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

Percentage Accurate: 77.0% → 99.8%

Time: 9.7s

Alternatives: 7

Speedup: 3.0×

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: 77.0% 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, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\tan \left(\frac{x}{2}\right)}{0.75} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (tan (/ x 2.0)) 0.75))

C

double code(double x) {
	return tan((x / 2.0)) / 0.75;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x / 2.0d0)) / 0.75d0
end function

Java

public static double code(double x) {
	return Math.tan((x / 2.0)) / 0.75;
}

Python

def code(x):
	return math.tan((x / 2.0)) / 0.75

Julia

function code(x)
	return Float64(tan(Float64(x / 2.0)) / 0.75)
end

MATLAB

function tmp = code(x)
	tmp = tan((x / 2.0)) / 0.75;
end

Wolfram

code[x_] := N[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{8}{3}} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\color{blue}{8}}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}}{\sin \color{blue}{x}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{\frac{8}{3}}{\sin x}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{\frac{8}{3}}{\sin x}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{\color{blue}{8}}{3}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{8}{3}\right), \color{blue}{\sin x}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \sin \color{blue}{x}\right)\right)\right) \]

   15. sin-lowering-sin.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{sin.f64}\left(x\right)\right)\right)\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. sin-mult N/A

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

   5. associate-/l/ N/A

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

   6. div-inv N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right), \color{blue}{\left(\frac{1}{\frac{\sin x}{\frac{8}{3}} \cdot 2}\right)}\right) \]

6. Applied egg-rr 51.4%

   \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{\sin x \cdot 0.75}} \]
7. Step-by-step derivation

   1. un-div-inv N/A

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

   2. associate-/r* N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 - \cos x}{\sin x}\right), \color{blue}{\frac{3}{4}}\right) \]

   4. hang-p0-tan N/A

      \[\leadsto \mathsf{/.f64}\left(\tan \left(\frac{x}{2}\right), \frac{3}{4}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right), \frac{3}{4}\right) \]

   6. /-lowering-/.f64 99.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right), \frac{3}{4}\right) \]

8. Applied egg-rr 99.8%

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

Alternative 2: 99.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \tan \left(\frac{x}{2}\right) \cdot 1.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (tan (/ x 2.0)) 1.3333333333333333))

C

double code(double x) {
	return tan((x / 2.0)) * 1.3333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x / 2.0d0)) * 1.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.tan((x / 2.0)) * 1.3333333333333333;
}

Python

def code(x):
	return math.tan((x / 2.0)) * 1.3333333333333333

Julia

function code(x)
	return Float64(tan(Float64(x / 2.0)) * 1.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = tan((x / 2.0)) * 1.3333333333333333;
end

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{8}{3}} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\color{blue}{8}}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}}{\sin \color{blue}{x}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{\frac{8}{3}}{\sin x}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{\frac{8}{3}}{\sin x}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{\color{blue}{8}}{3}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{8}{3}\right), \color{blue}{\sin x}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \sin \color{blue}{x}\right)\right)\right) \]

   15. sin-lowering-sin.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{sin.f64}\left(x\right)\right)\right)\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. sin-mult N/A

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

   5. associate-/l/ N/A

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

   6. div-inv N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right), \color{blue}{\left(\frac{1}{\frac{\sin x}{\frac{8}{3}} \cdot 2}\right)}\right) \]

6. Applied egg-rr 51.4%

   \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{\sin x \cdot 0.75}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. times-frac N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 - \cos x}{\sin x}\right), \color{blue}{\left({\frac{3}{4}}^{-1}\right)}\right) \]

   6. hang-p0-tan N/A

      \[\leadsto \mathsf{*.f64}\left(\tan \left(\frac{x}{2}\right), \left({\color{blue}{\frac{3}{4}}}^{-1}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right), \left({\color{blue}{\frac{3}{4}}}^{-1}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right), \left({\frac{3}{4}}^{-1}\right)\right) \]

   9. metadata-eval 99.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right), \frac{4}{3}\right) \]

8. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right) \cdot 1.3333333333333333} \]
9. Add Preprocessing

Alternative 3: 55.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{8}{3}} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\color{blue}{8}}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}}{\sin \color{blue}{x}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{\frac{8}{3}}{\sin x}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{\frac{8}{3}}{\sin x}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{\color{blue}{8}}{3}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{8}{3}\right), \color{blue}{\sin x}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \sin \color{blue}{x}\right)\right)\right) \]

   15. sin-lowering-sin.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{sin.f64}\left(x\right)\right)\right)\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \color{blue}{\frac{4}{3}}\right) \]
6. Step-by-step derivation

