tanhf (example 3.4)

Percentage Accurate: 52.1% → 100.0%

Time: 7.0s

Alternatives: 8

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{\sin x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (sin x)))

C

double code(double x) {
	return (1.0 - cos(x)) / sin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / sin(x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / Math.sin(x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / math.sin(x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / sin(x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / sin(x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{\sin x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (sin x)))

C

double code(double x) {
	return (1.0 - cos(x)) / sin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / sin(x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / Math.sin(x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / math.sin(x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / sin(x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / sin(x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.5%

   \[\frac{1 - \cos x}{\sin x} \]
2. Step-by-step derivation

   1. hang-p0-tan N/A

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

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

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

   3. /-lowering-/.f64 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 51.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right) + \left(x \cdot \left(0.004166666666666667 + \left(x \cdot x\right) \cdot 0.00042162698412698415\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   (+ 0.5 (* x (* x 0.041666666666666664)))
   (*
    (* x (+ 0.004166666666666667 (* (* x x) 0.00042162698412698415)))
    (* x (* x x))))))

C

double code(double x) {
	return x * ((0.5 + (x * (x * 0.041666666666666664))) + ((x * (0.004166666666666667 + ((x * x) * 0.00042162698412698415))) * (x * (x * x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * ((0.5d0 + (x * (x * 0.041666666666666664d0))) + ((x * (0.004166666666666667d0 + ((x * x) * 0.00042162698412698415d0))) * (x * (x * x))))
end function

Java

public static double code(double x) {
	return x * ((0.5 + (x * (x * 0.041666666666666664))) + ((x * (0.004166666666666667 + ((x * x) * 0.00042162698412698415))) * (x * (x * x))));
}

Python

def code(x):
	return x * ((0.5 + (x * (x * 0.041666666666666664))) + ((x * (0.004166666666666667 + ((x * x) * 0.00042162698412698415))) * (x * (x * x))))

Julia

function code(x)
	return Float64(x * Float64(Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664))) + Float64(Float64(x * Float64(0.004166666666666667 + Float64(Float64(x * x) * 0.00042162698412698415))) * Float64(x * Float64(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = x * ((0.5 + (x * (x * 0.041666666666666664))) + ((x * (0.004166666666666667 + ((x * x) * 0.00042162698412698415))) * (x * (x * x))));
end

Wolfram

code[x_] := N[(x * N[(N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(0.004166666666666667 + N[(N[(x * x), $MachinePrecision] * 0.00042162698412698415), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

   \[\frac{1 - \cos x}{\sin x} \]
2. Step-by-step derivation

   1. hang-p0-tan N/A

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

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

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

   3. /-lowering-/.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{240}} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \left(\left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \color{blue}{\left(\frac{17}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \left({x}^{2} \cdot \color{blue}{\frac{17}{40320}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{17}{40320}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{17}{40320}\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 53.3%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{17}{40320}\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 53.3%

   \[\leadsto \color{blue}{x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot \left(0.004166666666666667 + \left(x \cdot x\right) \cdot 0.00042162698412698415\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} + \left(\frac{1}{24} \cdot \left(x \cdot x\right) + \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{240} + \left(x \cdot x\right) \cdot \frac{17}{40320}\right)\right)\right) \cdot \left(x \cdot x\right)}\right)\right)\right) \]

