Jmat.Real.dawson

Percentage Accurate: 54.8% → 100.0%

Time: 18.2s

Alternatives: 8

Speedup: 23.0×

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: 54.8% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 6000:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.2909738639 \cdot x\_m, x\_m, 0.7715471019\right) + \mathsf{fma}\left(\mathsf{fma}\left(0.0003579942 \cdot x\_m, x\_m, 0.0008327945\right), t\_0 \cdot t\_0, \mathsf{fma}\left(0.0140005442 \cdot x\_m, x\_m, 0.0694555761\right) \cdot t\_0\right), 1\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001789971 \cdot x\_m, x\_m, 0.0005064034\right) \cdot \left(\left(t\_0 \cdot x\_m\right) \cdot x\_m\right)\right) \cdot x\_m, x\_m, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0072644182 \cdot x\_m, x\_m, 0.0424060604\right) \cdot x\_m, x\_m, 0.1049934947\right), x\_m \cdot x\_m, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{x\_m \cdot x\_m} - -0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

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_m)))
   (*
    x_s
    (if (<= x_m 6000.0)
      (*
       (/
        x_m
        (fma
         (* x_m x_m)
         (+
          (fma (* 0.2909738639 x_m) x_m 0.7715471019)
          (fma
           (fma (* 0.0003579942 x_m) x_m 0.0008327945)
           (* t_0 t_0)
           (* (fma (* 0.0140005442 x_m) x_m 0.0694555761) t_0)))
         1.0))
       (fma
        (*
         (* (fma (* 0.0001789971 x_m) x_m 0.0005064034) (* (* t_0 x_m) x_m))
         x_m)
        x_m
        (fma
         (fma
          (* (fma (* 0.0072644182 x_m) x_m 0.0424060604) x_m)
          x_m
          0.1049934947)
         (* x_m x_m)
         1.0)))
      (/ (- (/ 0.2514179000665374 (* x_m 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 t_0 = ((x_m * x_m) * x_m) * x_m;
	double tmp;
	if (x_m <= 6000.0) {
		tmp = (x_m / fma((x_m * x_m), (fma((0.2909738639 * x_m), x_m, 0.7715471019) + fma(fma((0.0003579942 * x_m), x_m, 0.0008327945), (t_0 * t_0), (fma((0.0140005442 * x_m), x_m, 0.0694555761) * t_0))), 1.0)) * fma(((fma((0.0001789971 * x_m), x_m, 0.0005064034) * ((t_0 * x_m) * x_m)) * x_m), x_m, fma(fma((fma((0.0072644182 * x_m), x_m, 0.0424060604) * x_m), x_m, 0.1049934947), (x_m * x_m), 1.0));
	} else {
		tmp = ((0.2514179000665374 / (x_m * x_m)) - -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(Float64(Float64(x_m * x_m) * x_m) * x_m)
	tmp = 0.0
	if (x_m <= 6000.0)
		tmp = Float64(Float64(x_m / fma(Float64(x_m * x_m), Float64(fma(Float64(0.2909738639 * x_m), x_m, 0.7715471019) + fma(fma(Float64(0.0003579942 * x_m), x_m, 0.0008327945), Float64(t_0 * t_0), Float64(fma(Float64(0.0140005442 * x_m), x_m, 0.0694555761) * t_0))), 1.0)) * fma(Float64(Float64(fma(Float64(0.0001789971 * x_m), x_m, 0.0005064034) * Float64(Float64(t_0 * x_m) * x_m)) * x_m), x_m, fma(fma(Float64(fma(Float64(0.0072644182 * x_m), x_m, 0.0424060604) * x_m), x_m, 0.1049934947), Float64(x_m * x_m), 1.0)));
	else
		tmp = Float64(Float64(Float64(0.2514179000665374 / Float64(x_m * x_m)) - -0.5) / x_m);
	end
	return Float64(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[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 6000.0], N[(N[(x$95$m / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(0.2909738639 * x$95$m), $MachinePrecision] * x$95$m + 0.7715471019), $MachinePrecision] + N[(N[(N[(0.0003579942 * x$95$m), $MachinePrecision] * x$95$m + 0.0008327945), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(N[(0.0140005442 * x$95$m), $MachinePrecision] * x$95$m + 0.0694555761), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(0.0001789971 * x$95$m), $MachinePrecision] * x$95$m + 0.0005064034), $MachinePrecision] * N[(N[(t$95$0 * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + N[(N[(N[(N[(N[(0.0072644182 * x$95$m), $MachinePrecision] * x$95$m + 0.0424060604), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 0.1049934947), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6e3

