ENA, Section 1.4, Exercise 4a

Percentage Accurate: 53.3% → 99.6%

Time: 13.8s

Alternatives: 10

Speedup: 41.0×

Specification

\[-1 \leq x \land x \leq 1\]

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (- x (sin x)) (tan x)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(Float64(x - sin(x)) / tan(x))
end

MATLAB

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

Wolfram

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

Initial Program: 53.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (- x (sin x)) (tan x)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(Float64(x - sin(x)) / tan(x))
end

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\\ t_1 := x \cdot t\_0\\ x \cdot \frac{x \cdot \left(0.004629629629629629 + t\_1 \cdot \left(\left(x \cdot x\right) \cdot \left(t\_0 \cdot t\_0\right)\right)\right)}{0.027777777777777776 + t\_1 \cdot \left(t\_1 - 0.16666666666666666\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          x
          (+
           -0.06388888888888888
           (*
            (* x x)
            (+ -0.0007275132275132275 (* (* x x) -0.00023644179894179894))))))
        (t_1 (* x t_0)))
   (*
    x
    (/
     (* x (+ 0.004629629629629629 (* t_1 (* (* x x) (* t_0 t_0)))))
     (+ 0.027777777777777776 (* t_1 (- t_1 0.16666666666666666)))))))

C

double code(double x) {
	double t_0 = x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))));
	double t_1 = x * t_0;
	return x * ((x * (0.004629629629629629 + (t_1 * ((x * x) * (t_0 * t_0))))) / (0.027777777777777776 + (t_1 * (t_1 - 0.16666666666666666))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = x * ((-0.06388888888888888d0) + ((x * x) * ((-0.0007275132275132275d0) + ((x * x) * (-0.00023644179894179894d0)))))
    t_1 = x * t_0
    code = x * ((x * (0.004629629629629629d0 + (t_1 * ((x * x) * (t_0 * t_0))))) / (0.027777777777777776d0 + (t_1 * (t_1 - 0.16666666666666666d0))))
end function

Java

public static double code(double x) {
	double t_0 = x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))));
	double t_1 = x * t_0;
	return x * ((x * (0.004629629629629629 + (t_1 * ((x * x) * (t_0 * t_0))))) / (0.027777777777777776 + (t_1 * (t_1 - 0.16666666666666666))));
}

Python

def code(x):
	t_0 = x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))))
	t_1 = x * t_0
	return x * ((x * (0.004629629629629629 + (t_1 * ((x * x) * (t_0 * t_0))))) / (0.027777777777777776 + (t_1 * (t_1 - 0.16666666666666666))))

Julia

function code(x)
	t_0 = Float64(x * Float64(-0.06388888888888888 + Float64(Float64(x * x) * Float64(-0.0007275132275132275 + Float64(Float64(x * x) * -0.00023644179894179894)))))
	t_1 = Float64(x * t_0)
	return Float64(x * Float64(Float64(x * Float64(0.004629629629629629 + Float64(t_1 * Float64(Float64(x * x) * Float64(t_0 * t_0))))) / Float64(0.027777777777777776 + Float64(t_1 * Float64(t_1 - 0.16666666666666666)))))
end

MATLAB

function tmp = code(x)
	t_0 = x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))));
	t_1 = x * t_0;
	tmp = x * ((x * (0.004629629629629629 + (t_1 * ((x * x) * (t_0 * t_0))))) / (0.027777777777777776 + (t_1 * (t_1 - 0.16666666666666666))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(-0.06388888888888888 + N[(N[(x * x), $MachinePrecision] * N[(-0.0007275132275132275 + N[(N[(x * x), $MachinePrecision] * -0.00023644179894179894), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, N[(x * N[(N[(x * N[(0.004629629629629629 + N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.027777777777777776 + N[(t$95$1 * N[(t$95$1 - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 51.8%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right)} - \frac{23}{360}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)}\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} + x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \color{blue}{x}\right)\right) \]

   2. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{{\frac{1}{6}}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)}^{3}}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) - \frac{1}{6} \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)\right)} \cdot x\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left({\frac{1}{6}}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)}^{3}\right) \cdot x}{\color{blue}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) - \frac{1}{6} \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left({\frac{1}{6}}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)}^{3}\right) \cdot x\right), \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) - \frac{1}{6} \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)\right)\right)}\right)\right) \]

