Jmat.Real.dawson

Percentage Accurate: 54.2% → 100.0%

Time: 14.2s

Alternatives: 11

Speedup: 21.6×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := t\_0 \cdot \left(x \cdot x\right)\\ t_2 := t\_1 \cdot \left(x \cdot x\right)\\ t_3 := t\_2 \cdot \left(x \cdot x\right)\\ \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t\_0\right) + 0.0072644182 \cdot t\_1\right) + 0.0005064034 \cdot t\_2\right) + 0.0001789971 \cdot t\_3}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot t\_0\right) + 0.0694555761 \cdot t\_1\right) + 0.0140005442 \cdot t\_2\right) + 0.0008327945 \cdot t\_3\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(t\_3 \cdot \left(x \cdot x\right)\right)} \cdot x \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x)))
        (t_1 (* t_0 (* x x)))
        (t_2 (* t_1 (* x x)))
        (t_3 (* t_2 (* x x))))
   (*
    (/
     (+
      (+
       (+
        (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 t_0))
        (* 0.0072644182 t_1))
       (* 0.0005064034 t_2))
      (* 0.0001789971 t_3))
     (+
      (+
       (+
        (+
         (+ (+ 1.0 (* 0.7715471019 (* x x))) (* 0.2909738639 t_0))
         (* 0.0694555761 t_1))
        (* 0.0140005442 t_2))
       (* 0.0008327945 t_3))
      (* (* 2.0 0.0001789971) (* t_3 (* x x)))))
    x)))

C

double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = t_0 * (x * x);
	double t_2 = t_1 * (x * x);
	double t_3 = t_2 * (x * x);
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = (x * x) * (x * x)
    t_1 = t_0 * (x * x)
    t_2 = t_1 * (x * x)
    t_3 = t_2 * (x * x)
    code = ((((((1.0d0 + (0.1049934947d0 * (x * x))) + (0.0424060604d0 * t_0)) + (0.0072644182d0 * t_1)) + (0.0005064034d0 * t_2)) + (0.0001789971d0 * t_3)) / ((((((1.0d0 + (0.7715471019d0 * (x * x))) + (0.2909738639d0 * t_0)) + (0.0694555761d0 * t_1)) + (0.0140005442d0 * t_2)) + (0.0008327945d0 * t_3)) + ((2.0d0 * 0.0001789971d0) * (t_3 * (x * x))))) * x
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = t_0 * (x * x);
	double t_2 = t_1 * (x * x);
	double t_3 = t_2 * (x * x);
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
}

Python

def code(x):
	t_0 = (x * x) * (x * x)
	t_1 = t_0 * (x * x)
	t_2 = t_1 * (x * x)
	t_3 = t_2 * (x * x)
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	t_1 = Float64(t_0 * Float64(x * x))
	t_2 = Float64(t_1 * Float64(x * x))
	t_3 = Float64(t_2 * Float64(x * x))
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(0.0424060604 * t_0)) + Float64(0.0072644182 * t_1)) + Float64(0.0005064034 * t_2)) + Float64(0.0001789971 * t_3)) / Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.7715471019 * Float64(x * x))) + Float64(0.2909738639 * t_0)) + Float64(0.0694555761 * t_1)) + Float64(0.0140005442 * t_2)) + Float64(0.0008327945 * t_3)) + Float64(Float64(2.0 * 0.0001789971) * Float64(t_3 * Float64(x * x))))) * x)
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * (x * x);
	t_1 = t_0 * (x * x);
	t_2 = t_1 * (x * x);
	t_3 = t_2 * (x * x);
	tmp = ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 + N[(0.7715471019 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(t$95$3 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]

Initial Program: 54.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := t\_0 \cdot \left(x \cdot x\right)\\ t_2 := t\_1 \cdot \left(x \cdot x\right)\\ t_3 := t\_2 \cdot \left(x \cdot x\right)\\ \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t\_0\right) + 0.0072644182 \cdot t\_1\right) + 0.0005064034 \cdot t\_2\right) + 0.0001789971 \cdot t\_3}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot t\_0\right) + 0.0694555761 \cdot t\_1\right) + 0.0140005442 \cdot t\_2\right) + 0.0008327945 \cdot t\_3\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(t\_3 \cdot \left(x \cdot x\right)\right)} \cdot x \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x)))
        (t_1 (* t_0 (* x x)))
        (t_2 (* t_1 (* x x)))
        (t_3 (* t_2 (* x x))))
   (*
    (/
     (+
      (+
       (+
        (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 t_0))
        (* 0.0072644182 t_1))
       (* 0.0005064034 t_2))
      (* 0.0001789971 t_3))
     (+
      (+
       (+
        (+
         (+ (+ 1.0 (* 0.7715471019 (* x x))) (* 0.2909738639 t_0))
         (* 0.0694555761 t_1))
        (* 0.0140005442 t_2))
       (* 0.0008327945 t_3))
      (* (* 2.0 0.0001789971) (* t_3 (* x x)))))
    x)))

C

double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = t_0 * (x * x);
	double t_2 = t_1 * (x * x);
	double t_3 = t_2 * (x * x);
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = (x * x) * (x * x)
    t_1 = t_0 * (x * x)
    t_2 = t_1 * (x * x)
    t_3 = t_2 * (x * x)
    code = ((((((1.0d0 + (0.1049934947d0 * (x * x))) + (0.0424060604d0 * t_0)) + (0.0072644182d0 * t_1)) + (0.0005064034d0 * t_2)) + (0.0001789971d0 * t_3)) / ((((((1.0d0 + (0.7715471019d0 * (x * x))) + (0.2909738639d0 * t_0)) + (0.0694555761d0 * t_1)) + (0.0140005442d0 * t_2)) + (0.0008327945d0 * t_3)) + ((2.0d0 * 0.0001789971d0) * (t_3 * (x * x))))) * x
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = t_0 * (x * x);
	double t_2 = t_1 * (x * x);
	double t_3 = t_2 * (x * x);
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
}

Python

def code(x):
	t_0 = (x * x) * (x * x)
	t_1 = t_0 * (x * x)
	t_2 = t_1 * (x * x)
	t_3 = t_2 * (x * x)
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	t_1 = Float64(t_0 * Float64(x * x))
	t_2 = Float64(t_1 * Float64(x * x))
	t_3 = Float64(t_2 * Float64(x * x))
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(0.0424060604 * t_0)) + Float64(0.0072644182 * t_1)) + Float64(0.0005064034 * t_2)) + Float64(0.0001789971 * t_3)) / Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.7715471019 * Float64(x * x))) + Float64(0.2909738639 * t_0)) + Float64(0.0694555761 * t_1)) + Float64(0.0140005442 * t_2)) + Float64(0.0008327945 * t_3)) + Float64(Float64(2.0 * 0.0001789971) * Float64(t_3 * Float64(x * x))))) * x)
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * (x * x);
	t_1 = t_0 * (x * x);
	t_2 = t_1 * (x * x);
	t_3 = t_2 * (x * x);
	tmp = ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 + N[(0.7715471019 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(t$95$3 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]