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   if 6e3 < x

   1. Initial program 8.7%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   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. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}}}{x} \]

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{600041}{2386628}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{600041}{2386628}}{{x}^{2}}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      10. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628}}{\color{blue}{x \cdot x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628}}{\color{blue}{x \cdot x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      12. metadata-eval 100.0

          \[\leadsto \frac{\frac{0.2514179000665374}{x \cdot x} - \color{blue}{-0.5}}{x} \]

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.2909738639 \cdot x, x, 0.7715471019\right) + \mathsf{fma}\left(\mathsf{fma}\left(0.0003579942 \cdot x, x, 0.0008327945\right), \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0140005442 \cdot x, x, 0.0694555761\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right), 1\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001789971 \cdot x, x, 0.0005064034\right) \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0072644182 \cdot x, x, 0.0424060604\right) \cdot x, x, 0.1049934947\right), x \cdot x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{x \cdot x} - -0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot x\_m\\ t_1 := t\_0 \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_0 \cdot t\_0\right) \cdot 0.0001789971, t\_1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0072644182 \cdot x\_m, x\_m, 0.0424060604\right) \cdot x\_m, x\_m, 0.1049934947\right) \cdot x\_m, x\_m, 1\right)\right) \cdot x\_m}{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.0140005442 \cdot x\_m, x\_m, 0.0694555761\right) \cdot t\_1, \mathsf{fma}\left(\mathsf{fma}\left(0.2909738639 \cdot x\_m, x\_m, 0.7715471019\right), x\_m \cdot x\_m, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{x\_m \cdot x\_m} - -0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

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)) (t_1 (* t_0 x_m)))
   (*
    x_s
    (if (<= x_m 1.9)
      (/
       (*
        (fma
         (* (* t_0 t_0) 0.0001789971)
         t_1
         (fma
          (*
           (fma
            (* (fma (* 0.0072644182 x_m) x_m 0.0424060604) x_m)
            x_m
            0.1049934947)
           x_m)
          x_m
          1.0))
        x_m)
       (fma
        (* x_m x_m)
        (* (fma (* 0.0140005442 x_m) x_m 0.0694555761) t_1)
        (fma (fma (* 0.2909738639 x_m) x_m 0.7715471019) (* x_m x_m) 1.0)))
      (/ (- (/ 0.2514179000665374 (* x_m 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 t_0 = (x_m * x_m) * x_m;
	double t_1 = t_0 * x_m;
	double tmp;
	if (x_m <= 1.9) {
		tmp = (fma(((t_0 * t_0) * 0.0001789971), t_1, fma((fma((fma((0.0072644182 * x_m), x_m, 0.0424060604) * x_m), x_m, 0.1049934947) * x_m), x_m, 1.0)) * x_m) / fma((x_m * x_m), (fma((0.0140005442 * x_m), x_m, 0.0694555761) * t_1), fma(fma((0.2909738639 * x_m), x_m, 0.7715471019), (x_m * x_m), 1.0));
	} else {
		tmp = ((0.2514179000665374 / (x_m * x_m)) - -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(Float64(x_m * x_m) * x_m)
	t_1 = Float64(t_0 * x_m)
	tmp = 0.0
	if (x_m <= 1.9)
		tmp = Float64(Float64(fma(Float64(Float64(t_0 * t_0) * 0.0001789971), t_1, fma(Float64(fma(Float64(fma(Float64(0.0072644182 * x_m), x_m, 0.0424060604) * x_m), x_m, 0.1049934947) * x_m), x_m, 1.0)) * x_m) / fma(Float64(x_m * x_m), Float64(fma(Float64(0.0140005442 * x_m), x_m, 0.0694555761) * t_1), fma(fma(Float64(0.2909738639 * x_m), x_m, 0.7715471019), Float64(x_m * x_m), 1.0)));
	else
		tmp = Float64(Float64(Float64(0.2514179000665374 / Float64(x_m * x_m)) - -0.5) / x_m);
	end
	return Float64(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[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 1.9], N[(N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.0001789971), $MachinePrecision] * t$95$1 + N[(N[(N[(N[(N[(N[(0.0072644182 * x$95$m), $MachinePrecision] * x$95$m + 0.0424060604), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 0.1049934947), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(0.0140005442 * x$95$m), $MachinePrecision] * x$95$m + 0.0694555761), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(N[(0.2909738639 * x$95$m), $MachinePrecision] * x$95$m + 0.7715471019), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)} + \frac{1789971}{10000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right) + 1\right)} + \frac{1789971}{10000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) + \frac{1789971}{10000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      3. unpow2 N/A

