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

Percentage Accurate: 76.4% → 99.5%

Time: 11.1s

Alternatives: 6

Speedup: 3.0×

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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{t_0}{\frac{0.375}{\frac{t_0}{\sin x}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ t_0 (/ 0.375 (/ t_0 (sin x))))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return t_0 / (0.375 / (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 = t_0 / (0.375d0 / (t_0 / sin(x)))
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return t_0 / (0.375 / (t_0 / Math.sin(x)));
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return t_0 / (0.375 / (t_0 / math.sin(x)))

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(t_0 / Float64(0.375 / Float64(t_0 / sin(x))))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = t_0 / (0.375 / (t_0 / sin(x)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 / N[(0.375 / N[(t$95$0 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.0%

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

   1. associate-*r/ 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* 99.2%

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

   2. metadata-eval 99.2%

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

   3. associate-*r/ 75.0%

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

   4. *-commutative 75.0%

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

   5. associate-/l* 99.2%

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

   6. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. clear-num 99.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{1}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x}}}} \]

   2. inv-pow 99.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{{\left(\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\right)}^{-1}}} \]

   3. *-un-lft-identity 99.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{{\left(\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\color{blue}{1 \cdot \sin x}}\right)}^{-1}} \]

   4. times-frac 99.5%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{{\color{blue}{\left(\frac{2.6666666666666665}{1} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right)}}^{-1}} \]

   5. metadata-eval 99.5%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{{\left(\color{blue}{2.6666666666666665} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right)}^{-1}} \]

7. Applied egg-rr 99.5%

   \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{{\left(2.6666666666666665 \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right)}^{-1}}} \]
8. Step-by-step derivation

   1. unpow-1 99.5%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{1}{2.6666666666666665 \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}}}} \]

   2. associate-/r* 99.6%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{\frac{1}{2.6666666666666665}}{\frac{\sin \left(x \cdot 0.5\right)}{\sin x}}}} \]

   3. metadata-eval 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 1.3333333333333333 \cdot \tan \left(\frac{x}{2}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 1.3333333333333333 (tan (/ x 2.0))))

C

double code(double x) {
	return 1.3333333333333333 * tan((x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.3333333333333333d0 * tan((x / 2.0d0))
end function

Java

public static double code(double x) {
	return 1.3333333333333333 * Math.tan((x / 2.0));
}

Python

def code(x):
	return 1.3333333333333333 * math.tan((x / 2.0))

Julia

function code(x)
	return Float64(1.3333333333333333 * tan(Float64(x / 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.3333333333333333 * tan((x / 2.0));
end

Wolfram

code[x_] := N[(1.3333333333333333 * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. associate-*r/ 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r/ 74.9%

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

   2. associate-*r/ 75.0%

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

   3. expm1-log1p-u 63.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2.6666666666666665 \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}{\sin x}\right)\right)} \]

   4. associate-*l/ 62.9%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{2.6666666666666665}{\sin x} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}\right)\right) \]

   5. expm1-udef 39.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2.6666666666666665}{\sin x} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)\right)} - 1} \]

5. Applied egg-rr 39.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2.6666666666666665 \cdot \frac{0.5 + -0.5 \cdot \cos x}{\sin x}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 39.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2.6666666666666665 \cdot \frac{0.5 + -0.5 \cdot \cos x}{\sin x}\right)\right)} \]

   2. expm1-log1p 51.1%

      \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{0.5 + -0.5 \cdot \cos x}{\sin x}} \]

   3. associate-*r/ 51.1%

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

   4. associate-*l/ 51.1%

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

   5. metadata-eval 51.1%

      \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \left(\color{blue}{0.5 \cdot 1} + -0.5 \cdot \cos x\right) \]

   6. metadata-eval 51.1%

      \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \left(0.5 \cdot 1 + \color{blue}{\left(-0.5\right)} \cdot \cos x\right) \]

   7. distribute-lft-neg-in 51.1%

      \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \left(0.5 \cdot 1 + \color{blue}{\left(-0.5 \cdot \cos x\right)}\right) \]

   8. distribute-rgt-neg-in 51.1%

      \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \left(0.5 \cdot 1 + \color{blue}{0.5 \cdot \left(-\cos x\right)}\right) \]

   9. distribute-lft-in 51.1%

      \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \color{blue}{\left(0.5 \cdot \left(1 + \left(-\cos x\right)\right)\right)} \]

   10. sub-neg 51.1%

       \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \left(0.5 \cdot \color{blue}{\left(1 - \cos x\right)}\right) \]

   11. cos-0 51.1%

       \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \left(0.5 \cdot \left(\color{blue}{\cos 0} - \cos x\right)\right) \]