Alternative 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 60000000:\\ \;\;\;\;\frac{x\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.1049934947 + \left(x\_m \cdot x\_m\right) \cdot \left(\left(0.0424060604 + x\_m \cdot \left(x\_m \cdot 0.0072644182\right)\right) + x\_m \cdot \left(t\_0 \cdot \left(0.0005064034 + x\_m \cdot \left(x\_m \cdot 0.0001789971\right)\right)\right)\right)\right)\right)}{1 + x\_m \cdot \left(x\_m \cdot \left(0.7715471019 + \left(\left(x\_m \cdot x\_m\right) \cdot 0.2909738639 + \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(0.0694555761 + x\_m \cdot \left(x\_m \cdot 0.0140005442\right)\right) + \left(\left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot t\_0\right)\right) \cdot \left(0.0008327945 + x\_m \cdot \left(x\_m \cdot 0.0003579942\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* x_m (* x_m x_m))))
   (*
    x_s
    (if (<= x_m 60000000.0)
      (/
       (*
        x_m
        (+
         1.0
         (*
          (* x_m x_m)
          (+
           0.1049934947
           (*
            (* x_m x_m)
            (+
             (+ 0.0424060604 (* x_m (* x_m 0.0072644182)))
             (*
              x_m
              (* t_0 (+ 0.0005064034 (* x_m (* x_m 0.0001789971)))))))))))
       (+
        1.0
        (*
         x_m
         (*
          x_m
          (+
           0.7715471019
           (+
            (* (* x_m x_m) 0.2909738639)
            (*
             (* x_m x_m)
             (+
              (* (* x_m x_m) (+ 0.0694555761 (* x_m (* x_m 0.0140005442))))
              (*
               (* (* x_m x_m) (* x_m t_0))
               (+ 0.0008327945 (* x_m (* x_m 0.0003579942))))))))))))
      (/ 0.5 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = x_m * (x_m * x_m);
	double tmp;
	if (x_m <= 60000000.0) {
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * ((0.0424060604 + (x_m * (x_m * 0.0072644182))) + (x_m * (t_0 * (0.0005064034 + (x_m * (x_m * 0.0001789971))))))))))) / (1.0 + (x_m * (x_m * (0.7715471019 + (((x_m * x_m) * 0.2909738639) + ((x_m * x_m) * (((x_m * x_m) * (0.0694555761 + (x_m * (x_m * 0.0140005442)))) + (((x_m * x_m) * (x_m * t_0)) * (0.0008327945 + (x_m * (x_m * 0.0003579942)))))))))));
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m * (x_m * x_m)
    if (x_m <= 60000000.0d0) then
        tmp = (x_m * (1.0d0 + ((x_m * x_m) * (0.1049934947d0 + ((x_m * x_m) * ((0.0424060604d0 + (x_m * (x_m * 0.0072644182d0))) + (x_m * (t_0 * (0.0005064034d0 + (x_m * (x_m * 0.0001789971d0))))))))))) / (1.0d0 + (x_m * (x_m * (0.7715471019d0 + (((x_m * x_m) * 0.2909738639d0) + ((x_m * x_m) * (((x_m * x_m) * (0.0694555761d0 + (x_m * (x_m * 0.0140005442d0)))) + (((x_m * x_m) * (x_m * t_0)) * (0.0008327945d0 + (x_m * (x_m * 0.0003579942d0)))))))))))
    else
        tmp = 0.5d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = x_m * (x_m * x_m);
	double tmp;
	if (x_m <= 60000000.0) {
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * ((0.0424060604 + (x_m * (x_m * 0.0072644182))) + (x_m * (t_0 * (0.0005064034 + (x_m * (x_m * 0.0001789971))))))))))) / (1.0 + (x_m * (x_m * (0.7715471019 + (((x_m * x_m) * 0.2909738639) + ((x_m * x_m) * (((x_m * x_m) * (0.0694555761 + (x_m * (x_m * 0.0140005442)))) + (((x_m * x_m) * (x_m * t_0)) * (0.0008327945 + (x_m * (x_m * 0.0003579942)))))))))));
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = x_m * (x_m * x_m)
	tmp = 0
	if x_m <= 60000000.0:
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * ((0.0424060604 + (x_m * (x_m * 0.0072644182))) + (x_m * (t_0 * (0.0005064034 + (x_m * (x_m * 0.0001789971))))))))))) / (1.0 + (x_m * (x_m * (0.7715471019 + (((x_m * x_m) * 0.2909738639) + ((x_m * x_m) * (((x_m * x_m) * (0.0694555761 + (x_m * (x_m * 0.0140005442)))) + (((x_m * x_m) * (x_m * t_0)) * (0.0008327945 + (x_m * (x_m * 0.0003579942)))))))))))
	else:
		tmp = 0.5 / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(x_m * Float64(x_m * x_m))
	tmp = 0.0
	if (x_m <= 60000000.0)
		tmp = Float64(Float64(x_m * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.1049934947 + Float64(Float64(x_m * x_m) * Float64(Float64(0.0424060604 + Float64(x_m * Float64(x_m * 0.0072644182))) + Float64(x_m * Float64(t_0 * Float64(0.0005064034 + Float64(x_m * Float64(x_m * 0.0001789971))))))))))) / Float64(1.0 + Float64(x_m * Float64(x_m * Float64(0.7715471019 + Float64(Float64(Float64(x_m * x_m) * 0.2909738639) + Float64(Float64(x_m * x_m) * Float64(Float64(Float64(x_m * x_m) * Float64(0.0694555761 + Float64(x_m * Float64(x_m * 0.0140005442)))) + Float64(Float64(Float64(x_m * x_m) * Float64(x_m * t_0)) * Float64(0.0008327945 + Float64(x_m * Float64(x_m * 0.0003579942))))))))))));
	else
		tmp = Float64(0.5 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = x_m * (x_m * x_m);
	tmp = 0.0;
	if (x_m <= 60000000.0)
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * ((0.0424060604 + (x_m * (x_m * 0.0072644182))) + (x_m * (t_0 * (0.0005064034 + (x_m * (x_m * 0.0001789971))))))))))) / (1.0 + (x_m * (x_m * (0.7715471019 + (((x_m * x_m) * 0.2909738639) + ((x_m * x_m) * (((x_m * x_m) * (0.0694555761 + (x_m * (x_m * 0.0140005442)))) + (((x_m * x_m) * (x_m * t_0)) * (0.0008327945 + (x_m * (x_m * 0.0003579942)))))))))));
	else
		tmp = 0.5 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 60000000.0], N[(N[(x$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.1049934947 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.0424060604 + N[(x$95$m * N[(x$95$m * 0.0072644182), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(t$95$0 * N[(0.0005064034 + N[(x$95$m * N[(x$95$m * 0.0001789971), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x$95$m * N[(x$95$m * N[(0.7715471019 + N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.2909738639), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0694555761 + N[(x$95$m * N[(x$95$m * 0.0140005442), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(0.0008327945 + N[(x$95$m * N[(x$95$m * 0.0003579942), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6e7