         \[\leadsto \frac{\left(\left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 1\right) + \frac{1789971}{10000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right) \cdot x\right) \cdot x} + 1\right) + \frac{1789971}{10000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right) \cdot x, x, 1\right)} + \frac{1789971}{10000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0072644182 \cdot x, x, 0.0424060604\right) \cdot x, x, 0.1049934947\right) \cdot x, x, 1\right)} + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{4} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right) \cdot {x}^{4}}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right) \cdot {x}^{4}}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{70002721}{5000000000} \cdot {x}^{2} + \frac{694555761}{10000000000}\right)} \cdot {x}^{4}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \left(\frac{70002721}{5000000000} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{694555761}{10000000000}\right) \cdot {x}^{4}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(\frac{70002721}{5000000000} \cdot x\right) \cdot x} + \frac{694555761}{10000000000}\right) \cdot {x}^{4}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(x \cdot \frac{70002721}{5000000000}\right)} \cdot x + \frac{694555761}{10000000000}\right) \cdot {x}^{4}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot \frac{70002721}{5000000000}, x, \frac{694555761}{10000000000}\right)} \cdot {x}^{4}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{\frac{70002721}{5000000000} \cdot x}, x, \frac{694555761}{10000000000}\right) \cdot {x}^{4}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{\frac{70002721}{5000000000} \cdot x}, x, \frac{694555761}{10000000000}\right) \cdot {x}^{4}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]

       10. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{70002721}{5000000000} \cdot x, x, \frac{694555761}{10000000000}\right) \cdot {x}^{\color{blue}{\left(3 + 1\right)}}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]

       11. pow-plus N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{70002721}{5000000000} \cdot x, x, \frac{694555761}{10000000000}\right) \cdot \color{blue}{\left({x}^{3} \cdot x\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{70002721}{5000000000} \cdot x, x, \frac{694555761}{10000000000}\right) \cdot \color{blue}{\left({x}^{3} \cdot x\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]

       13. unpow3 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{70002721}{5000000000} \cdot x, x, \frac{694555761}{10000000000}\right) \cdot \left(\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{1789971}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{36322091}{5000000000} \cdot x, x, \frac{106015151}{2500000000}\right) \cdot x, x, \frac{1049934947}{10000000000}\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{70002721}{5000000000} \cdot x, x, \frac{694555761}{10000000000}\right) \cdot \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{2909738639}{10000000000} \cdot x, x, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)} \]

       17. lower-*.f64 99.7

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

   12. Applied rewrites 99.7%

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

   if 1.8999999999999999 < x

   1. Initial program 8.8%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   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. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}}}{x} \]

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{600041}{2386628}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{600041}{2386628}}{{x}^{2}}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      10. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628}}{\color{blue}{x \cdot x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628}}{\color{blue}{x \cdot x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      12. metadata-eval 99.5

          \[\leadsto \frac{\frac{0.2514179000665374}{x \cdot x} - \color{blue}{-0.5}}{x} \]

   5. Applied rewrites 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.0001789971, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0072644182 \cdot x, x, 0.0424060604\right) \cdot x, x, 0.1049934947\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0140005442 \cdot x, x, 0.0694555761\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(0.2909738639 \cdot x, x, 0.7715471019\right), x \cdot x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{x \cdot x} - -0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(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))