Jmat.Real.dawson

Percentage Accurate: 53.2% → 100.0%

Time: 17.4s

Alternatives: 18

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: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5000:\\ \;\;\;\;x\_m \cdot \frac{1}{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, x\_m \cdot \left(t\_0 \cdot 0.0008327945\right), \left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(t\_0 \cdot 0.0003579942\right)\right)\right)\right), \mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, \left(x\_m \cdot x\_m\right) \cdot 0.0140005442, x\_m \cdot 0.0694555761\right), x\_m \cdot 0.2909738639\right), 0.7715471019\right)\right), 1\right)}{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, t\_0 \cdot \left(x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.0001789971, 0.0005064034\right)\right), \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.0072644182, 0.0424060604\right), 0.1049934947\right)\right), 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m} \cdot \left(0.5 + \frac{0.2514179000665374}{x\_m \cdot x\_m}\right)\\ \end{array} \end{array} \end{array} \]

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 5000.0)
      (*
       x_m
       (/
        1.0
        (/
         (fma
          (* x_m x_m)
          (fma
           x_m
           (fma
            (* x_m x_m)
            (* x_m (* t_0 0.0008327945))
            (* (* x_m x_m) (* x_m (* (* x_m x_m) (* t_0 0.0003579942)))))
           (fma
            x_m
            (fma
             x_m
             (*
              x_m
              (fma x_m (* (* x_m x_m) 0.0140005442) (* x_m 0.0694555761)))
             (* x_m 0.2909738639))
            0.7715471019))
          1.0)
         (fma
          (* x_m x_m)
          (fma
           x_m
           (* t_0 (* x_m (fma x_m (* x_m 0.0001789971) 0.0005064034)))
           (fma
            (* x_m x_m)
            (fma x_m (* x_m 0.0072644182) 0.0424060604)
            0.1049934947))
          1.0))))
      (* (/ 1.0 x_m) (+ 0.5 (/ 0.2514179000665374 (* 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 tmp;
	if (x_m <= 5000.0) {
		tmp = x_m * (1.0 / (fma((x_m * x_m), fma(x_m, fma((x_m * x_m), (x_m * (t_0 * 0.0008327945)), ((x_m * x_m) * (x_m * ((x_m * x_m) * (t_0 * 0.0003579942))))), fma(x_m, fma(x_m, (x_m * fma(x_m, ((x_m * x_m) * 0.0140005442), (x_m * 0.0694555761))), (x_m * 0.2909738639)), 0.7715471019)), 1.0) / fma((x_m * x_m), fma(x_m, (t_0 * (x_m * fma(x_m, (x_m * 0.0001789971), 0.0005064034))), fma((x_m * x_m), fma(x_m, (x_m * 0.0072644182), 0.0424060604), 0.1049934947)), 1.0)));
	} else {
		tmp = (1.0 / x_m) * (0.5 + (0.2514179000665374 / (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)))
	tmp = 0.0
	if (x_m <= 5000.0)
		tmp = Float64(x_m * Float64(1.0 / Float64(fma(Float64(x_m * x_m), fma(x_m, fma(Float64(x_m * x_m), Float64(x_m * Float64(t_0 * 0.0008327945)), Float64(Float64(x_m * x_m) * Float64(x_m * Float64(Float64(x_m * x_m) * Float64(t_0 * 0.0003579942))))), fma(x_m, fma(x_m, Float64(x_m * fma(x_m, Float64(Float64(x_m * x_m) * 0.0140005442), Float64(x_m * 0.0694555761))), Float64(x_m * 0.2909738639)), 0.7715471019)), 1.0) / fma(Float64(x_m * x_m), fma(x_m, Float64(t_0 * Float64(x_m * fma(x_m, Float64(x_m * 0.0001789971), 0.0005064034))), fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.0072644182), 0.0424060604), 0.1049934947)), 1.0))));
	else
		tmp = Float64(Float64(1.0 / x_m) * Float64(0.5 + Float64(0.2514179000665374 / Float64(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]}, N[(x$95$s * If[LessEqual[x$95$m, 5000.0], N[(x$95$m * N[(1.0 / N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(t$95$0 * 0.0008327945), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(t$95$0 * 0.0003579942), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0140005442), $MachinePrecision] + N[(x$95$m * 0.0694555761), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 0.2909738639), $MachinePrecision]), $MachinePrecision] + 0.7715471019), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(t$95$0 * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.0001789971), $MachinePrecision] + 0.0005064034), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(0.5 + N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e3