   1. Initial program 61.4%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   4. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.1049934947 + \left(x \cdot x\right) \cdot \left(\left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182\right)\right) + x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)\right)\right)}{1 + x \cdot \left(x \cdot \left(0.7715471019 + \left(\left(x \cdot x\right) \cdot 0.2909738639 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.0694555761 + x \cdot \left(x \cdot 0.0140005442\right)\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(0.0008327945 + x \cdot \left(x \cdot 0.0003579942\right)\right)\right)\right)\right)\right)}} \]

   if 6e7 < x

   1. Initial program 1.6%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 100.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5000:\\ \;\;\;\;\frac{x\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.1049934947 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0424060604 + x\_m \cdot \left(x\_m \cdot \left(0.0072644182 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0005064034 + \left(x\_m \cdot x\_m\right) \cdot 0.0001789971\right)\right)\right)\right)\right)\right)}{1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.7715471019 + \left(x\_m \cdot x\_m\right) \cdot \left(0.2909738639 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0694555761 + x\_m \cdot \left(x\_m \cdot 0.0140005442 + \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(0.0008327945 + \left(x\_m \cdot x\_m\right) \cdot 0.0003579942\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5000.0)
    (/
     (*
      x_m
      (+
       1.0
       (*
        (* x_m x_m)
        (+
         0.1049934947
         (*
          (* x_m x_m)
          (+
           0.0424060604
           (*
            x_m
            (*
             x_m
             (+
              0.0072644182
              (*
               (* x_m x_m)
               (+ 0.0005064034 (* (* x_m x_m) 0.0001789971))))))))))))
     (+
      1.0
      (*
       (* x_m x_m)
       (+
        0.7715471019
        (*
         (* x_m x_m)
         (+
          0.2909738639
          (*
           (* x_m x_m)
           (+
            0.0694555761
            (*
             x_m
             (+
              (* x_m 0.0140005442)
              (*
               (* x_m (* x_m x_m))
               (+ 0.0008327945 (* (* x_m x_m) 0.0003579942)))))))))))))
    (/ (+ 0.5 (/ 0.2514179000665374 (* x_m x_m))) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5000.0) {
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + (x_m * (x_m * (0.0072644182 + ((x_m * x_m) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))))))))))) / (1.0 + ((x_m * x_m) * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + (x_m * ((x_m * 0.0140005442) + ((x_m * (x_m * x_m)) * (0.0008327945 + ((x_m * x_m) * 0.0003579942))))))))))));
	} else {
		tmp = (0.5 + (0.2514179000665374 / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 5000.0d0) then
        tmp = (x_m * (1.0d0 + ((x_m * x_m) * (0.1049934947d0 + ((x_m * x_m) * (0.0424060604d0 + (x_m * (x_m * (0.0072644182d0 + ((x_m * x_m) * (0.0005064034d0 + ((x_m * x_m) * 0.0001789971d0)))))))))))) / (1.0d0 + ((x_m * x_m) * (0.7715471019d0 + ((x_m * x_m) * (0.2909738639d0 + ((x_m * x_m) * (0.0694555761d0 + (x_m * ((x_m * 0.0140005442d0) + ((x_m * (x_m * x_m)) * (0.0008327945d0 + ((x_m * x_m) * 0.0003579942d0))))))))))))
    else
        tmp = (0.5d0 + (0.2514179000665374d0 / (x_m * x_m))) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5000.0) {
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + (x_m * (x_m * (0.0072644182 + ((x_m * x_m) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))))))))))) / (1.0 + ((x_m * x_m) * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + (x_m * ((x_m * 0.0140005442) + ((x_m * (x_m * x_m)) * (0.0008327945 + ((x_m * x_m) * 0.0003579942))))))))))));
	} else {
		tmp = (0.5 + (0.2514179000665374 / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 5000.0:
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + (x_m * (x_m * (0.0072644182 + ((x_m * x_m) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))))))))))) / (1.0 + ((x_m * x_m) * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + (x_m * ((x_m * 0.0140005442) + ((x_m * (x_m * x_m)) * (0.0008327945 + ((x_m * x_m) * 0.0003579942))))))))))))
	else:
		tmp = (0.5 + (0.2514179000665374 / (x_m * x_m))) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 5000.0)
		tmp = Float64(Float64(x_m * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.1049934947 + Float64(Float64(x_m * x_m) * Float64(0.0424060604 + Float64(x_m * Float64(x_m * Float64(0.0072644182 + Float64(Float64(x_m * x_m) * Float64(0.0005064034 + Float64(Float64(x_m * x_m) * 0.0001789971)))))))))))) / Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.7715471019 + Float64(Float64(x_m * x_m) * Float64(0.2909738639 + Float64(Float64(x_m * x_m) * Float64(0.0694555761 + Float64(x_m * Float64(Float64(x_m * 0.0140005442) + Float64(Float64(x_m * Float64(x_m * x_m)) * Float64(0.0008327945 + Float64(Float64(x_m * x_m) * 0.0003579942)))))))))))));
	else
		tmp = Float64(Float64(0.5 + Float64(0.2514179000665374 / Float64(x_m * x_m))) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 5000.0)
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + (x_m * (x_m * (0.0072644182 + ((x_m * x_m) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))))))))))) / (1.0 + ((x_m * x_m) * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + (x_m * ((x_m * 0.0140005442) + ((x_m * (x_m * x_m)) * (0.0008327945 + ((x_m * x_m) * 0.0003579942))))))))))));
	else
		tmp = (0.5 + (0.2514179000665374 / (x_m * x_m))) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 5000.0], N[(N[(x$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.1049934947 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0424060604 + N[(x$95$m * N[(x$95$m * N[(0.0072644182 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0005064034 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0001789971), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.7715471019 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.2909738639 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0694555761 + N[(x$95$m * N[(N[(x$95$m * 0.0140005442), $MachinePrecision] + N[(N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.0008327945 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0003579942), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e3