   2. associate-+r+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} + \frac{1}{24} \cdot \left(x \cdot x\right)\right) + \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{240} + \left(x \cdot x\right) \cdot \frac{17}{40320}\right)\right)\right) \cdot \left(x \cdot x\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} + \frac{1}{24} \cdot \left(x \cdot x\right)\right), \color{blue}{\left(\left(x \cdot \left(x \cdot \left(\frac{1}{240} + \left(x \cdot x\right) \cdot \frac{17}{40320}\right)\right)\right) \cdot \left(x \cdot x\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{24} \cdot \left(x \cdot x\right)\right)\right), \left(\color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{240} + \left(x \cdot x\right) \cdot \frac{17}{40320}\right)\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{24} \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{240} + \left(x \cdot x\right) \cdot \frac{17}{40320}\right)\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{1}{24} \cdot x\right)\right)\right), \left(\left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{240} + \left(x \cdot x\right) \cdot \frac{17}{40320}\right)\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot x\right)\right)\right), \left(\left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{240} + \left(x \cdot x\right) \cdot \frac{17}{40320}\right)\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{24}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{240} + \left(x \cdot x\right) \cdot \frac{17}{40320}\right)}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{240} + \left(x \cdot x\right) \cdot \frac{17}{40320}\right)}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \left(\left(\left(x \cdot \left(\frac{1}{240} + \left(x \cdot x\right) \cdot \frac{17}{40320}\right)\right) \cdot x\right) \cdot \left(\color{blue}{x} \cdot x\right)\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \left(\left(x \cdot \left(\frac{1}{240} + \left(x \cdot x\right) \cdot \frac{17}{40320}\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

   12. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \left(\left(x \cdot \left(\frac{1}{240} + \left(x \cdot x\right) \cdot \frac{17}{40320}\right)\right) \cdot {x}^{\color{blue}{3}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{240} + \left(x \cdot x\right) \cdot \frac{17}{40320}\right)\right), \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{240} + \left(x \cdot x\right) \cdot \frac{17}{40320}\right)\right), \left({\color{blue}{x}}^{3}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \left(\left(x \cdot x\right) \cdot \frac{17}{40320}\right)\right)\right), \left({x}^{3}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{17}{40320}\right)\right)\right), \left({x}^{3}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{17}{40320}\right)\right)\right), \left({x}^{3}\right)\right)\right)\right) \]

   18. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{17}{40320}\right)\right)\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{17}{40320}\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

9. Applied egg-rr 53.3%

   \[\leadsto x \cdot \color{blue}{\left(\left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right) + \left(x \cdot \left(0.004166666666666667 + \left(x \cdot x\right) \cdot 0.00042162698412698415\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]
10. Add Preprocessing

Alternative 3: 51.8% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot \left(0.004166666666666667 + \left(x \cdot x\right) \cdot 0.00042162698412698415\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   0.5
   (*
    (* x x)
    (+
     0.041666666666666664
     (*
      x
      (* x (+ 0.004166666666666667 (* (* x x) 0.00042162698412698415)))))))))

C

double code(double x) {
	return x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * (0.004166666666666667 + ((x * x) * 0.00042162698412698415)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.5d0 + ((x * x) * (0.041666666666666664d0 + (x * (x * (0.004166666666666667d0 + ((x * x) * 0.00042162698412698415d0)))))))
end function

Java

public static double code(double x) {
	return x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * (0.004166666666666667 + ((x * x) * 0.00042162698412698415)))))));
}

Python

def code(x):
	return x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * (0.004166666666666667 + ((x * x) * 0.00042162698412698415)))))))

Julia

function code(x)
	return Float64(x * Float64(0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(x * Float64(x * Float64(0.004166666666666667 + Float64(Float64(x * x) * 0.00042162698412698415))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * (0.004166666666666667 + ((x * x) * 0.00042162698412698415)))))));
end

Wolfram

code[x_] := N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(x * N[(x * N[(0.004166666666666667 + N[(N[(x * x), $MachinePrecision] * 0.00042162698412698415), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

   \[\frac{1 - \cos x}{\sin x} \]
2. Step-by-step derivation

   1. hang-p0-tan N/A

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

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

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

   3. /-lowering-/.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{240}} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \left(\left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \color{blue}{\left(\frac{17}{40320} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \left({x}^{2} \cdot \color{blue}{\frac{17}{40320}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{17}{40320}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{17}{40320}\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 53.3%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{240}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{17}{40320}\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 53.3%

   \[\leadsto \color{blue}{x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot \left(0.004166666666666667 + \left(x \cdot x\right) \cdot 0.00042162698412698415\right)\right)\right)\right)} \]
8. Add Preprocessing