   1. Initial program 71.5%

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

   4. Applied rewrites 71.5%

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

   6. Applied rewrites 71.5%

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

   8. Applied rewrites 71.5%

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

   if 5e3 < x

   1. Initial program 0.1%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374, 0.5\right)}{x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{\color{blue}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \color{blue}{\frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\color{blue}{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{\color{blue}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\color{blue}{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \color{blue}{\left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right)} + \frac{1}{2}}{x} \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}{x} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}} \]

      11. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)} \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(0.5 + \frac{\frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374}{x \cdot x}\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

         \[\leadsto \frac{1}{x} \cdot \left(0.5 + \frac{0.2514179000665374}{\color{blue}{x \cdot x}}\right) \]

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(0.5 + \frac{0.2514179000665374}{x \cdot x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5000:\\ \;\;\;\;x \cdot \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0008327945\right), \left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0003579942\right)\right)\right)\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.0140005442, x \cdot 0.0694555761\right), x \cdot 0.2909738639\right), 0.7715471019\right)\right), 1\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0001789971, 0.0005064034\right)\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0072644182, 0.0424060604\right), 0.1049934947\right)\right), 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(0.5 + \frac{0.2514179000665374}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, t\_0 \cdot \left(x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.0001789971, 0.0005064034\right)\right), \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.0072644182, 0.0424060604\right), 0.1049934947\right)\right), 1\right) \cdot \frac{x\_m}{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, x\_m \cdot \left(t\_0 \cdot 0.0008327945\right), \left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(t\_0 \cdot 0.0003579942\right)\right)\right)\right), \mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, \left(x\_m \cdot x\_m\right) \cdot 0.0140005442, x\_m \cdot 0.0694555761\right), x\_m \cdot 0.2909738639\right), 0.7715471019\right)\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m} \cdot \left(0.5 + \frac{0.2514179000665374}{x\_m \cdot x\_m}\right)\\ \end{array} \end{array} \end{array} \]

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 5000.0)
      (*
       (fma
        (* x_m x_m)
        (fma
         x_m
         (* t_0 (* x_m (fma x_m (* x_m 0.0001789971) 0.0005064034)))
         (fma
          (* x_m x_m)
          (fma x_m (* x_m 0.0072644182) 0.0424060604)
          0.1049934947))
        1.0)
       (/
        x_m
        (fma
         (* x_m x_m)
         (fma
          x_m
          (fma
           (* x_m x_m)
           (* x_m (* t_0 0.0008327945))
           (* (* x_m x_m) (* x_m (* (* x_m x_m) (* t_0 0.0003579942)))))
          (fma
           x_m
           (fma
            x_m
            (* x_m (fma x_m (* (* x_m x_m) 0.0140005442) (* x_m 0.0694555761)))
            (* x_m 0.2909738639))
           0.7715471019))
         1.0)))
      (* (/ 1.0 x_m) (+ 0.5 (/ 0.2514179000665374 (* 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 tmp;
	if (x_m <= 5000.0) {
		tmp = fma((x_m * x_m), fma(x_m, (t_0 * (x_m * fma(x_m, (x_m * 0.0001789971), 0.0005064034))), fma((x_m * x_m), fma(x_m, (x_m * 0.0072644182), 0.0424060604), 0.1049934947)), 1.0) * (x_m / fma((x_m * x_m), fma(x_m, fma((x_m * x_m), (x_m * (t_0 * 0.0008327945)), ((x_m * x_m) * (x_m * ((x_m * x_m) * (t_0 * 0.0003579942))))), fma(x_m, fma(x_m, (x_m * fma(x_m, ((x_m * x_m) * 0.0140005442), (x_m * 0.0694555761))), (x_m * 0.2909738639)), 0.7715471019)), 1.0));
	} else {
		tmp = (1.0 / x_m) * (0.5 + (0.2514179000665374 / (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)))
	tmp = 0.0
	if (x_m <= 5000.0)
		tmp = Float64(fma(Float64(x_m * x_m), fma(x_m, Float64(t_0 * Float64(x_m * fma(x_m, Float64(x_m * 0.0001789971), 0.0005064034))), fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.0072644182), 0.0424060604), 0.1049934947)), 1.0) * Float64(x_m / fma(Float64(x_m * x_m), fma(x_m, fma(Float64(x_m * x_m), Float64(x_m * Float64(t_0 * 0.0008327945)), Float64(Float64(x_m * x_m) * Float64(x_m * Float64(Float64(x_m * x_m) * Float64(t_0 * 0.0003579942))))), fma(x_m, fma(x_m, Float64(x_m * fma(x_m, Float64(Float64(x_m * x_m) * 0.0140005442), Float64(x_m * 0.0694555761))), Float64(x_m * 0.2909738639)), 0.7715471019)), 1.0)));
	else
		tmp = Float64(Float64(1.0 / x_m) * Float64(0.5 + Float64(0.2514179000665374 / Float64(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]}, N[(x$95$s * If[LessEqual[x$95$m, 5000.0], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(t$95$0 * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.0001789971), $MachinePrecision] + 0.0005064034), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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] * N[(x$95$m / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(t$95$0 * 0.0008327945), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(t$95$0 * 0.0003579942), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0140005442), $MachinePrecision] + N[(x$95$m * 0.0694555761), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 0.2909738639), $MachinePrecision]), $MachinePrecision] + 0.7715471019), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(0.5 + N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e3