7. Simplified 56.4%

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

8. Final simplification 56.4%

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

Alternative 4: 52.2% accurate, 13.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((1.5d0 + ((x * x) * ((-0.125d0) + (x * (x * ((-0.0020833333333333333d0) + (x * (x * (-4.96031746031746d-5))))))))) / x)
end function

Java

public static double code(double x) {
	return 1.0 / ((1.5 + ((x * x) * (-0.125 + (x * (x * (-0.0020833333333333333 + (x * (x * -4.96031746031746e-5)))))))) / x);
}

Python

def code(x):
	return 1.0 / ((1.5 + ((x * x) * (-0.125 + (x * (x * (-0.0020833333333333333 + (x * (x * -4.96031746031746e-5)))))))) / x)

Julia

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

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((1.5 + ((x * x) * (-0.125 + (x * (x * (-0.0020833333333333333 + (x * (x * -4.96031746031746e-5)))))))) / x);
end

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{8}{3}} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\color{blue}{8}}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}}{\sin \color{blue}{x}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{\frac{8}{3}}{\sin x}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{\frac{8}{3}}{\sin x}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{\color{blue}{8}}{3}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{8}{3}\right), \color{blue}{\sin x}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \sin \color{blue}{x}\right)\right)\right) \]

   15. sin-lowering-sin.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{sin.f64}\left(x\right)\right)\right)\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. sin-mult N/A

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

   5. associate-/l/ N/A

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

   6. div-inv N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right), \color{blue}{\left(\frac{1}{\frac{\sin x}{\frac{8}{3}} \cdot 2}\right)}\right) \]

6. Applied egg-rr 51.4%

   \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{\sin x \cdot 0.75}} \]
7. Step-by-step derivation

   1. un-div-inv N/A

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

   2. associate-/r* N/A

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

   3. clear-num N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{3}{4}}{\frac{1 - \cos x}{\sin x}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{3}{4}, \color{blue}{\left(\frac{1 - \cos x}{\sin x}\right)}\right)\right) \]

   6. hang-p0-tan N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{3}{4}, \tan \left(\frac{x}{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{3}{4}, \mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 99.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{3}{4}, \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right)\right)\right) \]

8. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{0.75}{\tan \left(\frac{x}{2}\right)}}} \]

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{3}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{20160} \cdot {x}^{2} - \frac{1}{480}\right) - \frac{1}{8}\right)}{x}\right)}\right) \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{3}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{20160} \cdot {x}^{2} - \frac{1}{480}\right) - \frac{1}{8}\right)\right), \color{blue}{x}\right)\right) \]

11. Simplified 53.9%

    \[\leadsto \frac{1}{\color{blue}{\frac{1.5 + \left(x \cdot x\right) \cdot \left(-0.125 + x \cdot \left(x \cdot \left(-0.0020833333333333333 + x \cdot \left(x \cdot -4.96031746031746 \cdot 10^{-5}\right)\right)\right)\right)}{x}}} \]
12. Add Preprocessing

Alternative 5: 52.3% accurate, 28.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	return 1.0 / ((1.5 + (x * (x * -0.125))) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((1.5d0 + (x * (x * (-0.125d0)))) / x)
end function

Java

public static double code(double x) {
	return 1.0 / ((1.5 + (x * (x * -0.125))) / x);
}

Python

def code(x):
	return 1.0 / ((1.5 + (x * (x * -0.125))) / x)

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(1.5 + Float64(x * Float64(x * -0.125))) / x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((1.5 + (x * (x * -0.125))) / x);
end

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{8}{3}} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\color{blue}{8}}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}}{\sin \color{blue}{x}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{\frac{8}{3}}{\sin x}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{\frac{8}{3}}{\sin x}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{\color{blue}{8}}{3}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{8}{3}\right), \color{blue}{\sin x}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \sin \color{blue}{x}\right)\right)\right) \]

   15. sin-lowering-sin.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{sin.f64}\left(x\right)\right)\right)\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. sin-mult N/A

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

   5. associate-/l/ N/A

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

   6. div-inv N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right), \color{blue}{\left(\frac{1}{\frac{\sin x}{\frac{8}{3}} \cdot 2}\right)}\right) \]

6. Applied egg-rr 51.4%

   \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{\sin x \cdot 0.75}} \]
7. Step-by-step derivation

   1. un-div-inv N/A

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

   2. associate-/r* N/A

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

   3. clear-num N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{3}{4}}{\frac{1 - \cos x}{\sin x}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{3}{4}, \color{blue}{\left(\frac{1 - \cos x}{\sin x}\right)}\right)\right) \]

