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

Percentage Accurate: 76.3% → 99.8%

Time: 8.7s

Alternatives: 7

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: 76.3% 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: 99.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

4. Applied rewrites 55.0%

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

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. lower-*.f64 N/A

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

   4. metadata-eval 99.8

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.5% 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 80.2%

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

4. Applied rewrites 55.0%

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

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. *-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

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

7. Applied rewrites 99.4%

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

Alternative 3: 51.5% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.2%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 80.1

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

   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 55.0%

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

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. 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 49.2

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

7. Applied rewrites 49.2%

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

8. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lft-mult-inverse N/A

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

   3. associate-*l* N/A

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

   4. distribute-rgt-in N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. *-inverses N/A

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

   12. *-rgt-identity N/A

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

   13. sub-neg N/A

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

   14. metadata-eval N/A

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

   15. distribute-rgt-in N/A

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

   16. +-commutative N/A

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

10. Applied rewrites 49.2%

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

Alternative 4: 51.4% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \frac{-1.3333333333333333}{\mathsf{fma}\left(0.16666666666666666, x, \frac{-2}{x}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ -1.3333333333333333 (fma 0.16666666666666666 x (/ -2.0 x))))

C

double code(double x) {
	return -1.3333333333333333 / fma(0.16666666666666666, x, (-2.0 / x));
}

Julia

function code(x)
	return Float64(-1.3333333333333333 / fma(0.16666666666666666, x, Float64(-2.0 / x)))
end

Wolfram

code[x_] := N[(-1.3333333333333333 / N[(0.16666666666666666 * x + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

4. Applied rewrites 55.0%

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

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

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. frac-2neg N/A

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

   7. distribute-frac-neg2 N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. distribute-frac-neg2 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 99.3

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

   14. lift-/.f64 N/A

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

   15. clear-num N/A

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

   16. associate-/r/ N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval 99.3

       \[\leadsto \frac{-1.3333333333333333}{\frac{-1}{\tan \left(\color{blue}{0.5} \cdot x\right)}} \]

8. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{-1.3333333333333333}{\frac{-1}{\tan \left(0.5 \cdot x\right)}}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{-4}{3}}{\color{blue}{\frac{\frac{1}{6} \cdot {x}^{2} - 2}{x}}} \]
10. Step-by-step derivation

    1. div-sub N/A

       \[\leadsto \frac{\frac{-4}{3}}{\color{blue}{\frac{\frac{1}{6} \cdot {x}^{2}}{x} - \frac{2}{x}}} \]

    2. associate-/l* N/A

       \[\leadsto \frac{\frac{-4}{3}}{\color{blue}{\frac{1}{6} \cdot \frac{{x}^{2}}{x}} - \frac{2}{x}} \]

    3. unpow2 N/A

       \[\leadsto \frac{\frac{-4}{3}}{\frac{1}{6} \cdot \frac{\color{blue}{x \cdot x}}{x} - \frac{2}{x}} \]

    4. associate-*l/ N/A

       \[\leadsto \frac{\frac{-4}{3}}{\frac{1}{6} \cdot \color{blue}{\left(\frac{x}{x} \cdot x\right)} - \frac{2}{x}} \]

    5. *-inverses N/A

       \[\leadsto \frac{\frac{-4}{3}}{\frac{1}{6} \cdot \left(\color{blue}{1} \cdot x\right) - \frac{2}{x}} \]

    6. *-lft-identity N/A

       \[\leadsto \frac{\frac{-4}{3}}{\frac{1}{6} \cdot \color{blue}{x} - \frac{2}{x}} \]

    7. sub-neg N/A

       \[\leadsto \frac{\frac{-4}{3}}{\color{blue}{\frac{1}{6} \cdot x + \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}} \]

    8. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{-4}{3}}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, x, \mathsf{neg}\left(\frac{2}{x}\right)\right)}} \]

    9. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{-4}{3}}{\mathsf{fma}\left(\frac{1}{6}, x, \color{blue}{\frac{\mathsf{neg}\left(2\right)}{x}}\right)} \]

    10. metadata-eval N/A

        \[\leadsto \frac{\frac{-4}{3}}{\mathsf{fma}\left(\frac{1}{6}, x, \frac{\color{blue}{-2}}{x}\right)} \]

    11. lower-/.f64 49.1

        \[\leadsto \frac{-1.3333333333333333}{\mathsf{fma}\left(0.16666666666666666, x, \color{blue}{\frac{-2}{x}}\right)} \]

11. Applied rewrites 49.1%

    \[\leadsto \frac{-1.3333333333333333}{\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, \frac{-2}{x}\right)}} \]
12. Add Preprocessing

Alternative 5: 51.1% accurate, 20.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

4. Applied rewrites 55.0%

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

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. lower-*.f64 48.8

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

9. Applied rewrites 48.8%

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

Alternative 6: 50.8% accurate, 31.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return (0.5 * x) * 1.3333333333333333

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

4. Applied rewrites 55.0%

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

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. lower-*.f64 48.8

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

9. Applied rewrites 48.8%

   \[\leadsto \frac{\color{blue}{0.5 \cdot x}}{0.75} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

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

    2. div-inv N/A

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

    3. metadata-eval N/A

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

    4. lower-*.f64 48.5

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

11. Applied rewrites 48.5%

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

Alternative 7: 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 80.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 48.5

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

5. Applied rewrites 48.5%

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

6. Final simplification 48.5%

   \[\leadsto 0.6666666666666666 \cdot x \]
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 2024254 
(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)))