   12. metadata-eval 51.1%

       \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \left(\color{blue}{\frac{1}{2}} \cdot \left(\cos 0 - \cos x\right)\right) \]

   13. associate-/r/ 51.1%

       \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \color{blue}{\frac{1}{\frac{2}{\cos 0 - \cos x}}} \]

   14. associate-/l* 51.1%

       \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \color{blue}{\frac{1 \cdot \left(\cos 0 - \cos x\right)}{2}} \]

   15. *-lft-identity 51.1%

       \[\leadsto \frac{2.6666666666666665}{\sin x} \cdot \frac{\color{blue}{\cos 0 - \cos x}}{2} \]

   16. times-frac 51.1%

       \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \left(\cos 0 - \cos x\right)}{\sin x \cdot 2}} \]

   17. *-commutative 51.1%

       \[\leadsto \frac{2.6666666666666665 \cdot \left(\cos 0 - \cos x\right)}{\color{blue}{2 \cdot \sin x}} \]

   18. times-frac 51.1%

       \[\leadsto \color{blue}{\frac{2.6666666666666665}{2} \cdot \frac{\cos 0 - \cos x}{\sin x}} \]

   19. metadata-eval 51.1%

       \[\leadsto \color{blue}{1.3333333333333333} \cdot \frac{\cos 0 - \cos x}{\sin x} \]

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 51.7% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot -0.125 + 1.5 \cdot \frac{1}{x}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((x * (-0.125d0)) + (1.5d0 * (1.0d0 / x)))
end function

Java

public static double code(double x) {
	return 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
}

Python

def code(x):
	return 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
end

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. associate-*r/ 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r/ 74.9%

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

   2. associate-*r/ 75.0%

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

   3. clear-num 74.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{2.6666666666666665 \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}}} \]

   4. associate-/r* 74.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin x}{2.6666666666666665}}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}} \]

   5. div-inv 75.0%

      \[\leadsto \frac{1}{\frac{\color{blue}{\sin x \cdot \frac{1}{2.6666666666666665}}}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}} \]

   6. metadata-eval 75.0%

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

   7. sqr-sin-a 51.0%

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

   8. cancel-sign-sub-inv 51.0%

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

   9. metadata-eval 51.0%

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

   10. *-commutative 51.0%

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

   11. associate-*r* 51.0%

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

   12. metadata-eval 51.0%

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

   13. *-un-lft-identity 51.0%

       \[\leadsto \frac{1}{\frac{\sin x \cdot 0.375}{0.5 + -0.5 \cdot \cos \color{blue}{x}}} \]

5. Applied egg-rr 51.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sin x \cdot 0.375}{0.5 + -0.5 \cdot \cos x}}} \]

6. Taylor expanded in x around 0 54.0%

   \[\leadsto \frac{1}{\color{blue}{-0.125 \cdot x + 1.5 \cdot \frac{1}{x}}} \]

7. Final simplification 54.0%

   \[\leadsto \frac{1}{x \cdot -0.125 + 1.5 \cdot \frac{1}{x}} \]

Alternative 4: 51.7% accurate, 31.3× speedup

Math

\[\begin{array}{l} \\ \frac{2.6666666666666665}{x \cdot -0.3333333333333333 + \frac{-4}{-x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.6666666666666665 (+ (* x -0.3333333333333333) (/ -4.0 (- x)))))

C

double code(double x) {
	return 2.6666666666666665 / ((x * -0.3333333333333333) + (-4.0 / -x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.6666666666666665d0 / ((x * (-0.3333333333333333d0)) + ((-4.0d0) / -x))
end function

Java

public static double code(double x) {
	return 2.6666666666666665 / ((x * -0.3333333333333333) + (-4.0 / -x));
}

Python

def code(x):
	return 2.6666666666666665 / ((x * -0.3333333333333333) + (-4.0 / -x))

Julia

function code(x)
	return Float64(2.6666666666666665 / Float64(Float64(x * -0.3333333333333333) + Float64(-4.0 / Float64(-x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.6666666666666665 / ((x * -0.3333333333333333) + (-4.0 / -x));
end

Wolfram

code[x_] := N[(2.6666666666666665 / N[(N[(x * -0.3333333333333333), $MachinePrecision] + N[(-4.0 / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. associate-*r/ 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r/ 74.9%

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

   2. associate-*r/ 75.0%

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

   3. associate-/l* 75.0%

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

   4. sqr-sin-a 51.1%

      \[\leadsto \frac{2.6666666666666665}{\frac{\sin x}{\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(x \cdot 0.5\right)\right)}}} \]

   5. cancel-sign-sub-inv 51.1%

      \[\leadsto \frac{2.6666666666666665}{\frac{\sin x}{\color{blue}{0.5 + \left(-0.5\right) \cdot \cos \left(2 \cdot \left(x \cdot 0.5\right)\right)}}} \]

