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

Percentage Accurate: 76.9% → 99.8%

Time: 10.9s

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: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

   1. clear-num N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. sin-mult N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. frac-2neg N/A

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

   9. neg-mul-1 N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. times-frac N/A

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

4. Applied egg-rr 51.5%

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

5. Taylor expanded in x around inf

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

   1. hang-p0-tan N/A

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

   2. *-rgt-identity N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. tan-lowering-tan.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 99.4

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

7. Simplified 99.4%

   \[\leadsto \color{blue}{\tan \left(x \cdot 0.5\right)} \cdot 1.3333333333333333 \]
8. Step-by-step derivation

   1. tan-quot N/A

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

   2. associate-*l/ N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. *-lowering-*.f64 99.4

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

9. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 1.3333333333333333}{\cos \left(x \cdot 0.5\right)}} \]
10. Step-by-step derivation

    1. associate-*l/ N/A

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

    2. div-inv N/A

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

    3. div-inv N/A

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

    4. tan-quot N/A

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

    5. metadata-eval N/A

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

    6. div-inv N/A

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

    7. hang-p0-tan N/A

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

    8. metadata-eval N/A

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

    9. div-inv N/A

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

    10. hang-p0-tan N/A

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

    11. div-inv N/A

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

    12. metadata-eval N/A

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

    13. /-lowering-/.f64 N/A

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

    14. tan-lowering-tan.f64 N/A

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

    15. *-lowering-*.f64 99.8

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

11. Applied egg-rr 99.8%

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

Alternative 2: 99.4% accurate, 3.1× speedup

Math

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

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 74.4%

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

   1. clear-num N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. sin-mult N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. frac-2neg N/A

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

   9. neg-mul-1 N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. times-frac N/A

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

4. Applied egg-rr 51.5%

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

5. Taylor expanded in x around inf

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

   1. hang-p0-tan N/A

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

   2. *-rgt-identity N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. tan-lowering-tan.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 99.4

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

7. Simplified 99.4%

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

Alternative 3: 51.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ 1.3333333333333333 \cdot \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -6.613756613756614 \cdot 10^{-5}, -0.002777777777777778\right), -0.16666666666666666\right), 2\right)}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  1.3333333333333333
  (/
   1.0
   (/
    (fma
     (* x x)
     (fma
      (* x x)
      (fma x (* x -6.613756613756614e-5) -0.002777777777777778)
      -0.16666666666666666)
     2.0)
    x))))

C

double code(double x) {
	return 1.3333333333333333 * (1.0 / (fma((x * x), fma((x * x), fma(x, (x * -6.613756613756614e-5), -0.002777777777777778), -0.16666666666666666), 2.0) / x));
}

Julia

function code(x)
	return Float64(1.3333333333333333 * Float64(1.0 / Float64(fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * -6.613756613756614e-5), -0.002777777777777778), -0.16666666666666666), 2.0) / x)))
end

Wolfram

code[x_] := N[(1.3333333333333333 * N[(1.0 / N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -6.613756613756614e-5), $MachinePrecision] + -0.002777777777777778), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.4%

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

   1. clear-num N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. sin-mult N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. frac-2neg N/A

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

   9. neg-mul-1 N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. times-frac N/A

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

4. Applied egg-rr 51.5%

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

5. Taylor expanded in x around inf

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

   1. hang-p0-tan N/A

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

   2. *-rgt-identity N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. tan-lowering-tan.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 99.4

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

7. Simplified 99.4%

   \[\leadsto \color{blue}{\tan \left(x \cdot 0.5\right)} \cdot 1.3333333333333333 \]
8. Step-by-step derivation

   1. tan-quot N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. tan-quot N/A

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

   6. /-lowering-/.f64 N/A

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

   7. tan-lowering-tan.f64 N/A

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

   8. *-lowering-*.f64 99.4

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

9. Applied egg-rr 99.4%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{1}{\color{blue}{\frac{2 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{15120} \cdot {x}^{2} - \frac{1}{360}\right) - \frac{1}{6}\right)}{x}}} \cdot \frac{4}{3} \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{2 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{15120} \cdot {x}^{2} - \frac{1}{360}\right) - \frac{1}{6}\right)}{x}}} \cdot \frac{4}{3} \]

    2. +-commutative N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{15120} \cdot {x}^{2} - \frac{1}{360}\right) - \frac{1}{6}\right) + 2}}{x}} \cdot \frac{4}{3} \]

    3. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-1}{15120} \cdot {x}^{2} - \frac{1}{360}\right) - \frac{1}{6}, 2\right)}}{x}} \cdot \frac{4}{3} \]

    4. unpow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{15120} \cdot {x}^{2} - \frac{1}{360}\right) - \frac{1}{6}, 2\right)}{x}} \cdot \frac{4}{3} \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{15120} \cdot {x}^{2} - \frac{1}{360}\right) - \frac{1}{6}, 2\right)}{x}} \cdot \frac{4}{3} \]

    6. sub-neg N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{15120} \cdot {x}^{2} - \frac{1}{360}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 2\right)}{x}} \cdot \frac{4}{3} \]

    7. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{-1}{15120} \cdot {x}^{2} - \frac{1}{360}\right) + \color{blue}{\frac{-1}{6}}, 2\right)}{x}} \cdot \frac{4}{3} \]

    8. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{15120} \cdot {x}^{2} - \frac{1}{360}, \frac{-1}{6}\right)}, 2\right)}{x}} \cdot \frac{4}{3} \]