   1. Initial program 61.4%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   4. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.1049934947 + \left(x \cdot x\right) \cdot \left(\left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182\right)\right) + x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)\right)\right)}{1 + x \cdot \left(x \cdot \left(0.7715471019 + \left(\left(x \cdot x\right) \cdot 0.2909738639 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.0694555761 + x \cdot \left(x \cdot 0.0140005442\right)\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(0.0008327945 + x \cdot \left(x \cdot 0.0003579942\right)\right)\right)\right)\right)\right)}} \]

   6. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182 + \left(x \cdot x\right) \cdot \left(x \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(0.7715471019 + \left(x \cdot x\right) \cdot \left(0.2909738639 + \left(x \cdot x\right) \cdot \left(\left(0.0694555761 + \left(x \cdot x\right) \cdot 0.0140005442\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(0.0008327945 + \left(x \cdot x\right) \cdot 0.0003579942\right)\right)\right)\right)\right)}} \]

   8. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + x \cdot \left(x \cdot \left(0.0072644182 + \left(x \cdot x\right) \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(0.7715471019 + \left(x \cdot x\right) \cdot \left(0.2909738639 + \left(x \cdot x\right) \cdot \left(0.0694555761 + x \cdot \left(x \cdot 0.0140005442 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0008327945 + \left(x \cdot x\right) \cdot 0.0003579942\right)\right)\right)\right)\right)}} \]

   if 5e3 < x

   1. Initial program 1.6%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{x}} \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{600041}{2386628} \cdot 1}{{x}^{2}}\right)\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{600041}{2386628}}{{x}^{2}}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{600041}{2386628}, \left({x}^{2}\right)\right)\right), x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{600041}{2386628}, \left(x \cdot x\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{600041}{2386628}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.2514179000665374}{x \cdot x}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.92:\\ \;\;\;\;\frac{x\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.1049934947 + x\_m \cdot \left(x\_m \cdot \left(0.0424060604 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0072644182 + \left(x\_m \cdot x\_m\right) \cdot 0.0005064034\right)\right)\right)\right)\right)}{1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.7715471019 + \left(x\_m \cdot x\_m\right) \cdot \left(0.2909738639 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0694555761 + \left(x\_m \cdot x\_m\right) \cdot 0.0140005442\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x\_m \cdot x\_m}}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.92)
    (/
     (*
      x_m
      (+
       1.0
       (*
        (* x_m x_m)
        (+
         0.1049934947
         (*
          x_m
          (*
           x_m
           (+
            0.0424060604
            (*
             (* x_m x_m)
             (+ 0.0072644182 (* (* x_m x_m) 0.0005064034))))))))))
     (+
      1.0
      (*
       (* x_m x_m)
       (+
        0.7715471019
        (*
         (* x_m x_m)
         (+
          0.2909738639
          (* (* x_m x_m) (+ 0.0694555761 (* (* x_m x_m) 0.0140005442)))))))))
    (/
     (+
      0.5
      (/
       (+ 0.2514179000665374 (/ 0.15298196345929074 (* x_m x_m)))
       (* x_m x_m)))
     x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.92) {
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * (0.0424060604 + ((x_m * x_m) * (0.0072644182 + ((x_m * x_m) * 0.0005064034)))))))))) / (1.0 + ((x_m * x_m) * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * 0.0140005442))))))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.92d0) then
        tmp = (x_m * (1.0d0 + ((x_m * x_m) * (0.1049934947d0 + (x_m * (x_m * (0.0424060604d0 + ((x_m * x_m) * (0.0072644182d0 + ((x_m * x_m) * 0.0005064034d0)))))))))) / (1.0d0 + ((x_m * x_m) * (0.7715471019d0 + ((x_m * x_m) * (0.2909738639d0 + ((x_m * x_m) * (0.0694555761d0 + ((x_m * x_m) * 0.0140005442d0))))))))
    else
        tmp = (0.5d0 + ((0.2514179000665374d0 + (0.15298196345929074d0 / (x_m * x_m))) / (x_m * x_m))) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.92) {
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * (0.0424060604 + ((x_m * x_m) * (0.0072644182 + ((x_m * x_m) * 0.0005064034)))))))))) / (1.0 + ((x_m * x_m) * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * 0.0140005442))))))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.92:
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * (0.0424060604 + ((x_m * x_m) * (0.0072644182 + ((x_m * x_m) * 0.0005064034)))))))))) / (1.0 + ((x_m * x_m) * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * 0.0140005442))))))))
	else:
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.92)
		tmp = Float64(Float64(x_m * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.1049934947 + Float64(x_m * Float64(x_m * Float64(0.0424060604 + Float64(Float64(x_m * x_m) * Float64(0.0072644182 + Float64(Float64(x_m * x_m) * 0.0005064034)))))))))) / Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.7715471019 + Float64(Float64(x_m * x_m) * Float64(0.2909738639 + Float64(Float64(x_m * x_m) * Float64(0.0694555761 + Float64(Float64(x_m * x_m) * 0.0140005442)))))))));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.2514179000665374 + Float64(0.15298196345929074 / Float64(x_m * x_m))) / Float64(x_m * x_m))) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.92)
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * (0.0424060604 + ((x_m * x_m) * (0.0072644182 + ((x_m * x_m) * 0.0005064034)))))))))) / (1.0 + ((x_m * x_m) * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * 0.0140005442))))))));
	else
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.92], N[(N[(x$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.1049934947 + N[(x$95$m * N[(x$95$m * N[(0.0424060604 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0072644182 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0005064034), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.7715471019 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.2909738639 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0694555761 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0140005442), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(0.2514179000665374 + N[(0.15298196345929074 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.9199999999999999