7. Applied egg-rr 99.4%

   \[\leadsto x \cdot \color{blue}{\frac{\left(0.004629629629629629 + \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right) \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right)\right)\right) \cdot x}{0.027777777777777776 + \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right) - 0.16666666666666666\right)}} \]

8. Final simplification 99.4%

   \[\leadsto x \cdot \frac{x \cdot \left(0.004629629629629629 + \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right) \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right)\right)\right)}{0.027777777777777776 + \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right) - 0.16666666666666666\right)} \]
9. Add Preprocessing

Alternative 2: 99.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\\ x \cdot \frac{x \cdot \left(0.027777777777777776 - \left(x \cdot x\right) \cdot \left(t\_0 \cdot t\_0\right)\right)}{0.16666666666666666 - x \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          x
          (+
           -0.06388888888888888
           (*
            (* x x)
            (+ -0.0007275132275132275 (* (* x x) -0.00023644179894179894)))))))
   (*
    x
    (/
     (* x (- 0.027777777777777776 (* (* x x) (* t_0 t_0))))
     (- 0.16666666666666666 (* x t_0))))))

C

double code(double x) {
	double t_0 = x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))));
	return x * ((x * (0.027777777777777776 - ((x * x) * (t_0 * t_0)))) / (0.16666666666666666 - (x * t_0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = x * ((-0.06388888888888888d0) + ((x * x) * ((-0.0007275132275132275d0) + ((x * x) * (-0.00023644179894179894d0)))))
    code = x * ((x * (0.027777777777777776d0 - ((x * x) * (t_0 * t_0)))) / (0.16666666666666666d0 - (x * t_0)))
end function

Java

public static double code(double x) {
	double t_0 = x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))));
	return x * ((x * (0.027777777777777776 - ((x * x) * (t_0 * t_0)))) / (0.16666666666666666 - (x * t_0)));
}

Python

def code(x):
	t_0 = x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))))
	return x * ((x * (0.027777777777777776 - ((x * x) * (t_0 * t_0)))) / (0.16666666666666666 - (x * t_0)))

Julia

function code(x)
	t_0 = Float64(x * Float64(-0.06388888888888888 + Float64(Float64(x * x) * Float64(-0.0007275132275132275 + Float64(Float64(x * x) * -0.00023644179894179894)))))
	return Float64(x * Float64(Float64(x * Float64(0.027777777777777776 - Float64(Float64(x * x) * Float64(t_0 * t_0)))) / Float64(0.16666666666666666 - Float64(x * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))));
	tmp = x * ((x * (0.027777777777777776 - ((x * x) * (t_0 * t_0)))) / (0.16666666666666666 - (x * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(-0.06388888888888888 + N[(N[(x * x), $MachinePrecision] * N[(-0.0007275132275132275 + N[(N[(x * x), $MachinePrecision] * -0.00023644179894179894), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x * N[(N[(x * N[(0.027777777777777776 - N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.16666666666666666 - N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 51.8%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right)} - \frac{23}{360}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)}\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} + x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \color{blue}{x}\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)}{\frac{1}{6} - x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)} \cdot x\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)\right) \cdot x}{\color{blue}{\frac{1}{6} - x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)\right) \cdot x\right), \color{blue}{\left(\frac{1}{6} - x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)}\right)\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto x \cdot \color{blue}{\frac{\left(0.027777777777777776 - \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right) \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right)\right) \cdot x}{0.16666666666666666 - x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)}} \]

8. Final simplification 99.3%

   \[\leadsto x \cdot \frac{x \cdot \left(0.027777777777777776 - \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right) \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right)\right)}{0.16666666666666666 - x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)} \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \left(x \cdot x\right) \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (*
   x
   (*
    (+
     -0.06388888888888888
     (*
      (* x x)
      (+ -0.0007275132275132275 (* (* x x) -0.00023644179894179894))))
    (* x (* x x))))
  (* (* x x) 0.16666666666666666)))