   1. Initial program 71.5%

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

   4. Applied rewrites 71.5%

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

   6. Applied rewrites 71.5%

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

   8. Applied rewrites 71.5%

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

   if 5e3 < x

   1. Initial program 0.1%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374, 0.5\right)}{x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{\color{blue}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \color{blue}{\frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\color{blue}{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{\color{blue}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\color{blue}{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \color{blue}{\left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right)} + \frac{1}{2}}{x} \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}{x} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}} \]

      11. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)} \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(0.5 + \frac{\frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374}{x \cdot x}\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

         \[\leadsto \frac{1}{x} \cdot \left(0.5 + \frac{0.2514179000665374}{\color{blue}{x \cdot x}}\right) \]

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(0.5 + \frac{0.2514179000665374}{x \cdot x}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := 0.2514179000665374 + \frac{0.15298196345929074 + \frac{11.259630434457211}{x\_m \cdot x\_m}}{x\_m \cdot x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m} \cdot \frac{\frac{t\_0 \cdot t\_0}{x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)} - 0.25}{\frac{t\_0}{x\_m \cdot x\_m} - 0.5}\\ \end{array} \end{array} \end{array} \]

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
         (+
          0.2514179000665374
          (/
           (+ 0.15298196345929074 (/ 11.259630434457211 (* x_m x_m)))
           (* x_m x_m)))))
   (*
    x_s
    (if (<= x_m 2.1)
      (/
       x_m
       (fma
        x_m
        (*
         x_m
         (fma
          (* x_m x_m)
          (fma x_m (* x_m 0.015175085973910875) 0.17858401087518092)
          0.6665536072))
        1.0))
      (*
       (/ 1.0 x_m)
       (/
        (- (/ (* t_0 t_0) (* x_m (* x_m (* x_m x_m)))) 0.25)
        (- (/ t_0 (* x_m x_m)) 0.5)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = 0.2514179000665374 + ((0.15298196345929074 + (11.259630434457211 / (x_m * x_m))) / (x_m * x_m));
	double tmp;
	if (x_m <= 2.1) {
		tmp = x_m / fma(x_m, (x_m * fma((x_m * x_m), fma(x_m, (x_m * 0.015175085973910875), 0.17858401087518092), 0.6665536072)), 1.0);
	} else {
		tmp = (1.0 / x_m) * ((((t_0 * t_0) / (x_m * (x_m * (x_m * x_m)))) - 0.25) / ((t_0 / (x_m * x_m)) - 0.5));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(0.2514179000665374 + Float64(Float64(0.15298196345929074 + Float64(11.259630434457211 / Float64(x_m * x_m))) / Float64(x_m * x_m)))
	tmp = 0.0
	if (x_m <= 2.1)
		tmp = Float64(x_m / fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.015175085973910875), 0.17858401087518092), 0.6665536072)), 1.0));
	else
		tmp = Float64(Float64(1.0 / x_m) * Float64(Float64(Float64(Float64(t_0 * t_0) / Float64(x_m * Float64(x_m * Float64(x_m * x_m)))) - 0.25) / Float64(Float64(t_0 / Float64(x_m * x_m)) - 0.5)));
	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[(0.2514179000665374 + N[(N[(0.15298196345929074 + N[(11.259630434457211 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 2.1], 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 * 0.015175085973910875), $MachinePrecision] + 0.17858401087518092), $MachinePrecision] + 0.6665536072), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] / N[(N[(t$95$0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 71.3%