   6. hang-p0-tan N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{3}{4}, \tan \left(\frac{x}{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{3}{4}, \mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 99.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{3}{4}, \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right)\right)\right) \]

8. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{0.75}{\tan \left(\frac{x}{2}\right)}}} \]

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{3}{2} + \frac{-1}{8} \cdot {x}^{2}}{x}\right)}\right) \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{3}{2} + \frac{-1}{8} \cdot {x}^{2}\right), \color{blue}{x}\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \left(\frac{-1}{8} \cdot {x}^{2}\right)\right), x\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \left({x}^{2} \cdot \frac{-1}{8}\right)\right), x\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \left(\left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right), x\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \left(x \cdot \left(x \cdot \frac{-1}{8}\right)\right)\right), x\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{8}\right)\right)\right), x\right)\right) \]

    7. *-lowering-*.f64 53.7%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{8}\right)\right)\right), x\right)\right) \]

11. Simplified 53.7%

    \[\leadsto \frac{1}{\color{blue}{\frac{1.5 + x \cdot \left(x \cdot -0.125\right)}{x}}} \]
12. Add Preprocessing

Alternative 6: 51.9% accurate, 104.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x 1.5))

C

double code(double x) {
	return x / 1.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / 1.5d0
end function

Java

public static double code(double x) {
	return x / 1.5;
}

Python

def code(x):
	return x / 1.5

Julia

function code(x)
	return Float64(x / 1.5)
end

MATLAB

function tmp = code(x)
	tmp = x / 1.5;
end

Wolfram

code[x_] := N[(x / 1.5), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{8}{3}} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\color{blue}{8}}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}}{\sin \color{blue}{x}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{\frac{8}{3}}{\sin x}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{\frac{8}{3}}{\sin x}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{\color{blue}{8}}{3}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{8}{3}\right), \color{blue}{\sin x}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \sin \color{blue}{x}\right)\right)\right) \]

   15. sin-lowering-sin.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{sin.f64}\left(x\right)\right)\right)\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. sin-mult N/A

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

   5. associate-/l/ N/A

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

   6. div-inv N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\cos \left(x \cdot \frac{1}{2} - x \cdot \frac{1}{2}\right) - \cos \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right), \color{blue}{\left(\frac{1}{\frac{\sin x}{\frac{8}{3}} \cdot 2}\right)}\right) \]

6. Applied egg-rr 51.4%

   \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{\sin x \cdot 0.75}} \]
7. Step-by-step derivation

   1. un-div-inv N/A

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

   2. associate-/r* N/A

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

   3. clear-num N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{3}{4}}{\frac{1 - \cos x}{\sin x}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{3}{4}, \color{blue}{\left(\frac{1 - \cos x}{\sin x}\right)}\right)\right) \]

   6. hang-p0-tan N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{3}{4}, \tan \left(\frac{x}{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{3}{4}, \mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 99.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{3}{4}, \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right)\right)\right) \]

8. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{0.75}{\tan \left(\frac{x}{2}\right)}}} \]

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{3}{2}}{x}\right)}\right) \]
10. Step-by-step derivation

    1. /-lowering-/.f64 53.0%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{3}{2}, \color{blue}{x}\right)\right) \]

11. Simplified 53.0%

    \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
12. Step-by-step derivation

    1. clear-num N/A

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

    2. /-lowering-/.f64 53.2%

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\frac{3}{2}}\right) \]

13. Applied egg-rr 53.2%

    \[\leadsto \color{blue}{\frac{x}{1.5}} \]
14. Add Preprocessing

Alternative 7: 51.6% accurate, 104.3× 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 76.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{8}{3}} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\color{blue}{8}}{3} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}}{\sin \color{blue}{x}}\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{\frac{8}{3}}{\sin x}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\sin \left(x \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{\frac{8}{3}}{\sin x}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(x \cdot \frac{1}{2}\right)\right), \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \left(\frac{\frac{\color{blue}{8}}{3}}{\sin x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{8}{3}\right), \color{blue}{\sin x}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \sin \color{blue}{x}\right)\right)\right) \]

   15. sin-lowering-sin.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(\frac{8}{3}, \mathsf{sin.f64}\left(x\right)\right)\right)\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{2.6666666666666665}{\sin x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 52.9%

      \[\leadsto \mathsf{*.f64}\left(\frac{2}{3}, \color{blue}{x}\right) \]

7. Simplified 52.9%

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

8. Final simplification 52.9%

   \[\leadsto x \cdot 0.6666666666666666 \]
9. 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 2024155 
(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)))