C

double code(double x) {
	return (x * ((-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894)))) * (x * (x * x)))) + ((x * x) * 0.16666666666666666);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (((-0.06388888888888888d0) + ((x * x) * ((-0.0007275132275132275d0) + ((x * x) * (-0.00023644179894179894d0))))) * (x * (x * x)))) + ((x * x) * 0.16666666666666666d0)
end function

Java

public static double code(double x) {
	return (x * ((-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894)))) * (x * (x * x)))) + ((x * x) * 0.16666666666666666);
}

Python

def code(x):
	return (x * ((-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894)))) * (x * (x * x)))) + ((x * x) * 0.16666666666666666)

Julia

function code(x)
	return Float64(Float64(x * Float64(Float64(-0.06388888888888888 + Float64(Float64(x * x) * Float64(-0.0007275132275132275 + Float64(Float64(x * x) * -0.00023644179894179894)))) * Float64(x * Float64(x * x)))) + Float64(Float64(x * x) * 0.16666666666666666))
end

MATLAB

function tmp = code(x)
	tmp = (x * ((-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894)))) * (x * (x * x)))) + ((x * x) * 0.16666666666666666);
end

Wolfram

code[x_] := N[(N[(x * N[(N[(-0.06388888888888888 + N[(N[(x * x), $MachinePrecision] * N[(-0.0007275132275132275 + N[(N[(x * x), $MachinePrecision] * -0.00023644179894179894), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.8%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right)} - \frac{23}{360}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)}\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{6} + x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right) + \color{blue}{\frac{1}{6}}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot x\right) + \color{blue}{\frac{1}{6} \cdot \left(x \cdot x\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right)}\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right) \cdot \left(x \cdot x\right) + 0.16666666666666666 \cdot \left(x \cdot x\right)} \]
8. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{6}}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot x\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{6}}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right) \cdot \left(x \cdot x\right)\right), x\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{6}}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right) \cdot x\right) \cdot \left(x \cdot x\right)\right), x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right), \left(x \cdot \left(x \cdot x\right)\right)\right), x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \left(\left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right), \left(x \cdot \left(x \cdot x\right)\right)\right), x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right), \left(x \cdot \left(x \cdot x\right)\right)\right), x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right), \left(x \cdot \left(x \cdot x\right)\right)\right), x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-11}{15120}, \left(\left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right), \left(x \cdot \left(x \cdot x\right)\right)\right), x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-11}{15120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-143}{604800}\right)\right)\right)\right), \left(x \cdot \left(x \cdot x\right)\right)\right), x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-11}{15120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-143}{604800}\right)\right)\right)\right), \left(x \cdot \left(x \cdot x\right)\right)\right), x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-11}{15120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-143}{604800}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   14. *-lowering-*.f64 99.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-11}{15120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-143}{604800}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

9. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(\left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x} + 0.16666666666666666 \cdot \left(x \cdot x\right) \]

10. Final simplification 99.3%

    \[\leadsto x \cdot \left(\left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \left(x \cdot x\right) \cdot 0.16666666666666666 \]
11. Add Preprocessing

Alternative 4: 99.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot 0.16666666666666666 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (* (* x x) 0.16666666666666666)
  (*
   (* x x)
   (*
    x
    (*
     x
     (+
      -0.06388888888888888
      (*
       (* x x)
       (+ -0.0007275132275132275 (* (* x x) -0.00023644179894179894)))))))))

C

double code(double x) {
	return ((x * x) * 0.16666666666666666) + ((x * x) * (x * (x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) * 0.16666666666666666d0) + ((x * x) * (x * (x * ((-0.06388888888888888d0) + ((x * x) * ((-0.0007275132275132275d0) + ((x * x) * (-0.00023644179894179894d0))))))))
end function

Java

public static double code(double x) {
	return ((x * x) * 0.16666666666666666) + ((x * x) * (x * (x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894)))))));
}

Python

def code(x):
	return ((x * x) * 0.16666666666666666) + ((x * x) * (x * (x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894)))))))

