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

Percentage Accurate: 77.3% → 99.4%

Time: 6.1s

Alternatives: 4

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: 99.4% 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 77.1%

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

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.3

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.3

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.3

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.3

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

   16. lift-/.f64 N/A

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

   17. metadata-eval 99.3

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

4. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \left(\sin \left(0.5 \cdot x\right) \cdot 2.6666666666666665\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. sin-mult N/A

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

   18. lift-/.f64 N/A

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

6. Applied rewrites 53.8%

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

7. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{4}{3} \cdot \frac{1 - \cos x}{\sin x}} \]
8. 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 99.4

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

9. Applied rewrites 99.4%

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

Alternative 2: 50.7% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.4444444444444444 \cdot x}{\mathsf{fma}\left(-0.05555555555555555, x \cdot x, 0.6666666666666666\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* 0.4444444444444444 x)
  (fma -0.05555555555555555 (* x x) 0.6666666666666666)))

C

double code(double x) {
	return (0.4444444444444444 * x) / fma(-0.05555555555555555, (x * x), 0.6666666666666666);
}

Julia

function code(x)
	return Float64(Float64(0.4444444444444444 * x) / fma(-0.05555555555555555, Float64(x * x), 0.6666666666666666))
end

Wolfram

code[x_] := N[(N[(0.4444444444444444 * x), $MachinePrecision] / N[(-0.05555555555555555 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 49.4

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

5. Applied rewrites 49.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.005555555555555556, x \cdot x, 0.05555555555555555\right), x \cdot x, 0.6666666666666666\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 48.7%

   \[\leadsto \frac{\left(0.4444444444444444 - {\left(\mathsf{fma}\left(x \cdot x, 0.005555555555555556, 0.05555555555555555\right)\right)}^{2} \cdot {x}^{4}\right) \cdot x}{\color{blue}{0.6666666666666666 - \left(\mathsf{fma}\left(x \cdot x, 0.005555555555555556, 0.05555555555555555\right) \cdot x\right) \cdot x}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{4}{9} \cdot x}{\color{blue}{\frac{2}{3}} - \left(\mathsf{fma}\left(x \cdot x, \frac{1}{180}, \frac{1}{18}\right) \cdot x\right) \cdot x} \]
9. Step-by-step derivation

10. Applied rewrites 49.9%

    \[\leadsto \frac{0.4444444444444444 \cdot x}{\color{blue}{0.6666666666666666} - \left(\mathsf{fma}\left(x \cdot x, 0.005555555555555556, 0.05555555555555555\right) \cdot x\right) \cdot x} \]

11. Taylor expanded in x around 0

    \[\leadsto \frac{\frac{4}{9} \cdot x}{\frac{2}{3} + \color{blue}{\frac{-1}{18} \cdot {x}^{2}}} \]
12. Step-by-step derivation

13. Applied rewrites 50.1%

    \[\leadsto \frac{0.4444444444444444 \cdot x}{\mathsf{fma}\left(-0.05555555555555555, \color{blue}{x \cdot x}, 0.6666666666666666\right)} \]
14. Add Preprocessing

Alternative 3: 50.2% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.4444444444444444 \cdot x}{0.6666666666666666} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 0.4444444444444444 x) 0.6666666666666666))

C

double code(double x) {
	return (0.4444444444444444 * x) / 0.6666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.4444444444444444d0 * x) / 0.6666666666666666d0
end function

Java

public static double code(double x) {
	return (0.4444444444444444 * x) / 0.6666666666666666;
}

Python

def code(x):
	return (0.4444444444444444 * x) / 0.6666666666666666

Julia

function code(x)
	return Float64(Float64(0.4444444444444444 * x) / 0.6666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = (0.4444444444444444 * x) / 0.6666666666666666;
end

Wolfram

code[x_] := N[(N[(0.4444444444444444 * x), $MachinePrecision] / 0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 77.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 49.4

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

5. Applied rewrites 49.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.005555555555555556, x \cdot x, 0.05555555555555555\right), x \cdot x, 0.6666666666666666\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 48.7%

   \[\leadsto \frac{\left(0.4444444444444444 - {\left(\mathsf{fma}\left(x \cdot x, 0.005555555555555556, 0.05555555555555555\right)\right)}^{2} \cdot {x}^{4}\right) \cdot x}{\color{blue}{0.6666666666666666 - \left(\mathsf{fma}\left(x \cdot x, 0.005555555555555556, 0.05555555555555555\right) \cdot x\right) \cdot x}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{4}{9} \cdot x}{\color{blue}{\frac{2}{3}} - \left(\mathsf{fma}\left(x \cdot x, \frac{1}{180}, \frac{1}{18}\right) \cdot x\right) \cdot x} \]
9. Step-by-step derivation

10. Applied rewrites 49.9%

    \[\leadsto \frac{0.4444444444444444 \cdot x}{\color{blue}{0.6666666666666666} - \left(\mathsf{fma}\left(x \cdot x, 0.005555555555555556, 0.05555555555555555\right) \cdot x\right) \cdot x} \]

11. Taylor expanded in x around 0

    \[\leadsto \frac{\frac{4}{9} \cdot x}{\frac{2}{3}} \]
12. Step-by-step derivation

13. Applied rewrites 49.8%

    \[\leadsto \frac{0.4444444444444444 \cdot x}{0.6666666666666666} \]
14. Add Preprocessing

Alternative 4: 50.2% 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 77.1%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 49.8

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

5. Applied rewrites 49.8%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
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 2024320 
(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)))