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

Percentage Accurate: 76.9% → 99.8%

Time: 7.8s

Alternatives: 5

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: 76.9% 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 77.6%

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

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

4. Applied rewrites 49.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. associate-*l/ N/A

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \tan \left(\frac{x}{2}\right)}{0.375}} \]

8. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{-0.75}{\tan \left(x \cdot -0.5\right)}}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 99.8

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

    \[\leadsto \frac{\tan \left(x \cdot -0.5\right)}{-0.75} \]
12. 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 77.6%

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

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

   7. clear-num N/A

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

   8. clear-num N/A

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

   9. sin-mult N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. metadata-eval N/A

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

4. Applied rewrites 49.0%

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

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%

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

Alternative 3: 51.4% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

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

4. Applied rewrites 49.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. associate-*l/ N/A

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \tan \left(\frac{x}{2}\right)}{0.375}} \]

8. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{-0.75}{\tan \left(x \cdot -0.5\right)}}} \]

9. Taylor expanded in x around 0

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

    1. lower-/.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 54.8

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

11. Applied rewrites 54.8%

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

Alternative 4: 51.1% 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 77.6%

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

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

4. Applied rewrites 49.0%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. associate-/r* N/A

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

6. Applied rewrites 49.0%

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

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 54.6

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

9. Applied rewrites 54.6%

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

Alternative 5: 50.8% 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 77.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 54.4

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

5. Applied rewrites 54.4%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
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 2024331 
(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)))