Julia

function code(x)
	return Float64(Float64(Float64(x * x) * 0.16666666666666666) + Float64(Float64(x * x) * Float64(x * Float64(x * Float64(-0.06388888888888888 + Float64(Float64(x * x) * Float64(-0.0007275132275132275 + Float64(Float64(x * x) * -0.00023644179894179894))))))))
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) * 0.16666666666666666) + ((x * x) * (x * (x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894)))))));
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(-0.06388888888888888 + N[(N[(x * x), $MachinePrecision] * N[(-0.0007275132275132275 + N[(N[(x * x), $MachinePrecision] * -0.00023644179894179894), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.8%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right)} - \frac{23}{360}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)}\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{6} + x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right) + \color{blue}{\frac{1}{6}}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot x\right) + \color{blue}{\frac{1}{6} \cdot \left(x \cdot x\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right)}\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right) \cdot \left(x \cdot x\right) + 0.16666666666666666 \cdot \left(x \cdot x\right)} \]

8. Final simplification 99.3%

   \[\leadsto \left(x \cdot x\right) \cdot 0.16666666666666666 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 5: 99.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right) + 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (+
    (*
     x
     (*
      x
      (+
       -0.06388888888888888
       (*
        (* x x)
        (+ -0.0007275132275132275 (* (* x x) -0.00023644179894179894))))))
    0.16666666666666666))))

C

double code(double x) {
	return x * (x * ((x * (x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894)))))) + 0.16666666666666666));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * ((x * (x * ((-0.06388888888888888d0) + ((x * x) * ((-0.0007275132275132275d0) + ((x * x) * (-0.00023644179894179894d0))))))) + 0.16666666666666666d0))
end function

Java

public static double code(double x) {
	return x * (x * ((x * (x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894)))))) + 0.16666666666666666));
}

Python

def code(x):
	return x * (x * ((x * (x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894)))))) + 0.16666666666666666))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(Float64(x * Float64(x * Float64(-0.06388888888888888 + Float64(Float64(x * x) * Float64(-0.0007275132275132275 + Float64(Float64(x * x) * -0.00023644179894179894)))))) + 0.16666666666666666)))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * ((x * (x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894)))))) + 0.16666666666666666));
end

Wolfram

code[x_] := N[(x * N[(x * N[(N[(x * N[(x * N[(-0.06388888888888888 + N[(N[(x * x), $MachinePrecision] * N[(-0.0007275132275132275 + N[(N[(x * x), $MachinePrecision] * -0.00023644179894179894), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.8%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right)} - \frac{23}{360}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)}\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right)\right)} \]

6. Final simplification 99.3%

   \[\leadsto x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right) + 0.16666666666666666\right)\right) \]
7. Add Preprocessing

Alternative 6: 99.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot -0.0007275132275132275\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (+
    0.16666666666666666
    (* (* x x) (+ -0.06388888888888888 (* (* x x) -0.0007275132275132275)))))))

C

double code(double x) {
	return x * (x * (0.16666666666666666 + ((x * x) * (-0.06388888888888888 + ((x * x) * -0.0007275132275132275)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (0.16666666666666666d0 + ((x * x) * ((-0.06388888888888888d0) + ((x * x) * (-0.0007275132275132275d0))))))
end function

Java

public static double code(double x) {
	return x * (x * (0.16666666666666666 + ((x * x) * (-0.06388888888888888 + ((x * x) * -0.0007275132275132275)))));
}

Python

def code(x):
	return x * (x * (0.16666666666666666 + ((x * x) * (-0.06388888888888888 + ((x * x) * -0.0007275132275132275)))))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(0.16666666666666666 + Float64(Float64(x * x) * Float64(-0.06388888888888888 + Float64(Float64(x * x) * -0.0007275132275132275))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (0.16666666666666666 + ((x * x) * (-0.06388888888888888 + ((x * x) * -0.0007275132275132275)))));
end

Wolfram

code[x_] := N[(x * N[(x * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(-0.06388888888888888 + N[(N[(x * x), $MachinePrecision] * -0.0007275132275132275), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.8%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right)} - \frac{23}{360}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)}\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{-11}{15120} \cdot {x}^{2}} - \frac{23}{360}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{-11}{15120} \cdot {x}^{2}} - \frac{23}{360}\right)\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-11}{15120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-11}{15120} \cdot {x}^{2} + \frac{-23}{360}\right)\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-23}{360} + \color{blue}{\frac{-11}{15120} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-23}{360}, \color{blue}{\left(\frac{-11}{15120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-23}{360}, \left({x}^{2} \cdot \color{blue}{\frac{-11}{15120}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-11}{15120}}\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-11}{15120}\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 99.1%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-11}{15120}\right)\right)\right)\right)\right)\right) \]