   1. Initial program 61.4%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   3. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.0424060604 + \left(x \cdot x\right) \cdot 0.0072644182\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)\right)}{1 + \left(\left(x \cdot x\right) \cdot \left(0.7715471019 + \left(x \cdot x\right) \cdot 0.2909738639\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(0.0694555761 + \left(x \cdot x\right) \cdot 0.0140005442\right)\right)\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(0.0008327945 + x \cdot \left(x \cdot 0.0003579942\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{7715471019}{10000000000} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{7715471019}{10000000000} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{7715471019}{10000000000}} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{7715471019}{10000000000}} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \color{blue}{\left({x}^{2} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{2909738639}{10000000000}} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{2909738639}{10000000000}} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \color{blue}{\left({x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{694555761}{10000000000}} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{694555761}{10000000000}} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \color{blue}{\left(\frac{70002721}{5000000000} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \left({x}^{2} \cdot \color{blue}{\frac{70002721}{5000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{70002721}{5000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 57.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 57.5%

      \[\leadsto \frac{x \cdot \left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.0424060604 + \left(x \cdot x\right) \cdot 0.0072644182\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)\right)}{1 + \color{blue}{\left(x \cdot x\right) \cdot \left(0.7715471019 + \left(x \cdot x\right) \cdot \left(0.2909738639 + \left(x \cdot x\right) \cdot \left(0.0694555761 + \left(x \cdot x\right) \cdot 0.0140005442\right)\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + {x}^{2} \cdot \left(\frac{36322091}{5000000000} + \frac{2532017}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + {x}^{2} \cdot \left(\frac{36322091}{5000000000} + \frac{2532017}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + {x}^{2} \cdot \left(\frac{36322091}{5000000000} + \frac{2532017}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + {x}^{2} \cdot \left(\frac{36322091}{5000000000} + \frac{2532017}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \left({x}^{2} \cdot \left(\frac{106015151}{2500000000} + {x}^{2} \cdot \left(\frac{36322091}{5000000000} + \frac{2532017}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \left(\left(x \cdot x\right) \cdot \left(\frac{106015151}{2500000000} + {x}^{2} \cdot \left(\frac{36322091}{5000000000} + \frac{2532017}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \left(x \cdot \left(x \cdot \left(\frac{106015151}{2500000000} + {x}^{2} \cdot \left(\frac{36322091}{5000000000} + \frac{2532017}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{106015151}{2500000000} + {x}^{2} \cdot \left(\frac{36322091}{5000000000} + \frac{2532017}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{106015151}{2500000000} + {x}^{2} \cdot \left(\frac{36322091}{5000000000} + \frac{2532017}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \left({x}^{2} \cdot \left(\frac{36322091}{5000000000} + \frac{2532017}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{36322091}{5000000000} + \frac{2532017}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{36322091}{5000000000} + \frac{2532017}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{36322091}{5000000000} + \frac{2532017}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{36322091}{5000000000}, \left(\frac{2532017}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{36322091}{5000000000}, \left({x}^{2} \cdot \frac{2532017}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{36322091}{5000000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{2532017}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{36322091}{5000000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{2532017}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 57.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{36322091}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{2532017}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 57.6%

       \[\leadsto \frac{x \cdot \left(1 + \color{blue}{\left(x \cdot x\right) \cdot \left(0.1049934947 + x \cdot \left(x \cdot \left(0.0424060604 + \left(x \cdot x\right) \cdot \left(0.0072644182 + \left(x \cdot x\right) \cdot 0.0005064034\right)\right)\right)\right)}\right)}{1 + \left(x \cdot x\right) \cdot \left(0.7715471019 + \left(x \cdot x\right) \cdot \left(0.2909738639 + \left(x \cdot x\right) \cdot \left(0.0694555761 + \left(x \cdot x\right) \cdot 0.0140005442\right)\right)\right)} \]

   if 1.9199999999999999 < x

   1. Initial program 1.6%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}\right)}{\color{blue}{x}} \]

      2. sub-neg N/A

         \[\leadsto \frac{-1 \cdot \left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{x} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{-1 \cdot \left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right) + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right)\right) + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      6. remove-double-neg N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + -1 \cdot \frac{-1}{2}}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \frac{1}{2}}{x} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}{x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)}{x} \]

      11. sub-neg N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \color{blue}{x}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x \cdot x}}{x \cdot x}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.5:\\ \;\;\;\;\frac{x\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.1049934947 + x\_m \cdot \left(x\_m \cdot \left(0.0424060604 + \left(x\_m \cdot x\_m\right) \cdot 0.0072644182\right)\right)\right)\right)}{1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.7715471019 + \left(x\_m \cdot x\_m\right) \cdot \left(0.2909738639 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0694555761 + \left(x\_m \cdot x\_m\right) \cdot 0.0140005442\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x\_m \cdot x\_m}}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.5)
    (/
     (*
      x_m
      (+
       1.0
       (*
        (* x_m x_m)
        (+
         0.1049934947
         (* x_m (* x_m (+ 0.0424060604 (* (* x_m x_m) 0.0072644182))))))))
     (+
      1.0
      (*
       (* x_m x_m)
       (+
        0.7715471019
        (*
         (* x_m x_m)
         (+
          0.2909738639
          (* (* x_m x_m) (+ 0.0694555761 (* (* x_m x_m) 0.0140005442)))))))))
    (/
     (+
      0.5
      (/
       (+ 0.2514179000665374 (/ 0.15298196345929074 (* x_m x_m)))
       (* x_m x_m)))
     x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.5) {
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * (0.0424060604 + ((x_m * x_m) * 0.0072644182)))))))) / (1.0 + ((x_m * x_m) * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * 0.0140005442))))))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.5d0) then
        tmp = (x_m * (1.0d0 + ((x_m * x_m) * (0.1049934947d0 + (x_m * (x_m * (0.0424060604d0 + ((x_m * x_m) * 0.0072644182d0)))))))) / (1.0d0 + ((x_m * x_m) * (0.7715471019d0 + ((x_m * x_m) * (0.2909738639d0 + ((x_m * x_m) * (0.0694555761d0 + ((x_m * x_m) * 0.0140005442d0))))))))
    else
        tmp = (0.5d0 + ((0.2514179000665374d0 + (0.15298196345929074d0 / (x_m * x_m))) / (x_m * x_m))) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.5) {
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * (0.0424060604 + ((x_m * x_m) * 0.0072644182)))))))) / (1.0 + ((x_m * x_m) * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * 0.0140005442))))))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 2.5:
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * (0.0424060604 + ((x_m * x_m) * 0.0072644182)))))))) / (1.0 + ((x_m * x_m) * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * 0.0140005442))))))))
	else:
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 2.5)
		tmp = Float64(Float64(x_m * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.1049934947 + Float64(x_m * Float64(x_m * Float64(0.0424060604 + Float64(Float64(x_m * x_m) * 0.0072644182)))))))) / Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.7715471019 + Float64(Float64(x_m * x_m) * Float64(0.2909738639 + Float64(Float64(x_m * x_m) * Float64(0.0694555761 + Float64(Float64(x_m * x_m) * 0.0140005442)))))))));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.2514179000665374 + Float64(0.15298196345929074 / Float64(x_m * x_m))) / Float64(x_m * x_m))) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 2.5)
		tmp = (x_m * (1.0 + ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * (0.0424060604 + ((x_m * x_m) * 0.0072644182)))))))) / (1.0 + ((x_m * x_m) * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * 0.0140005442))))))));
	else
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 2.5], N[(N[(x$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.1049934947 + N[(x$95$m * N[(x$95$m * N[(0.0424060604 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0072644182), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.7715471019 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.2909738639 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0694555761 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0140005442), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(0.2514179000665374 + N[(0.15298196345929074 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 61.4%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   3. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.0424060604 + \left(x \cdot x\right) \cdot 0.0072644182\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)\right)}{1 + \left(\left(x \cdot x\right) \cdot \left(0.7715471019 + \left(x \cdot x\right) \cdot 0.2909738639\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(0.0694555761 + \left(x \cdot x\right) \cdot 0.0140005442\right)\right)\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(0.0008327945 + x \cdot \left(x \cdot 0.0003579942\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{7715471019}{10000000000} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{7715471019}{10000000000} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{7715471019}{10000000000}} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{7715471019}{10000000000}} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \color{blue}{\left({x}^{2} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{2909738639}{10000000000}} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{2909738639}{10000000000}} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \color{blue}{\left({x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{694555761}{10000000000}} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{694555761}{10000000000}} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \color{blue}{\left(\frac{70002721}{5000000000} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \left({x}^{2} \cdot \color{blue}{\frac{70002721}{5000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{70002721}{5000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 57.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 57.5%

