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

Percentage Accurate: 76.4% → 99.8%

Time: 7.9s

Alternatives: 6

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: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

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

   6. associate-/l* N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. sin-mult N/A

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

   10. clear-num N/A

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

   11. frac-times N/A

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

   12. lift-/.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lift-/.f64 N/A

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

4. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{2}{1 - \cos x} \cdot \sin x}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. clear-num N/A

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

   12. lower-/.f64 N/A

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\frac{\tan \left(\frac{x}{2}\right)}{2}}{0.375}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval 99.8

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

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

   6. associate-/l* N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. sin-mult N/A

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

   10. clear-num N/A

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

   11. frac-times N/A

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

   12. lift-/.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lift-/.f64 N/A

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

4. Applied rewrites 52.4%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. hang-p0-tan N/A

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

   4. *-rgt-identity N/A

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

   5. associate-/l* N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. lower-tan.f64 N/A

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

   9. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 52.0% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

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

   6. associate-/l* N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. sin-mult N/A

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

   10. clear-num N/A

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

   11. frac-times N/A

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

   12. lift-/.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lift-/.f64 N/A

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

4. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{2}{1 - \cos x} \cdot \sin x}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. clear-num N/A

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

   8. lift-/.f64 N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/r* N/A

       \[\leadsto \frac{-1}{\frac{-1}{\color{blue}{\frac{\frac{\frac{8}{3}}{\frac{2}{1 - \cos x}}}{\sin x}}}} \]

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{-1}{\frac{-1}{1.3333333333333333 \cdot \tan \left(\frac{x}{2}\right)}}} \]

7. Taylor expanded in x around 0

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r/ N/A

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

   7. *-inverses N/A

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

   8. *-rgt-identity N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

9. Applied rewrites 52.3%

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

Alternative 4: 52.1% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

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

   6. associate-/l* N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. sin-mult N/A

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

   10. clear-num N/A

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

   11. frac-times N/A

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

   12. lift-/.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lift-/.f64 N/A

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

4. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{2}{1 - \cos x} \cdot \sin x}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. clear-num N/A

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

   8. lift-/.f64 N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/r* N/A

       \[\leadsto \frac{-1}{\frac{-1}{\color{blue}{\frac{\frac{\frac{8}{3}}{\frac{2}{1 - \cos x}}}{\sin x}}}} \]

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{-1}{\frac{-1}{1.3333333333333333 \cdot \tan \left(\frac{x}{2}\right)}}} \]

7. Taylor expanded in x around 0

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. *-inverses N/A

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

   8. *-rgt-identity N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 52.1

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

9. Applied rewrites 52.1%

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

Alternative 5: 51.7% accurate, 20.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

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

   6. associate-/l* N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. sin-mult N/A

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

   10. clear-num N/A

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

   11. frac-times N/A

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

   12. lift-/.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lift-/.f64 N/A

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

4. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{2}{1 - \cos x} \cdot \sin x}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. clear-num N/A

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

   12. lower-/.f64 N/A

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\frac{\tan \left(\frac{x}{2}\right)}{2}}{0.375}} \]

7. Taylor expanded in x around 0

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

   1. lower-*.f64 51.7

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

9. Applied rewrites 51.7%

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

Alternative 6: 51.4% accurate, 57.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 0.6666666666666666 * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 51.5

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

5. Applied rewrites 51.5%

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

Developer Target 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024295 
(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)))