8. Simplified 99.1%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot -0.0007275132275132275\right)\right)\right)} \]
9. Add Preprocessing

Alternative 7: 99.3% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot 0.16666666666666666 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.06388888888888888\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (* (* x x) 0.16666666666666666)
  (* (* x x) (* x (* x -0.06388888888888888)))))

C

double code(double x) {
	return ((x * x) * 0.16666666666666666) + ((x * x) * (x * (x * -0.06388888888888888)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) * 0.16666666666666666d0) + ((x * x) * (x * (x * (-0.06388888888888888d0))))
end function

Java

public static double code(double x) {
	return ((x * x) * 0.16666666666666666) + ((x * x) * (x * (x * -0.06388888888888888)));
}

Python

def code(x):
	return ((x * x) * 0.16666666666666666) + ((x * x) * (x * (x * -0.06388888888888888)))

Julia

function code(x)
	return Float64(Float64(Float64(x * x) * 0.16666666666666666) + Float64(Float64(x * x) * Float64(x * Float64(x * -0.06388888888888888))))
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) * 0.16666666666666666) + ((x * x) * (x * (x * -0.06388888888888888)));
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.06388888888888888), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.8%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right)} - \frac{23}{360}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)}\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{6} + x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right) + \color{blue}{\frac{1}{6}}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot x\right) + \color{blue}{\frac{1}{6} \cdot \left(x \cdot x\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right)}\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right) \cdot \left(x \cdot x\right) + 0.16666666666666666 \cdot \left(x \cdot x\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{-23}{360} \cdot {x}^{2}\right)}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({x}^{2} \cdot \frac{-23}{360}\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \frac{-23}{360}\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{-23}{360} \cdot x\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{-23}{360} \cdot x\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \frac{-23}{360}\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   7. *-lowering-*.f64 98.8%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-23}{360}\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

10. Simplified 98.8%

    \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot -0.06388888888888888\right)\right)} \cdot \left(x \cdot x\right) + 0.16666666666666666 \cdot \left(x \cdot x\right) \]

11. Final simplification 98.8%

    \[\leadsto \left(x \cdot x\right) \cdot 0.16666666666666666 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.06388888888888888\right)\right) \]
12. Add Preprocessing

Alternative 8: 99.4% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(0.16666666666666666 + -0.06388888888888888 \cdot \left(x \cdot x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (* x (+ 0.16666666666666666 (* -0.06388888888888888 (* x x))))))

C

double code(double x) {
	return x * (x * (0.16666666666666666 + (-0.06388888888888888 * (x * x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (0.16666666666666666d0 + ((-0.06388888888888888d0) * (x * x))))
end function

Java

public static double code(double x) {
	return x * (x * (0.16666666666666666 + (-0.06388888888888888 * (x * x))));
}

Python

def code(x):
	return x * (x * (0.16666666666666666 + (-0.06388888888888888 * (x * x))))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(0.16666666666666666 + Float64(-0.06388888888888888 * Float64(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (0.16666666666666666 + (-0.06388888888888888 * (x * x))));
end

Wolfram

code[x_] := N[(x * N[(x * N[(0.16666666666666666 + N[(-0.06388888888888888 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.8%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right)} - \frac{23}{360}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)}\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)}\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-23}{360} \cdot {x}^{2}\right)}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{-23}{360}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-23}{360}}\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-23}{360}\right)\right)\right)\right) \]

   6. *-lowering-*.f64 98.8%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-23}{360}\right)\right)\right)\right) \]

8. Simplified 98.8%

   \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.06388888888888888\right)\right)} \]

9. Final simplification 98.8%

   \[\leadsto x \cdot \left(x \cdot \left(0.16666666666666666 + -0.06388888888888888 \cdot \left(x \cdot x\right)\right)\right) \]
10. Add Preprocessing

Alternative 9: 98.7% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x \cdot 0.004629629629629629}{0.027777777777777776} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (/ (* x 0.004629629629629629) 0.027777777777777776)))