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) + 1}} \cdot x \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)}} \cdot x \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}}, 1\right)} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x, \frac{833192009}{1250000000}\right)}, 1\right)} \cdot x \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2} + \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      14. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      15. associate-*l* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      17. lower-*.f64 71.2

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.015175085973910875}, 0.17858401087518092\right), 0.6665536072\right), 1\right)} \cdot x \]

   7. Applied rewrites 71.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \cdot x \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right) + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)\right)} + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right)} + 1} \cdot x \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \cdot x \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{x}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \]

      9. lower-/.f64 71.2

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right) + 1}} \]

   9. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

   if 2.10000000000000009 < x

   1. Initial program 3.7%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374, 0.5\right)}{x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{\color{blue}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \color{blue}{\frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\color{blue}{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{\color{blue}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\color{blue}{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \color{blue}{\left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right)} + \frac{1}{2}}{x} \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}{x} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}} \]

      11. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(0.5 + \frac{\frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374}{x \cdot x}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{\color{blue}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \frac{\frac{\frac{1307076337763}{8543989815576} + \color{blue}{\frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \frac{\frac{\color{blue}{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{\color{blue}{x \cdot x}} + \frac{600041}{2386628}}{x \cdot x}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \frac{\color{blue}{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x}} + \frac{600041}{2386628}}{x \cdot x}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \frac{\color{blue}{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}}{x \cdot x}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{\color{blue}{x \cdot x}}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \color{blue}{\frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x}}\right) \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x} + \frac{1}{2}\right)} \]

      10. flip-+ N/A

          \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{\frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x} \cdot \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x} - \frac{1}{2} \cdot \frac{1}{2}}{\frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x} - \frac{1}{2}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{\frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x} \cdot \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x} - \frac{1}{2} \cdot \frac{1}{2}}{\frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x} - \frac{1}{2}}} \]

   9. Applied rewrites 99.2%

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{\frac{\left(\frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374\right) \cdot \left(\frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} - 0.25}{\frac{\frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374}{x \cdot x} - 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{\left(0.2514179000665374 + \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x}\right) \cdot \left(0.2514179000665374 + \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x}\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} - 0.25}{\frac{0.2514179000665374 + \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x}}{x \cdot x} - 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.15298196345929074 + \frac{11.259630434457211}{x\_m \cdot x\_m}}{x\_m \cdot x\_m} - -0.2514179000665374}{x\_m \cdot \left(x\_m \cdot x\_m\right)} + \frac{0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.1)
    (/
     x_m
     (fma
      x_m
      (*
       x_m
       (fma
        (* x_m x_m)
        (fma x_m (* x_m 0.015175085973910875) 0.17858401087518092)
        0.6665536072))
      1.0))
    (+
     (/
      (-
       (/
        (+ 0.15298196345929074 (/ 11.259630434457211 (* x_m x_m)))
        (* x_m x_m))
       -0.2514179000665374)
      (* x_m (* 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 tmp;
	if (x_m <= 2.1) {
		tmp = x_m / fma(x_m, (x_m * fma((x_m * x_m), fma(x_m, (x_m * 0.015175085973910875), 0.17858401087518092), 0.6665536072)), 1.0);
	} else {
		tmp = ((((0.15298196345929074 + (11.259630434457211 / (x_m * x_m))) / (x_m * x_m)) - -0.2514179000665374) / (x_m * (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)
	tmp = 0.0
	if (x_m <= 2.1)
		tmp = Float64(x_m / fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.015175085973910875), 0.17858401087518092), 0.6665536072)), 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.15298196345929074 + Float64(11.259630434457211 / Float64(x_m * x_m))) / Float64(x_m * x_m)) - -0.2514179000665374) / Float64(x_m * Float64(x_m * x_m))) + Float64(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_] := N[(x$95$s * If[LessEqual[x$95$m, 2.1], 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 * 0.015175085973910875), $MachinePrecision] + 0.17858401087518092), $MachinePrecision] + 0.6665536072), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.15298196345929074 + N[(11.259630434457211 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - -0.2514179000665374), $MachinePrecision] / N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 71.3%

