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

Percentage Accurate: 76.8% → 99.8%

Time: 11.0s

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: 76.8% 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 83.0%

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

4. Applied rewrites 56.1%

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{0.75}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 99.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.0%

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

4. Applied rewrites 56.1%

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

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. lower-*.f64 N/A

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

   2. hang-p0-tan N/A

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

   3. *-rgt-identity N/A

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

   4. associate-/l* N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-tan.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 51.3% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.0%

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

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

   4. lower-/.f64 83.0

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}}} \]

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. sin-mult N/A

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

   11. associate-*r/ N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

4. Applied rewrites 56.1%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{\frac{3}{2} + \frac{-1}{8} \cdot {x}^{2}}{x}}} \]
6. 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. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 47.7

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

7. Applied rewrites 47.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, -0.125, 1.5\right)}{x}}} \]
8. Step-by-step derivation

9. Applied rewrites 47.8%

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

Alternative 4: 51.2% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.0%

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

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

   4. lower-/.f64 83.0

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}}} \]

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. sin-mult N/A

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

   11. associate-*r/ N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

4. Applied rewrites 56.1%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{\frac{3}{2} + \frac{-1}{8} \cdot {x}^{2}}{x}}} \]
6. 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. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 47.7

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

7. Applied rewrites 47.7%

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

Alternative 5: 50.8% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 0.5}{0.75} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return (x * 0.5) / 0.75

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.0%

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

4. Applied rewrites 56.1%

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{0.75}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot x}}{\frac{3}{4}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{1}{2}}}{\frac{3}{4}} \]

   2. lower-*.f64 47.5

      \[\leadsto \frac{\color{blue}{x \cdot 0.5}}{0.75} \]

9. Applied rewrites 47.5%

   \[\leadsto \frac{\color{blue}{x \cdot 0.5}}{0.75} \]
10. Add Preprocessing

Alternative 6: 50.5% accurate, 57.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * 0.6666666666666666

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.0%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 47.3

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

5. Applied rewrites 47.3%

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

6. Final simplification 47.3%

   \[\leadsto x \cdot 0.6666666666666666 \]
7. Add Preprocessing

Developer Target 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024221 
(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)))