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

Percentage Accurate: 76.8% → 99.8%

Time: 10.2s

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 78.6%

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

4. Applied egg-rr 53.7%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. associate-/l* N/A

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

   5. lift--.f64 N/A

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

   6. flip-- N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l/ N/A

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

   14. clear-num N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

6. Applied egg-rr 98.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. div-inv N/A

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

   10. associate-/r* N/A

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

   11. clear-num N/A

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

   12. lower-/.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. lift-cos.f64 N/A

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

   16. hang-0p-tan N/A

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

   17. lower-tan.f64 N/A

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

   18. lower-/.f64 N/A

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

   19. metadata-eval 99.8

       \[\leadsto \frac{\tan \left(\frac{x}{2}\right)}{\color{blue}{0.75}} \]

8. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{0.75}} \]
9. Step-by-step derivation

   1. div-inv N/A

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

   2. metadata-eval N/A

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

   3. lower-*.f64 99.8

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

10. Applied egg-rr 99.8%

    \[\leadsto \frac{\tan \color{blue}{\left(x \cdot 0.5\right)}}{0.75} \]
11. 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 78.6%

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

4. Applied egg-rr 53.7%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. hang-p0-tan N/A

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

   3. *-rgt-identity N/A

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

   4. associate-/l* N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-tan.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.5

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 52.0% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

4. Applied egg-rr 53.7%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. associate-/l* N/A

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

   5. lift--.f64 N/A

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

   6. flip-- N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l/ N/A

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

   14. clear-num N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

6. Applied egg-rr 98.8%

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

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. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 50.8

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

9. Simplified 50.8%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot -0.125, 1.5\right)}{x}}} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. clear-num N/A

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

    4. lower-/.f64 50.9

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

    5. lift-fma.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. associate-*r* N/A

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

    8. lift-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-fma.f64 50.9

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

11. Applied egg-rr 50.9%

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

Alternative 4: 51.5% accurate, 20.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

4. Applied egg-rr 53.7%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. associate-/l* N/A

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

   5. lift--.f64 N/A

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

   6. flip-- N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l/ N/A

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

   14. clear-num N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

6. Applied egg-rr 98.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. div-inv N/A

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

   10. associate-/r* N/A

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

   11. clear-num N/A

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

   12. lower-/.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. lift-cos.f64 N/A

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

   16. hang-0p-tan N/A

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

   17. lower-tan.f64 N/A

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

   18. lower-/.f64 N/A

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

   19. metadata-eval 99.8

       \[\leadsto \frac{\tan \left(\frac{x}{2}\right)}{\color{blue}{0.75}} \]

8. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{0.75}} \]

9. Taylor expanded in x around 0

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

    1. lower-*.f64 49.9

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

11. Simplified 49.9%

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

12. Final simplification 49.9%

    \[\leadsto \frac{x \cdot 0.5}{0.75} \]
13. Add Preprocessing

Alternative 5: 51.0% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(0.05555555555555555, x \cdot x, 0.6666666666666666\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (fma 0.05555555555555555 (* x x) 0.6666666666666666)))

C

double code(double x) {
	return x * fma(0.05555555555555555, (x * x), 0.6666666666666666);
}

Julia

function code(x)
	return Float64(x * fma(0.05555555555555555, Float64(x * x), 0.6666666666666666))
end

Wolfram

code[x_] := N[(x * N[(0.05555555555555555 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + \frac{1}{18} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + \frac{1}{18} \cdot {x}^{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{18} \cdot {x}^{2} + \frac{2}{3}\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{18}, {x}^{2}, \frac{2}{3}\right)} \]

   4. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\frac{1}{18}, \color{blue}{x \cdot x}, \frac{2}{3}\right) \]

   5. lower-*.f64 49.8

      \[\leadsto x \cdot \mathsf{fma}\left(0.05555555555555555, \color{blue}{x \cdot x}, 0.6666666666666666\right) \]

5. Simplified 49.8%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.05555555555555555, x \cdot x, 0.6666666666666666\right)} \]
6. 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 78.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 49.8

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

5. Simplified 49.8%

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

6. Final simplification 49.8%

   \[\leadsto x \cdot 0.6666666666666666 \]
7. Add Preprocessing

Developer Target 1: 99.6% 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 2024207 
(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)))