333)), -0.125), 1.5) / x);
}

Julia

function code(x)
	return Float64(1.0 / Float64(fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -4.96031746031746e-5), -0.0020833333333333333)), -0.125), 1.5) / x))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -4.96031746031746e-5), $MachinePrecision] + -0.0020833333333333333), $MachinePrecision]), $MachinePrecision] + -0.125), $MachinePrecision] + 1.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 99.5%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \frac{\sin \left(0.5 \cdot x\right)}{0.375}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. clear-num N/A

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

   6. *-commutative N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. frac-times N/A

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

   10. *-commutative N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. sqr-sin-a N/A

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

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

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

   17. lower-+.f64 N/A

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

6. Applied rewrites 50.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sin x \cdot 0.375}{0.5 + -0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot x\right)\right)}}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{\frac{3}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{20160} \cdot {x}^{2} - \frac{1}{480}\right) - \frac{1}{8}\right)}{x}}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{3}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{20160} \cdot {x}^{2} - \frac{1}{480}\right) - \frac{1}{8}\right)}{x}}} \]

9. Applied rewrites 54.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -4.96031746031746 \cdot 10^{-5}, -0.0020833333333333333\right), -0.125\right), 1.5\right)}{x}}} \]
10. Add Preprocessing

Alternative 4: 50.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0020833333333333333, -0.125\right), 1.5\right)}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (/ (fma (* x x) (fma x (* x -0.0020833333333333333) -0.125) 1.5) x)))

C

double code(double x) {
	return 1.0 / (fma((x * x), fma(x, (x * -0.0020833333333333333), -0.125), 1.5) / x);
}

Julia

function code(x)
	return Float64(1.0 / Float64(fma(Float64(x * x), fma(x, Float64(x * -0.0020833333333333333), -0.125), 1.5) / x))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.0020833333333333333), $MachinePrecision] + -0.125), $MachinePrecision] + 1.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 99.5%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \frac{\sin \left(0.5 \cdot x\right)}{0.375}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. clear-num N/A

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

   6. *-commutative N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. frac-times N/A

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

   10. *-commutative N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. sqr-sin-a N/A

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

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

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

   17. lower-+.f64 N/A

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

6. Applied rewrites 50.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sin x \cdot 0.375}{0.5 + -0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot x\right)\right)}}} \]

7. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 54.1

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot -0.0020833333333333333}, -0.125\right), 1.5\right)}{x}} \]

9. Applied rewrites 54.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0020833333333333333, -0.125\right), 1.5\right)}{x}}} \]
10. Add Preprocessing

Alternative 5: 50.2% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{0.5 \cdot x}{0.375} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (/ (* 0.5 x) 0.375)))

C

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

Fortran

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

Java

public static double code(double x) {
	return 0.5 * ((0.5 * x) / 0.375);
}

Python

def code(x):
	return 0.5 * ((0.5 * x) / 0.375)

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.5 * N[(N[(0.5 * x), $MachinePrecision] / 0.375), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 99.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 57.4%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{x \cdot \frac{1}{2}}}{\frac{3}{8}} \]

   2. lower-*.f64 54.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot 0.5}}{0.375} \]

10. Applied rewrites 54.0%

    \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot 0.5}}{0.375} \]

11. Final simplification 54.0%

    \[\leadsto 0.5 \cdot \frac{0.5 \cdot x}{0.375} \]
12. Add Preprocessing

Alternative 6: 49.8% 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 76.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 50.2%

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

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 53.8

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

7. Applied rewrites 53.8%

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

9. Applied rewrites 53.8%

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

Alternative 7: 49.8% 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 76.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 53.8

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

5. Applied rewrites 53.8%

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

Alternative 8: 49.9% 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 76.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 53.7

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

5. Applied rewrites 53.7%

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

6. Final simplification 53.7%

   \[\leadsto x \cdot 0.6666666666666666 \]
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 2024220 
(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)))