Jmat.Real.dawson

Percentage Accurate: 55.3% → 100.0%

Time: 15.5s

Alternatives: 9

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: 55.3% 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, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\\ t_1 := x\_m \cdot t\_0\\ t_2 := x\_m \cdot t\_1\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1500:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(t\_2, 0.0005064034, t\_2 \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.0001789971\right)\right), \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.0424060604, t\_0 \cdot 0.0072644182\right), \mathsf{fma}\left(x\_m \cdot x\_m, 0.1049934947, 1\right)\right)\right)}{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot t\_2, 0.0008327945, \left(t\_0 \cdot t\_2\right) \cdot 0.0003579942\right), \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(t\_0, 0.0694555761, x\_m \cdot \left(t\_1 \cdot 0.0140005442\right)\right), \mathsf{fma}\left(x\_m \cdot x\_m, 0.7715471019 + \left(x\_m \cdot x\_m\right) \cdot 0.2909738639, 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374 - \frac{-0.15298196345929074}{x\_m \cdot x\_m}}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \end{array} \]

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)))) (t_1 (* x_m t_0)) (t_2 (* x_m t_1)))
   (*
    x_s
    (if (<= x_m 1500.0)
      (/
       (*
        x_m
        (fma
         (* x_m x_m)
         (fma t_2 0.0005064034 (* t_2 (* (* x_m x_m) 0.0001789971)))
         (fma
          (* x_m x_m)
          (fma (* x_m x_m) 0.0424060604 (* t_0 0.0072644182))
          (fma (* x_m x_m) 0.1049934947 1.0))))
       (fma
        (* x_m x_m)
        (fma (* (* x_m x_m) t_2) 0.0008327945 (* (* t_0 t_2) 0.0003579942))
        (fma
         (* x_m x_m)
         (fma t_0 0.0694555761 (* x_m (* t_1 0.0140005442)))
         (fma (* x_m x_m) (+ 0.7715471019 (* (* x_m x_m) 0.2909738639)) 1.0))))
      (/
       (+
        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 t_0 = x_m * (x_m * (x_m * x_m));
	double t_1 = x_m * t_0;
	double t_2 = x_m * t_1;
	double tmp;
	if (x_m <= 1500.0) {
		tmp = (x_m * fma((x_m * x_m), fma(t_2, 0.0005064034, (t_2 * ((x_m * x_m) * 0.0001789971))), fma((x_m * x_m), fma((x_m * x_m), 0.0424060604, (t_0 * 0.0072644182)), fma((x_m * x_m), 0.1049934947, 1.0)))) / fma((x_m * x_m), fma(((x_m * x_m) * t_2), 0.0008327945, ((t_0 * t_2) * 0.0003579942)), fma((x_m * x_m), fma(t_0, 0.0694555761, (x_m * (t_1 * 0.0140005442))), fma((x_m * x_m), (0.7715471019 + ((x_m * x_m) * 0.2909738639)), 1.0)));
	} 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)
	t_0 = Float64(x_m * Float64(x_m * Float64(x_m * x_m)))
	t_1 = Float64(x_m * t_0)
	t_2 = Float64(x_m * t_1)
	tmp = 0.0
	if (x_m <= 1500.0)
		tmp = Float64(Float64(x_m * fma(Float64(x_m * x_m), fma(t_2, 0.0005064034, Float64(t_2 * Float64(Float64(x_m * x_m) * 0.0001789971))), fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.0424060604, Float64(t_0 * 0.0072644182)), fma(Float64(x_m * x_m), 0.1049934947, 1.0)))) / fma(Float64(x_m * x_m), fma(Float64(Float64(x_m * x_m) * t_2), 0.0008327945, Float64(Float64(t_0 * t_2) * 0.0003579942)), fma(Float64(x_m * x_m), fma(t_0, 0.0694555761, Float64(x_m * Float64(t_1 * 0.0140005442))), fma(Float64(x_m * x_m), Float64(0.7715471019 + Float64(Float64(x_m * x_m) * 0.2909738639)), 1.0))));
	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

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 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(x$95$m * t$95$1), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 1500.0], N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(t$95$2 * 0.0005064034 + N[(t$95$2 * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0001789971), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0424060604 + N[(t$95$0 * 0.0072644182), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.1049934947 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * 0.0008327945 + N[(N[(t$95$0 * t$95$2), $MachinePrecision] * 0.0003579942), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(t$95$0 * 0.0694555761 + N[(x$95$m * N[(t$95$1 * 0.0140005442), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.7715471019 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.2909738639), $MachinePrecision]), $MachinePrecision] + 1.0), $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 < 1500

   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 egg-rr 100.0%

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

   if 1500 < x

   1. Initial program 12.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \left(\frac{\frac{1307076337763}{8543989815576}}{{x}^{4}} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}} \]

   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 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1500:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), 0.0005064034, \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971\right)\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0424060604, \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0072644182\right), \mathsf{fma}\left(x \cdot x, 0.1049934947, 1\right)\right)\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right), 0.0008327945, \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \cdot 0.0003579942\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 0.0694555761, x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0140005442\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + \left(x \cdot x\right) \cdot 0.2909738639, 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374 - \frac{-0.15298196345929074}{x \cdot x}}{x \cdot x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.0072644182, 0.0424060604\right), 0.1049934947\right), 1\right)}{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.0694555761, 0.2909738639\right), 0.7715471019\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374 - \frac{-0.15298196345929074}{x\_m \cdot x\_m}}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

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 1.4)
    (/
     (*
      x_m
      (fma
       x_m
       (*
        x_m
        (fma
         (* x_m x_m)
         (fma x_m (* x_m 0.0072644182) 0.0424060604)
         0.1049934947))
       1.0))
     (fma
      (* x_m x_m)
      (fma
       (* x_m x_m)
       (fma x_m (* x_m 0.0694555761) 0.2909738639)
       0.7715471019)
      1.0))
    (/
     (+
      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 <= 1.4) {
		tmp = (x_m * fma(x_m, (x_m * fma((x_m * x_m), fma(x_m, (x_m * 0.0072644182), 0.0424060604), 0.1049934947)), 1.0)) / fma((x_m * x_m), fma((x_m * x_m), fma(x_m, (x_m * 0.0694555761), 0.2909738639), 0.7715471019), 1.0);
	} 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 <= 1.4)
		tmp = Float64(Float64(x_m * fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.0072644182), 0.0424060604), 0.1049934947)), 1.0)) / fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.0694555761), 0.2909738639), 0.7715471019), 1.0));
	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

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, 1.4], N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.0072644182), $MachinePrecision] + 0.0424060604), $MachinePrecision] + 0.1049934947), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.0694555761), $MachinePrecision] + 0.2909738639), $MachinePrecision] + 0.7715471019), $MachinePrecision] + 1.0), $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 < 1.3999999999999999

   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 egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.8

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

   10. Simplified 99.8%

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

   if 1.3999999999999999 < x

   1. Initial program 8.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \left(\frac{\frac{1307076337763}{8543989815576}}{{x}^{4}} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}} \]

   5. Simplified 99.5%

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

Reproduce

herbie shell --seed 2024218 
(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))