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

Percentage Accurate: 76.5% → 99.7%

Time: 8.2s

Alternatives: 4

Speedup: 3.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% 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: 99.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. clear-num N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. sin-mult N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. neg-mul-1 N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 54.2%

   \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x} \cdot 1.3333333333333333} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. hang-p0-tan N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. lift-*.f64 N/A

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

   9. lower-tan.f64 99.4

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.4

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

6. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 99.3

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.3

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

8. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval 99.7

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

    \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{0.75 \cdot \cos \left(x \cdot 0.5\right)} \]
12. Add Preprocessing

Alternative 2: 99.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. clear-num N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. sin-mult N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. neg-mul-1 N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 54.2%

   \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x} \cdot 1.3333333333333333} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. hang-p0-tan N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. lift-*.f64 N/A

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

   9. lower-tan.f64 99.4

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.4

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

6. Applied rewrites 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 51.3% accurate, 20.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. clear-num N/A

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

   8. frac-2neg N/A

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

   9. neg-mul-1 N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. associate-/r* N/A

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

   13. lower-/.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 rewrites 54.3%

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

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. lower-*.f64 49.7

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

7. Applied rewrites 49.7%

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

Alternative 4: 51.0% 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 74.5%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 49.4

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

5. Applied rewrites 49.4%

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

6. Final simplification 49.4%

   \[\leadsto 0.6666666666666666 \cdot x \]
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 2024249 
(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)))