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) + 1}} \cdot x \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)}} \cdot x \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}}, 1\right)} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x, \frac{833192009}{1250000000}\right)}, 1\right)} \cdot x \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2} + \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      14. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      15. associate-*l* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      17. lower-*.f64 71.2

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.015175085973910875}, 0.17858401087518092\right), 0.6665536072\right), 1\right)} \cdot x \]

   7. Applied rewrites 71.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \cdot x \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right) + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)\right)} + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right)} + 1} \cdot x \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \cdot x \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{x}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \]

      9. lower-/.f64 71.2

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right) + 1}} \]

   9. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

   if 2.10000000000000009 < x

   1. Initial program 3.7%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374, 0.5\right)}{x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{\color{blue}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \color{blue}{\frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\color{blue}{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{\color{blue}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\color{blue}{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \color{blue}{\left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right)} + \frac{1}{2}}{x} \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}{x} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}} \]

      11. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(0.5 + \frac{\frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374}{x \cdot x}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x}} \cdot \left(\frac{1}{2} + \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{\color{blue}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \frac{\frac{\frac{1307076337763}{8543989815576} + \color{blue}{\frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x}\right) \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \frac{\frac{\color{blue}{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{\color{blue}{x \cdot x}} + \frac{600041}{2386628}}{x \cdot x}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \frac{\color{blue}{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x}} + \frac{600041}{2386628}}{x \cdot x}\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \frac{\color{blue}{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}}{x \cdot x}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{\color{blue}{x \cdot x}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{2} + \color{blue}{\frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x}}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1}{2} + \frac{1}{x} \cdot \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x}} \]

      11. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{1}{2} + \frac{1}{x} \cdot \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x} \]

      12. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{2}}{x}} + \frac{1}{x} \cdot \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{x} + \frac{1}{x} \cdot \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x}} + \frac{1}{x} \cdot \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}}{x \cdot x} \]

   9. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{-0.2514179000665374 + \left(-\frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x}\right)}{\left(-x\right) \cdot \left(x \cdot x\right)} + \frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} - -0.2514179000665374}{x \cdot \left(x \cdot x\right)} + \frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m} \cdot \left(0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074 + \frac{11.259630434457211}{x\_m \cdot x\_m}}{x\_m \cdot x\_m}}{x\_m \cdot x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.1)
    (/
     x_m
     (fma
      x_m
      (*
       x_m
       (fma
        (* x_m x_m)
        (fma x_m (* x_m 0.015175085973910875) 0.17858401087518092)
        0.6665536072))
      1.0))
    (*
     (/ 1.0 x_m)
     (+
      0.5
      (/
       (+
        0.2514179000665374
        (/
         (+ 0.15298196345929074 (/ 11.259630434457211 (* x_m x_m)))
         (* x_m x_m)))
       (* x_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.1) {
		tmp = x_m / fma(x_m, (x_m * fma((x_m * x_m), fma(x_m, (x_m * 0.015175085973910875), 0.17858401087518092), 0.6665536072)), 1.0);
	} else {
		tmp = (1.0 / x_m) * (0.5 + ((0.2514179000665374 + ((0.15298196345929074 + (11.259630434457211 / (x_m * x_m))) / (x_m * x_m))) / (x_m * x_m)));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 2.1)
		tmp = Float64(x_m / fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.015175085973910875), 0.17858401087518092), 0.6665536072)), 1.0));
	else
		tmp = Float64(Float64(1.0 / x_m) * Float64(0.5 + Float64(Float64(0.2514179000665374 + Float64(Float64(0.15298196345929074 + Float64(11.259630434457211 / Float64(x_m * x_m))) / Float64(x_m * x_m))) / Float64(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, 2.1], 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 * 0.015175085973910875), $MachinePrecision] + 0.17858401087518092), $MachinePrecision] + 0.6665536072), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(0.5 + N[(N[(0.2514179000665374 + N[(N[(0.15298196345929074 + N[(11.259630434457211 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 71.3%