    9. unpow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{15120} \cdot {x}^{2} - \frac{1}{360}, \frac{-1}{6}\right), 2\right)}{x}} \cdot \frac{4}{3} \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{15120} \cdot {x}^{2} - \frac{1}{360}, \frac{-1}{6}\right), 2\right)}{x}} \cdot \frac{4}{3} \]

    11. sub-neg N/A

        \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{15120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{360}\right)\right)}, \frac{-1}{6}\right), 2\right)}{x}} \cdot \frac{4}{3} \]

    12. *-commutative N/A

        \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{15120}} + \left(\mathsf{neg}\left(\frac{1}{360}\right)\right), \frac{-1}{6}\right), 2\right)}{x}} \cdot \frac{4}{3} \]

    13. unpow2 N/A

        \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{15120} + \left(\mathsf{neg}\left(\frac{1}{360}\right)\right), \frac{-1}{6}\right), 2\right)}{x}} \cdot \frac{4}{3} \]

    14. associate-*l* N/A

        \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{-1}{15120}\right)} + \left(\mathsf{neg}\left(\frac{1}{360}\right)\right), \frac{-1}{6}\right), 2\right)}{x}} \cdot \frac{4}{3} \]

    15. metadata-eval N/A

        \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{-1}{15120}\right) + \color{blue}{\frac{-1}{360}}, \frac{-1}{6}\right), 2\right)}{x}} \cdot \frac{4}{3} \]

    16. accelerator-lowering-fma.f64 N/A

        \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{15120}, \frac{-1}{360}\right)}, \frac{-1}{6}\right), 2\right)}{x}} \cdot \frac{4}{3} \]

    17. *-lowering-*.f64 52.7

        \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot -6.613756613756614 \cdot 10^{-5}}, -0.002777777777777778\right), -0.16666666666666666\right), 2\right)}{x}} \cdot 1.3333333333333333 \]

12. Simplified 52.7%

    \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -6.613756613756614 \cdot 10^{-5}, -0.002777777777777778\right), -0.16666666666666666\right), 2\right)}{x}}} \cdot 1.3333333333333333 \]

13. Final simplification 52.7%

    \[\leadsto 1.3333333333333333 \cdot \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -6.613756613756614 \cdot 10^{-5}, -0.002777777777777778\right), -0.16666666666666666\right), 2\right)}{x}} \]
14. Add Preprocessing

Alternative 4: 51.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ 1.3333333333333333 \cdot \frac{1}{\frac{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 2\right)}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 1.3333333333333333 (/ 1.0 (/ (fma x (* x -0.16666666666666666) 2.0) x))))

C

double code(double x) {
	return 1.3333333333333333 * (1.0 / (fma(x, (x * -0.16666666666666666), 2.0) / x));
}

Julia

function code(x)
	return Float64(1.3333333333333333 * Float64(1.0 / Float64(fma(x, Float64(x * -0.16666666666666666), 2.0) / x)))
end

Wolfram

code[x_] := N[(1.3333333333333333 * N[(1.0 / N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.4%

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

   1. clear-num N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. sin-mult N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. frac-2neg N/A

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

   9. neg-mul-1 N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. times-frac N/A

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

4. Applied egg-rr 51.5%

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

5. Taylor expanded in x around inf

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

   1. hang-p0-tan N/A

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

   2. *-rgt-identity N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. tan-lowering-tan.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 99.4

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

7. Simplified 99.4%

   \[\leadsto \color{blue}{\tan \left(x \cdot 0.5\right)} \cdot 1.3333333333333333 \]
8. Step-by-step derivation

   1. tan-quot N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. tan-quot N/A

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

   6. /-lowering-/.f64 N/A

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

   7. tan-lowering-tan.f64 N/A

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

   8. *-lowering-*.f64 99.4

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

9. Applied egg-rr 99.4%

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

10. Taylor expanded in x around 0

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

    1. /-lowering-/.f64 N/A

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

    2. +-commutative N/A

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

    3. unpow2 N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. accelerator-lowering-fma.f64 N/A

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

    7. *-commutative N/A

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

    8. *-lowering-*.f64 52.7

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 2\right)}{x}} \cdot 1.3333333333333333 \]

12. Simplified 52.7%

    \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 2\right)}{x}}} \cdot 1.3333333333333333 \]

13. Final simplification 52.7%

    \[\leadsto 1.3333333333333333 \cdot \frac{1}{\frac{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 2\right)}{x}} \]
14. Add Preprocessing

Alternative 5: 51.5% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 0.25}{0.375} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* x 0.25) 0.375))

C

double code(double x) {
	return (x * 0.25) / 0.375;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 0.25d0) / 0.375d0
end function

Java

public static double code(double x) {
	return (x * 0.25) / 0.375;
}

Python

def code(x):
	return (x * 0.25) / 0.375

Julia

function code(x)
	return Float64(Float64(x * 0.25) / 0.375)
end

MATLAB

function tmp = code(x)
	tmp = (x * 0.25) / 0.375;
end

Wolfram

code[x_] := N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision]

Derivation

1. Initial program 74.4%

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

   1. clear-num N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. frac-2neg N/A

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

   6. neg-mul-1 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. associate-/r* N/A

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

   10. /-lowering-/.f64 N/A

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

4. Applied egg-rr 51.5%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{1}{4}}}{\frac{3}{8}} \]

   2. *-lowering-*.f64 52.4

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

7. Simplified 52.4%

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

Alternative 6: 51.2% 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 74.4%

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

3. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 52.1

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

5. Simplified 52.1%

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

6. Final simplification 52.1%

   \[\leadsto x \cdot 0.6666666666666666 \]
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 2024205 
(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)))