      \[\leadsto \frac{x \cdot \left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.0424060604 + \left(x \cdot x\right) \cdot 0.0072644182\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)\right)}{1 + \color{blue}{\left(x \cdot x\right) \cdot \left(0.7715471019 + \left(x \cdot x\right) \cdot \left(0.2909738639 + \left(x \cdot x\right) \cdot \left(0.0694555761 + \left(x \cdot x\right) \cdot 0.0140005442\right)\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \left({x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \left(\left(x \cdot x\right) \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \left(x \cdot \left(x \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \left(\frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \left({x}^{2} \cdot \frac{36322091}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 58.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 58.4%

       \[\leadsto \frac{\color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.1049934947 + x \cdot \left(x \cdot \left(0.0424060604 + \left(x \cdot x\right) \cdot 0.0072644182\right)\right)\right)\right)}}{1 + \left(x \cdot x\right) \cdot \left(0.7715471019 + \left(x \cdot x\right) \cdot \left(0.2909738639 + \left(x \cdot x\right) \cdot \left(0.0694555761 + \left(x \cdot x\right) \cdot 0.0140005442\right)\right)\right)} \]

   if 2.5 < x

   1. Initial program 1.6%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}\right)}{\color{blue}{x}} \]

      2. sub-neg N/A

         \[\leadsto \frac{-1 \cdot \left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{x} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{-1 \cdot \left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right) + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right)\right) + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      6. remove-double-neg N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + -1 \cdot \frac{-1}{2}}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \frac{1}{2}}{x} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}{x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)}{x} \]

      11. sub-neg N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \color{blue}{x}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x \cdot x}}{x \cdot x}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2:\\ \;\;\;\;x\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(-0.6665536072 + \left(x\_m \cdot x\_m\right) \cdot \left(0.265709700396151 + x\_m \cdot \left(x\_m \cdot -0.0732490286039007\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x\_m \cdot x\_m}}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.2)
    (*
     x_m
     (+
      1.0
      (*
       (* x_m x_m)
       (+
        -0.6665536072
        (*
         (* x_m x_m)
         (+ 0.265709700396151 (* x_m (* x_m -0.0732490286039007))))))))
    (/
     (+
      0.5
      (/
       (+ 0.2514179000665374 (/ 0.15298196345929074 (* x_m x_m)))
       (* x_m x_m)))
     x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.2) {
		tmp = x_m * (1.0 + ((x_m * x_m) * (-0.6665536072 + ((x_m * x_m) * (0.265709700396151 + (x_m * (x_m * -0.0732490286039007)))))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.2d0) then
        tmp = x_m * (1.0d0 + ((x_m * x_m) * ((-0.6665536072d0) + ((x_m * x_m) * (0.265709700396151d0 + (x_m * (x_m * (-0.0732490286039007d0))))))))
    else
        tmp = (0.5d0 + ((0.2514179000665374d0 + (0.15298196345929074d0 / (x_m * x_m))) / (x_m * x_m))) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.2) {
		tmp = x_m * (1.0 + ((x_m * x_m) * (-0.6665536072 + ((x_m * x_m) * (0.265709700396151 + (x_m * (x_m * -0.0732490286039007)))))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.2:
		tmp = x_m * (1.0 + ((x_m * x_m) * (-0.6665536072 + ((x_m * x_m) * (0.265709700396151 + (x_m * (x_m * -0.0732490286039007)))))))
	else:
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.2)
		tmp = Float64(x_m * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(-0.6665536072 + Float64(Float64(x_m * x_m) * Float64(0.265709700396151 + Float64(x_m * Float64(x_m * -0.0732490286039007))))))));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.2514179000665374 + Float64(0.15298196345929074 / Float64(x_m * x_m))) / Float64(x_m * x_m))) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.2)
		tmp = x_m * (1.0 + ((x_m * x_m) * (-0.6665536072 + ((x_m * x_m) * (0.265709700396151 + (x_m * (x_m * -0.0732490286039007)))))));
	else
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.2], N[(x$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(-0.6665536072 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.265709700396151 + N[(x$95$m * N[(x$95$m * -0.0732490286039007), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(0.2514179000665374 + N[(0.15298196345929074 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 61.4%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right)\right)}, x\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right)\right)\right), x\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right)\right)\right), x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right)\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{-833192009}{1250000000}\right)\right)\right), x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-833192009}{1250000000} + {x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3321371254951887171}{12500000000000000000}, \left(\frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3321371254951887171}{12500000000000000000}, \left({x}^{2} \cdot \frac{-9156128575487588197208397249}{125000000000000000000000000000}\right)\right)\right)\right)\right)\right), x\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3321371254951887171}{12500000000000000000}, \left(\left(x \cdot x\right) \cdot \frac{-9156128575487588197208397249}{125000000000000000000000000000}\right)\right)\right)\right)\right)\right), x\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3321371254951887171}{12500000000000000000}, \left(x \cdot \left(x \cdot \frac{-9156128575487588197208397249}{125000000000000000000000000000}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3321371254951887171}{12500000000000000000}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-9156128575487588197208397249}{125000000000000000000000000000}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. *-lowering-*.f64 57.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3321371254951887171}{12500000000000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-9156128575487588197208397249}{125000000000000000000000000000}\right)\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 57.6%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(-0.6665536072 + \left(x \cdot x\right) \cdot \left(0.265709700396151 + x \cdot \left(x \cdot -0.0732490286039007\right)\right)\right)\right)} \cdot x \]

   if 1.19999999999999996 < x

   1. Initial program 1.6%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}\right)}{\color{blue}{x}} \]

      2. sub-neg N/A

         \[\leadsto \frac{-1 \cdot \left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{x} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{-1 \cdot \left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right) + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right)\right) + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      6. remove-double-neg N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + -1 \cdot \frac{-1}{2}}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \frac{1}{2}}{x} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}{x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)}{x} \]

