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

Percentage Accurate: 76.8% → 99.8%

Time: 8.0s

Alternatives: 6

Speedup: 3.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right) \cdot 0.5}{0.375}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l/ N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 99.7

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 99.7

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

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right) \cdot 0.5}{0.375 \cdot \cos \left(0.5 \cdot x\right)}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. tan-quot N/A

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

   13. lift-tan.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. div-inv N/A

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

   17. lower-/.f64 99.8

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 50.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4.96031746031746 \cdot 10^{-5}, x \cdot x, -0.0020833333333333333\right), x \cdot x, -0.125\right), x \cdot x, 1.5\right)}{x}\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (/
   (fma
    (fma
     (fma -4.96031746031746e-5 (* x x) -0.0020833333333333333)
     (* x x)
     -0.125)
    (* x x)
    1.5)
   x)
  -1.0))

C

double code(double x) {
	return pow((fma(fma(fma(-4.96031746031746e-5, (x * x), -0.0020833333333333333), (x * x), -0.125), (x * x), 1.5) / x), -1.0);
}

Julia

function code(x)
	return Float64(fma(fma(fma(-4.96031746031746e-5, Float64(x * x), -0.0020833333333333333), Float64(x * x), -0.125), Float64(x * x), 1.5) / x) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(N[(N[(N[(-4.96031746031746e-5 * N[(x * x), $MachinePrecision] + -0.0020833333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.125), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.5), $MachinePrecision] / x), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 77.3%

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right) \cdot 0.5}{0.375}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 99.5

      \[\leadsto \frac{1}{\frac{\color{blue}{0.75}}{\tan \left(x \cdot 0.5\right)}} \]

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 99.5

       \[\leadsto \frac{1}{\frac{0.75}{\tan \color{blue}{\left(0.5 \cdot x\right)}}} \]

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{0.75}{\tan \left(0.5 \cdot x\right)}}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{\frac{3}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{20160} \cdot {x}^{2} - \frac{1}{480}\right) - \frac{1}{8}\right)}{x}}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{3}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{20160} \cdot {x}^{2} - \frac{1}{480}\right) - \frac{1}{8}\right)}{x}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{20160} \cdot {x}^{2} - \frac{1}{480}\right) - \frac{1}{8}\right) + \frac{3}{2}}}{x}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{20160} \cdot {x}^{2} - \frac{1}{480}\right) - \frac{1}{8}\right) \cdot {x}^{2}} + \frac{3}{2}}{x}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{-1}{20160} \cdot {x}^{2} - \frac{1}{480}\right) - \frac{1}{8}, {x}^{2}, \frac{3}{2}\right)}}{x}} \]

   5. sub-neg N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{20160} \cdot {x}^{2} - \frac{1}{480}\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, {x}^{2}, \frac{3}{2}\right)}{x}} \]

   6. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{20160} \cdot {x}^{2} - \frac{1}{480}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), {x}^{2}, \frac{3}{2}\right)}{x}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left(\frac{-1}{20160} \cdot {x}^{2} - \frac{1}{480}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{8}}, {x}^{2}, \frac{3}{2}\right)}{x}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{20160} \cdot {x}^{2} - \frac{1}{480}, {x}^{2}, \frac{-1}{8}\right)}, {x}^{2}, \frac{3}{2}\right)}{x}} \]

   9. sub-neg N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{20160} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{480}\right)\right)}, {x}^{2}, \frac{-1}{8}\right), {x}^{2}, \frac{3}{2}\right)}{x}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{20160} \cdot {x}^{2} + \color{blue}{\frac{-1}{480}}, {x}^{2}, \frac{-1}{8}\right), {x}^{2}, \frac{3}{2}\right)}{x}} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{20160}, {x}^{2}, \frac{-1}{480}\right)}, {x}^{2}, \frac{-1}{8}\right), {x}^{2}, \frac{3}{2}\right)}{x}} \]

   12. unpow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{20160}, \color{blue}{x \cdot x}, \frac{-1}{480}\right), {x}^{2}, \frac{-1}{8}\right), {x}^{2}, \frac{3}{2}\right)}{x}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{20160}, \color{blue}{x \cdot x}, \frac{-1}{480}\right), {x}^{2}, \frac{-1}{8}\right), {x}^{2}, \frac{3}{2}\right)}{x}} \]

   14. unpow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{20160}, x \cdot x, \frac{-1}{480}\right), \color{blue}{x \cdot x}, \frac{-1}{8}\right), {x}^{2}, \frac{3}{2}\right)}{x}} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{20160}, x \cdot x, \frac{-1}{480}\right), \color{blue}{x \cdot x}, \frac{-1}{8}\right), {x}^{2}, \frac{3}{2}\right)}{x}} \]

   16. unpow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{20160}, x \cdot x, \frac{-1}{480}\right), x \cdot x, \frac{-1}{8}\right), \color{blue}{x \cdot x}, \frac{3}{2}\right)}{x}} \]

   17. lower-*.f64 51.8

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4.96031746031746 \cdot 10^{-5}, x \cdot x, -0.0020833333333333333\right), x \cdot x, -0.125\right), \color{blue}{x \cdot x}, 1.5\right)}{x}} \]

9. Applied rewrites 51.8%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4.96031746031746 \cdot 10^{-5}, x \cdot x, -0.0020833333333333333\right), x \cdot x, -0.125\right), x \cdot x, 1.5\right)}{x}}} \]

10. Final simplification 51.8%

    \[\leadsto {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4.96031746031746 \cdot 10^{-5}, x \cdot x, -0.0020833333333333333\right), x \cdot x, -0.125\right), x \cdot x, 1.5\right)}{x}\right)}^{-1} \]
11. Add Preprocessing

Alternative 3: 50.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right) \cdot 0.5}{0.375}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 99.5

      \[\leadsto \frac{1}{\frac{\color{blue}{0.75}}{\tan \left(x \cdot 0.5\right)}} \]

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 99.5

       \[\leadsto \frac{1}{\frac{0.75}{\tan \color{blue}{\left(0.5 \cdot x\right)}}} \]

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{0.75}{\tan \left(0.5 \cdot x\right)}}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{\frac{3}{2} + \frac{-1}{8} \cdot {x}^{2}}{x}}} \]
8. 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 51.6

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

9. Applied rewrites 51.6%

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

10. Final simplification 51.6%

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

Alternative 4: 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 77.3%

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

4. Applied rewrites 99.5%

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

Alternative 5: 50.4% 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 77.3%

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

4. Applied rewrites 99.8%

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

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 51.2

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

7. Applied rewrites 51.2%

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

Alternative 6: 50.2% 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 77.3%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 51.0

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

5. Applied rewrites 51.0%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
6. 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 2024321 
(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)))