   6. metadata-eval 51.1%

      \[\leadsto \frac{2.6666666666666665}{\frac{\sin x}{0.5 + \color{blue}{-0.5} \cdot \cos \left(2 \cdot \left(x \cdot 0.5\right)\right)}} \]

   7. *-commutative 51.1%

      \[\leadsto \frac{2.6666666666666665}{\frac{\sin x}{0.5 + -0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(0.5 \cdot x\right)}\right)}} \]

   8. associate-*r* 51.1%

      \[\leadsto \frac{2.6666666666666665}{\frac{\sin x}{0.5 + -0.5 \cdot \cos \color{blue}{\left(\left(2 \cdot 0.5\right) \cdot x\right)}}} \]

   9. metadata-eval 51.1%

      \[\leadsto \frac{2.6666666666666665}{\frac{\sin x}{0.5 + -0.5 \cdot \cos \left(\color{blue}{1} \cdot x\right)}} \]

   10. *-un-lft-identity 51.1%

       \[\leadsto \frac{2.6666666666666665}{\frac{\sin x}{0.5 + -0.5 \cdot \cos \color{blue}{x}}} \]

5. Applied egg-rr 51.1%

   \[\leadsto \color{blue}{\frac{2.6666666666666665}{\frac{\sin x}{0.5 + -0.5 \cdot \cos x}}} \]

6. Taylor expanded in x around 0 53.9%

   \[\leadsto \frac{2.6666666666666665}{\color{blue}{-0.3333333333333333 \cdot x + 4 \cdot \frac{1}{x}}} \]
7. Step-by-step derivation

   1. div-inv 53.9%

      \[\leadsto \frac{2.6666666666666665}{-0.3333333333333333 \cdot x + \color{blue}{\frac{4}{x}}} \]

   2. frac-2neg 53.9%

      \[\leadsto \frac{2.6666666666666665}{-0.3333333333333333 \cdot x + \color{blue}{\frac{-4}{-x}}} \]

   3. metadata-eval 53.9%

      \[\leadsto \frac{2.6666666666666665}{-0.3333333333333333 \cdot x + \frac{\color{blue}{-4}}{-x}} \]

8. Applied egg-rr 53.9%

   \[\leadsto \frac{2.6666666666666665}{-0.3333333333333333 \cdot x + \color{blue}{\frac{-4}{-x}}} \]

9. Final simplification 53.9%

   \[\leadsto \frac{2.6666666666666665}{x \cdot -0.3333333333333333 + \frac{-4}{-x}} \]

Alternative 5: 51.3% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1.5}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ 1.5 x)))

C

double code(double x) {
	return 1.0 / (1.5 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.5d0 / x)
end function

Java

public static double code(double x) {
	return 1.0 / (1.5 / x);
}

Python

def code(x):
	return 1.0 / (1.5 / x)

Julia

function code(x)
	return Float64(1.0 / Float64(1.5 / x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.5 / x);
end

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. associate-*r/ 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r/ 74.9%

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

   2. associate-*r/ 75.0%

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

   3. clear-num 74.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{2.6666666666666665 \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}}} \]

   4. associate-/r* 74.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin x}{2.6666666666666665}}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}} \]

   5. div-inv 75.0%

      \[\leadsto \frac{1}{\frac{\color{blue}{\sin x \cdot \frac{1}{2.6666666666666665}}}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}} \]

   6. metadata-eval 75.0%

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

   7. sqr-sin-a 51.0%

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

   8. cancel-sign-sub-inv 51.0%

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

   9. metadata-eval 51.0%

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

   10. *-commutative 51.0%

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

   11. associate-*r* 51.0%

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

   12. metadata-eval 51.0%

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

   13. *-un-lft-identity 51.0%

       \[\leadsto \frac{1}{\frac{\sin x \cdot 0.375}{0.5 + -0.5 \cdot \cos \color{blue}{x}}} \]

5. Applied egg-rr 51.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sin x \cdot 0.375}{0.5 + -0.5 \cdot \cos x}}} \]

6. Taylor expanded in x around 0 53.0%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]

7. Final simplification 53.0%

   \[\leadsto \frac{1}{\frac{1.5}{x}} \]

Alternative 6: 51.2% accurate, 104.3× 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 75.0%

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

   1. associate-*r/ 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 53.0%

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

5. Final simplification 53.0%

   \[\leadsto x \cdot 0.6666666666666666 \]

Developer target: 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 2023192 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :herbie-target
  (/ (/ (* 8.0 (sin (* x 0.5))) 3.0) (/ (sin x) (sin (* x 0.5))))

  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))