      11. sub-neg N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \color{blue}{x}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x \cdot x}}{x \cdot x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.6665536072 + \left(x \cdot x\right) \cdot \left(0.265709700396151 + x \cdot \left(x \cdot -0.0732490286039007\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x \cdot x}}{x \cdot x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;x\_m \cdot \left(1 + x\_m \cdot \left(x\_m \cdot \left(-0.6665536072 + \left(x\_m \cdot x\_m\right) \cdot 0.265709700396151\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x\_m \cdot x\_m}}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.15)
    (*
     x_m
     (+
      1.0
      (* x_m (* x_m (+ -0.6665536072 (* (* x_m x_m) 0.265709700396151))))))
    (/
     (+
      0.5
      (/
       (+ 0.2514179000665374 (/ 0.15298196345929074 (* x_m x_m)))
       (* x_m x_m)))
     x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.15) {
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + ((x_m * x_m) * 0.265709700396151)))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.15d0) then
        tmp = x_m * (1.0d0 + (x_m * (x_m * ((-0.6665536072d0) + ((x_m * x_m) * 0.265709700396151d0)))))
    else
        tmp = (0.5d0 + ((0.2514179000665374d0 + (0.15298196345929074d0 / (x_m * x_m))) / (x_m * x_m))) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.15) {
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + ((x_m * x_m) * 0.265709700396151)))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.15:
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + ((x_m * x_m) * 0.265709700396151)))))
	else:
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.15)
		tmp = Float64(x_m * Float64(1.0 + Float64(x_m * Float64(x_m * Float64(-0.6665536072 + Float64(Float64(x_m * x_m) * 0.265709700396151))))));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.2514179000665374 + Float64(0.15298196345929074 / Float64(x_m * x_m))) / Float64(x_m * x_m))) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.15)
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + ((x_m * x_m) * 0.265709700396151)))));
	else
		tmp = (0.5 + ((0.2514179000665374 + (0.15298196345929074 / (x_m * x_m))) / (x_m * x_m))) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.15], N[(x$95$m * N[(1.0 + N[(x$95$m * N[(x$95$m * N[(-0.6665536072 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.265709700396151), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(0.2514179000665374 + N[(0.15298196345929074 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 61.4%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)}, x\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)\right), x\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)\right), x\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)\right)\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \frac{-833192009}{1250000000}\right)\right)\right)\right), x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-833192009}{1250000000} + \frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \left({x}^{2} \cdot \frac{3321371254951887171}{12500000000000000000}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{3321371254951887171}{12500000000000000000}\right)\right)\right)\right)\right), x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{3321371254951887171}{12500000000000000000}\right)\right)\right)\right)\right), x\right) \]

      13. *-lowering-*.f64 58.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{3321371254951887171}{12500000000000000000}\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 58.2%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.6665536072 + \left(x \cdot x\right) \cdot 0.265709700396151\right)\right)\right)} \cdot x \]

   if 1.1499999999999999 < x

   1. Initial program 1.6%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}\right)}{\color{blue}{x}} \]

      2. sub-neg N/A

         \[\leadsto \frac{-1 \cdot \left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{x} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{-1 \cdot \left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right) + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right)\right) + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      6. remove-double-neg N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + -1 \cdot \frac{-1}{2}}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \frac{1}{2}}{x} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}{x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)}{x} \]

      11. sub-neg N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \color{blue}{x}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x \cdot x}}{x \cdot x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.6665536072 + \left(x \cdot x\right) \cdot 0.265709700396151\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x \cdot x}}{x \cdot x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;x\_m \cdot \left(1 + x\_m \cdot \left(x\_m \cdot \left(-0.6665536072 + \left(x\_m \cdot x\_m\right) \cdot 0.265709700396151\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.1)
    (*
     x_m
     (+
      1.0
      (* x_m (* x_m (+ -0.6665536072 (* (* x_m x_m) 0.265709700396151))))))
    (/ (+ 0.5 (/ 0.2514179000665374 (* x_m x_m))) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + ((x_m * x_m) * 0.265709700396151)))));
	} else {
		tmp = (0.5 + (0.2514179000665374 / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.1d0) then
        tmp = x_m * (1.0d0 + (x_m * (x_m * ((-0.6665536072d0) + ((x_m * x_m) * 0.265709700396151d0)))))
    else
        tmp = (0.5d0 + (0.2514179000665374d0 / (x_m * x_m))) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + ((x_m * x_m) * 0.265709700396151)))));
	} else {
		tmp = (0.5 + (0.2514179000665374 / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.1:
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + ((x_m * x_m) * 0.265709700396151)))))
	else:
		tmp = (0.5 + (0.2514179000665374 / (x_m * x_m))) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(x_m * Float64(1.0 + Float64(x_m * Float64(x_m * Float64(-0.6665536072 + Float64(Float64(x_m * x_m) * 0.265709700396151))))));
	else
		tmp = Float64(Float64(0.5 + Float64(0.2514179000665374 / Float64(x_m * x_m))) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.1)
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + ((x_m * x_m) * 0.265709700396151)))));
	else
		tmp = (0.5 + (0.2514179000665374 / (x_m * x_m))) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.1], N[(x$95$m * N[(1.0 + N[(x$95$m * N[(x$95$m * N[(-0.6665536072 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.265709700396151), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 61.4%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)}, x\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)\right), x\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)\right), x\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)\right)\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \frac{-833192009}{1250000000}\right)\right)\right)\right), x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-833192009}{1250000000} + \frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \left({x}^{2} \cdot \frac{3321371254951887171}{12500000000000000000}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{3321371254951887171}{12500000000000000000}\right)\right)\right)\right)\right), x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{3321371254951887171}{12500000000000000000}\right)\right)\right)\right)\right), x\right) \]