C

double code(double x) {
	return x * ((x * 0.004629629629629629) / 0.027777777777777776);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * ((x * 0.004629629629629629d0) / 0.027777777777777776d0)
end function

Java

public static double code(double x) {
	return x * ((x * 0.004629629629629629) / 0.027777777777777776);
}

Python

def code(x):
	return x * ((x * 0.004629629629629629) / 0.027777777777777776)

Julia

function code(x)
	return Float64(x * Float64(Float64(x * 0.004629629629629629) / 0.027777777777777776))
end

MATLAB

function tmp = code(x)
	tmp = x * ((x * 0.004629629629629629) / 0.027777777777777776);
end

Wolfram

code[x_] := N[(x * N[(N[(x * 0.004629629629629629), $MachinePrecision] / 0.027777777777777776), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.8%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right)} - \frac{23}{360}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)}\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} + x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \color{blue}{x}\right)\right) \]

   2. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{{\frac{1}{6}}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)}^{3}}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) - \frac{1}{6} \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)\right)} \cdot x\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left({\frac{1}{6}}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)}^{3}\right) \cdot x}{\color{blue}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) - \frac{1}{6} \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left({\frac{1}{6}}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)}^{3}\right) \cdot x\right), \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) - \frac{1}{6} \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)\right)\right)}\right)\right) \]

7. Applied egg-rr 99.4%

   \[\leadsto x \cdot \color{blue}{\frac{\left(0.004629629629629629 + \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right) \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right)\right)\right) \cdot x}{0.027777777777777776 + \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right) - 0.16666666666666666\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{216} \cdot x\right)}, \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-11}{15120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-143}{604800}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-11}{15120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-143}{604800}\right)\right)\right)\right)\right)\right), \frac{1}{6}\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(x \cdot \frac{1}{216}\right), \mathsf{+.f64}\left(\color{blue}{\frac{1}{36}}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-11}{15120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-143}{604800}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-11}{15120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-143}{604800}\right)\right)\right)\right)\right)\right), \frac{1}{6}\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 99.0%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{216}\right), \mathsf{+.f64}\left(\color{blue}{\frac{1}{36}}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-11}{15120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-143}{604800}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-11}{15120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-143}{604800}\right)\right)\right)\right)\right)\right), \frac{1}{6}\right)\right)\right)\right)\right) \]

10. Simplified 99.0%

    \[\leadsto x \cdot \frac{\color{blue}{x \cdot 0.004629629629629629}}{0.027777777777777776 + \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right) - 0.16666666666666666\right)} \]

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{216}\right), \color{blue}{\frac{1}{36}}\right)\right) \]
12. Step-by-step derivation

13. Simplified 98.0%

    \[\leadsto x \cdot \frac{x \cdot 0.004629629629629629}{\color{blue}{0.027777777777777776}} \]
14. Add Preprocessing

Alternative 10: 98.7% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x x) 0.16666666666666666))

C

double code(double x) {
	return (x * x) * 0.16666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * 0.16666666666666666d0
end function

Java

public static double code(double x) {
	return (x * x) * 0.16666666666666666;
}

Python

def code(x):
	return (x * x) * 0.16666666666666666

Julia

function code(x)
	return Float64(Float64(x * x) * 0.16666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * 0.16666666666666666;
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 51.8%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f64 98.0%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

5. Simplified 98.0%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot x\right)} \]

6. Final simplification 98.0%

   \[\leadsto \left(x \cdot x\right) \cdot 0.16666666666666666 \]
7. Add Preprocessing

Developer Target 1: 98.7% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ 0.16666666666666666 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.16666666666666666 (* x x)))

C

double code(double x) {
	return 0.16666666666666666 * (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.16666666666666666d0 * (x * x)
end function

Java

public static double code(double x) {
	return 0.16666666666666666 * (x * x);
}

Python

def code(x):
	return 0.16666666666666666 * (x * x)

Julia

function code(x)
	return Float64(0.16666666666666666 * Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = 0.16666666666666666 * (x * x);
end

Wolfram

code[x_] := N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024158 
(FPCore (x)
  :name "ENA, Section 1.4, Exercise 4a"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :alt
  (! :herbie-platform default (* 1/6 (* x x)))

  (/ (- x (sin x)) (tan x)))