Alternative 4: 51.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 + 0.004166666666666667 \cdot \left(x \cdot x\right)\right) + x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (* (* x (* x x)) (+ 0.041666666666666664 (* 0.004166666666666667 (* x x))))
  (* x 0.5)))

C

double code(double x) {
	return ((x * (x * x)) * (0.041666666666666664 + (0.004166666666666667 * (x * x)))) + (x * 0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * (x * x)) * (0.041666666666666664d0 + (0.004166666666666667d0 * (x * x)))) + (x * 0.5d0)
end function

Java

public static double code(double x) {
	return ((x * (x * x)) * (0.041666666666666664 + (0.004166666666666667 * (x * x)))) + (x * 0.5);
}

Python

def code(x):
	return ((x * (x * x)) * (0.041666666666666664 + (0.004166666666666667 * (x * x)))) + (x * 0.5)

Julia

function code(x)
	return Float64(Float64(Float64(x * Float64(x * x)) * Float64(0.041666666666666664 + Float64(0.004166666666666667 * Float64(x * x)))) + Float64(x * 0.5))
end

MATLAB

function tmp = code(x)
	tmp = ((x * (x * x)) * (0.041666666666666664 + (0.004166666666666667 * (x * x)))) + (x * 0.5);
end

Wolfram

code[x_] := N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(0.004166666666666667 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

   \[\frac{1 - \cos x}{\sin x} \]
2. Step-by-step derivation

   1. hang-p0-tan N/A

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

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

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

   3. /-lowering-/.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{240} \cdot {x}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{240} \cdot {x}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{240} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{240}}\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(x \cdot x\right) \cdot \frac{1}{240}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{240}\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{240}\right)}\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 53.2%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{240}}\right)\right)\right)\right)\right)\right) \]

7. Simplified 53.2%

   \[\leadsto \color{blue}{x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.004166666666666667\right)\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{240}\right)\right) + \color{blue}{\frac{1}{2}}\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{240}\right)\right)\right) \cdot x + \color{blue}{\frac{1}{2} \cdot x} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{240}\right)\right)\right) \cdot x\right), \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{240}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot x\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{240}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot x\right)\right) \]

   6. pow3 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{240}\right)\right) \cdot {x}^{3}\right), \left(\frac{1}{2} \cdot x\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{24} + x \cdot \left(x \cdot \frac{1}{240}\right)\right), \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot x\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \left(x \cdot \frac{1}{240}\right)\right)\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(\left(x \cdot x\right) \cdot \frac{1}{240}\right)\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{240}\right)\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{240}\right)\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   12. cube-unmult N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{240}\right)\right), \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{240}\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{240}\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   15. *-lowering-*.f64 53.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{240}\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right)\right) \]

9. Applied egg-rr 53.3%

   \[\leadsto \color{blue}{\left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.004166666666666667\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + 0.5 \cdot x} \]

10. Final simplification 53.3%

    \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 + 0.004166666666666667 \cdot \left(x \cdot x\right)\right) + x \cdot 0.5 \]
11. Add Preprocessing

Alternative 5: 51.8% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.004166666666666667\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   0.5
   (* (* x x) (+ 0.041666666666666664 (* x (* x 0.004166666666666667)))))))

C

double code(double x) {
	return x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.004166666666666667)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.5d0 + ((x * x) * (0.041666666666666664d0 + (x * (x * 0.004166666666666667d0)))))
end function

Java

public static double code(double x) {
	return x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.004166666666666667)))));
}

Python

def code(x):
	return x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.004166666666666667)))))

Julia

function code(x)
	return Float64(x * Float64(0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(x * Float64(x * 0.004166666666666667))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.004166666666666667)))));
end

Wolfram

code[x_] := N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(x * N[(x * 0.004166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

   \[\frac{1 - \cos x}{\sin x} \]
2. Step-by-step derivation

   1. hang-p0-tan N/A

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

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

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

   3. /-lowering-/.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{240} \cdot {x}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{240} \cdot {x}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{240} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{240}}\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(x \cdot x\right) \cdot \frac{1}{240}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{240}\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{240}\right)}\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 53.2%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{240}}\right)\right)\right)\right)\right)\right) \]