      13. *-lowering-*.f64 58.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-833192009}{1250000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{3321371254951887171}{12500000000000000000}\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 58.2%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.6665536072 + \left(x \cdot x\right) \cdot 0.265709700396151\right)\right)\right)} \cdot x \]

   if 1.1000000000000001 < x

   1. Initial program 1.6%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{x}} \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{600041}{2386628} \cdot 1}{{x}^{2}}\right)\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{600041}{2386628}}{{x}^{2}}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{600041}{2386628}, \left({x}^{2}\right)\right)\right), x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{600041}{2386628}, \left(x \cdot x\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{600041}{2386628}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.2514179000665374}{x \cdot x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.6665536072 + \left(x \cdot x\right) \cdot 0.265709700396151\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374}{x \cdot x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.95:\\ \;\;\;\;x\_m \cdot \left(1 + x\_m \cdot \left(x\_m \cdot -0.6665536072\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.95)
    (* x_m (+ 1.0 (* x_m (* x_m -0.6665536072))))
    (/ (+ 0.5 (/ 0.2514179000665374 (* x_m x_m))) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.95) {
		tmp = x_m * (1.0 + (x_m * (x_m * -0.6665536072)));
	} else {
		tmp = (0.5 + (0.2514179000665374 / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.95d0) then
        tmp = x_m * (1.0d0 + (x_m * (x_m * (-0.6665536072d0))))
    else
        tmp = (0.5d0 + (0.2514179000665374d0 / (x_m * x_m))) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.95) {
		tmp = x_m * (1.0 + (x_m * (x_m * -0.6665536072)));
	} else {
		tmp = (0.5 + (0.2514179000665374 / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.95:
		tmp = x_m * (1.0 + (x_m * (x_m * -0.6665536072)))
	else:
		tmp = (0.5 + (0.2514179000665374 / (x_m * x_m))) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.95)
		tmp = Float64(x_m * Float64(1.0 + Float64(x_m * Float64(x_m * -0.6665536072))));
	else
		tmp = Float64(Float64(0.5 + Float64(0.2514179000665374 / Float64(x_m * x_m))) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.95)
		tmp = x_m * (1.0 + (x_m * (x_m * -0.6665536072)));
	else
		tmp = (0.5 + (0.2514179000665374 / (x_m * x_m))) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.95], N[(x$95$m * N[(1.0 + N[(x$95$m * N[(x$95$m * -0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   1. Initial program 61.4%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-833192009}{1250000000} \cdot {x}^{2}\right)}, x\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-833192009}{1250000000} \cdot {x}^{2}\right)\right), x\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-833192009}{1250000000} \cdot \left(x \cdot x\right)\right)\right), x\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{-833192009}{1250000000} \cdot x\right) \cdot x\right)\right), x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{-833192009}{1250000000} \cdot x\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-833192009}{1250000000} \cdot x\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-833192009}{1250000000}\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 57.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-833192009}{1250000000}\right)\right)\right), x\right) \]

   5. Simplified 57.5%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot -0.6665536072\right)\right)} \cdot x \]

   if 0.94999999999999996 < x

   1. Initial program 1.6%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{x}} \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{600041}{2386628} \cdot 1}{{x}^{2}}\right)\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{600041}{2386628}}{{x}^{2}}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{600041}{2386628}, \left({x}^{2}\right)\right)\right), x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{600041}{2386628}, \left(x \cdot x\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{600041}{2386628}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.2514179000665374}{x \cdot x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot -0.6665536072\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374}{x \cdot x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.3% accurate, 12.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.78:\\ \;\;\;\;x\_m \cdot \left(1 + x\_m \cdot \left(x\_m \cdot -0.6665536072\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.78)
    (* x_m (+ 1.0 (* x_m (* x_m -0.6665536072))))
    (/ 0.5 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.78) {
		tmp = x_m * (1.0 + (x_m * (x_m * -0.6665536072)));
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.78d0) then
        tmp = x_m * (1.0d0 + (x_m * (x_m * (-0.6665536072d0))))
    else
        tmp = 0.5d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.78) {
		tmp = x_m * (1.0 + (x_m * (x_m * -0.6665536072)));
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.78:
		tmp = x_m * (1.0 + (x_m * (x_m * -0.6665536072)))
	else:
		tmp = 0.5 / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.78)
		tmp = Float64(x_m * Float64(1.0 + Float64(x_m * Float64(x_m * -0.6665536072))));
	else
		tmp = Float64(0.5 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.78)
		tmp = x_m * (1.0 + (x_m * (x_m * -0.6665536072)));
	else
		tmp = 0.5 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.78], N[(x$95$m * N[(1.0 + N[(x$95$m * N[(x$95$m * -0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.78000000000000003

   1. Initial program 61.4%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-833192009}{1250000000} \cdot {x}^{2}\right)}, x\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-833192009}{1250000000} \cdot {x}^{2}\right)\right), x\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-833192009}{1250000000} \cdot \left(x \cdot x\right)\right)\right), x\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{-833192009}{1250000000} \cdot x\right) \cdot x\right)\right), x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{-833192009}{1250000000} \cdot x\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-833192009}{1250000000} \cdot x\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-833192009}{1250000000}\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 57.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-833192009}{1250000000}\right)\right)\right), x\right) \]

   5. Simplified 57.5%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot -0.6665536072\right)\right)} \cdot x \]

   if 0.78000000000000003 < x

   1. Initial program 1.6%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.78:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot -0.6665536072\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.1% accurate, 21.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.7:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 0.7) x_m (/ 0.5 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.7) {
		tmp = x_m;
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.7d0) then
        tmp = x_m
    else
        tmp = 0.5d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.7) {
		tmp = x_m;
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.7:
		tmp = x_m
	else:
		tmp = 0.5 / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.7)
		tmp = x_m;
	else
		tmp = Float64(0.5 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.7)
		tmp = x_m;
	else
		tmp = 0.5 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.7], x$95$m, N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 61.4%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 58.5%

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

   if 0.69999999999999996 < x

   1. Initial program 1.6%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.5% accurate, 173.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * x_m;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * x_m)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * x_m;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 46.5%

   \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Simplified 44.7%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x)))) (* 0.0072644182 (* (* (* x x) (* x x)) (* x x)))) (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1.0 (* 0.7715471019 (* x x))) (* 0.2909738639 (* (* x x) (* x x)))) (* 0.0694555761 (* (* (* x x) (* x x)) (* x x)))) (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2.0 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))