7. Simplified 53.2%

   \[\leadsto \color{blue}{x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.004166666666666667\right)\right)\right)} \]
8. Add Preprocessing

Alternative 6: 51.8% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 + x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (* x 0.5) (* x (* x (* x 0.041666666666666664)))))

C

double code(double x) {
	return (x * 0.5) + (x * (x * (x * 0.041666666666666664)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 0.5d0) + (x * (x * (x * 0.041666666666666664d0)))
end function

Java

public static double code(double x) {
	return (x * 0.5) + (x * (x * (x * 0.041666666666666664)));
}

Python

def code(x):
	return (x * 0.5) + (x * (x * (x * 0.041666666666666664)))

Julia

function code(x)
	return Float64(Float64(x * 0.5) + Float64(x * Float64(x * Float64(x * 0.041666666666666664))))
end

MATLAB

function tmp = code(x)
	tmp = (x * 0.5) + (x * (x * (x * 0.041666666666666664)));
end

Wolfram

code[x_] := N[(N[(x * 0.5), $MachinePrecision] + N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

   \[\frac{1 - \cos x}{\sin x} \]
2. Step-by-step derivation

   1. hang-p0-tan N/A

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

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

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

   3. /-lowering-/.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 53.2%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

7. Simplified 53.2%

   \[\leadsto \color{blue}{x \cdot \left(0.5 + 0.041666666666666664 \cdot \left(x \cdot x\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \left(\frac{1}{24} \cdot \left(x \cdot x\right) + \color{blue}{\frac{1}{2}}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto x \cdot \left(\frac{1}{24} \cdot \left(x \cdot x\right)\right) + \color{blue}{x \cdot \frac{1}{2}} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\frac{1}{24} \cdot \left(x \cdot x\right)\right) + \frac{1}{2} \cdot \color{blue}{x} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{24} \cdot \left(x \cdot x\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(x \cdot x\right)\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot x\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{24} \cdot x\right) \cdot x\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{24}\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   11. *-lowering-*.f64 53.2%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right)\right) \]

9. Applied egg-rr 53.2%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right) + 0.5 \cdot x} \]

10. Final simplification 53.2%

    \[\leadsto x \cdot 0.5 + x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right) \]
11. Add Preprocessing

Alternative 7: 51.8% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.5 + 0.041666666666666664 \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ 0.5 (* 0.041666666666666664 (* x x)))))

C

double code(double x) {
	return x * (0.5 + (0.041666666666666664 * (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.5d0 + (0.041666666666666664d0 * (x * x)))
end function

Java

public static double code(double x) {
	return x * (0.5 + (0.041666666666666664 * (x * x)));
}

Python

def code(x):
	return x * (0.5 + (0.041666666666666664 * (x * x)))

Julia

function code(x)
	return Float64(x * Float64(0.5 + Float64(0.041666666666666664 * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.5 + (0.041666666666666664 * (x * x)));
end

Wolfram

code[x_] := N[(x * N[(0.5 + N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

   \[\frac{1 - \cos x}{\sin x} \]
2. Step-by-step derivation

   1. hang-p0-tan N/A

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

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

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

   3. /-lowering-/.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 53.2%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

7. Simplified 53.2%

   \[\leadsto \color{blue}{x \cdot \left(0.5 + 0.041666666666666664 \cdot \left(x \cdot x\right)\right)} \]
8. Add Preprocessing

Alternative 8: 51.9% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * 0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.5%

   \[\frac{1 - \cos x}{\sin x} \]
2. Step-by-step derivation

   1. hang-p0-tan N/A

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

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

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

   3. /-lowering-/.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 53.2%

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

7. Simplified 53.2%

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

8. Final simplification 53.2%

   \[\leadsto x \cdot 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024155 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64

  :alt
  (! :herbie-platform default (tan (/ x 2)))

  (/ (- 1.0 (cos x)) (sin x)))