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) + 1}} \cdot x \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)}} \cdot x \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}}, 1\right)} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x, \frac{833192009}{1250000000}\right)}, 1\right)} \cdot x \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2} + \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      14. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      15. associate-*l* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      17. lower-*.f64 71.2

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.015175085973910875}, 0.17858401087518092\right), 0.6665536072\right), 1\right)} \cdot x \]

   7. Applied rewrites 71.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \cdot x \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right) + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)\right)} + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right)} + 1} \cdot x \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \cdot x \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{x}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \]

      9. lower-/.f64 71.2

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right) + 1}} \]

   9. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

   if 2.10000000000000009 < x

   1. Initial program 3.7%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374, 0.5\right)}{x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{\color{blue}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \color{blue}{\frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\color{blue}{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{\color{blue}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\color{blue}{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \color{blue}{\left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right)} + \frac{1}{2}}{x} \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}{x} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}} \]

      11. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(0.5 + \frac{\frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374}{x \cdot x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x}}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074 + \frac{11.259630434457211}{x\_m \cdot x\_m}}{x\_m \cdot x\_m}}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.1)
    (/
     x_m
     (fma
      x_m
      (*
       x_m
       (fma
        (* x_m x_m)
        (fma x_m (* x_m 0.015175085973910875) 0.17858401087518092)
        0.6665536072))
      1.0))
    (/
     (+
      0.5
      (/
       (+
        0.2514179000665374
        (/
         (+ 0.15298196345929074 (/ 11.259630434457211 (* x_m x_m)))
         (* x_m x_m)))
       (* x_m x_m)))
     x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.1) {
		tmp = x_m / fma(x_m, (x_m * fma((x_m * x_m), fma(x_m, (x_m * 0.015175085973910875), 0.17858401087518092), 0.6665536072)), 1.0);
	} else {
		tmp = (0.5 + ((0.2514179000665374 + ((0.15298196345929074 + (11.259630434457211 / (x_m * x_m))) / (x_m * x_m))) / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 2.1)
		tmp = Float64(x_m / fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.015175085973910875), 0.17858401087518092), 0.6665536072)), 1.0));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.2514179000665374 + Float64(Float64(0.15298196345929074 + Float64(11.259630434457211 / Float64(x_m * x_m))) / 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, 2.1], 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 * 0.015175085973910875), $MachinePrecision] + 0.17858401087518092), $MachinePrecision] + 0.6665536072), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(0.2514179000665374 + N[(N[(0.15298196345929074 + N[(11.259630434457211 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 71.3%

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) + 1}} \cdot x \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)}} \cdot x \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}}, 1\right)} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x, \frac{833192009}{1250000000}\right)}, 1\right)} \cdot x \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2} + \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      14. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      15. associate-*l* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      17. lower-*.f64 71.2

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.015175085973910875}, 0.17858401087518092\right), 0.6665536072\right), 1\right)} \cdot x \]

   7. Applied rewrites 71.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \cdot x \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right) + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)\right)} + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right)} + 1} \cdot x \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \cdot x \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{x}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \]

      9. lower-/.f64 71.2

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right) + 1}} \]

   9. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

   if 2.10000000000000009 < x

   1. Initial program 3.7%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374, 0.5\right)}{x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{\color{blue}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \color{blue}{\frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\color{blue}{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{\color{blue}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\color{blue}{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \color{blue}{\left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right)} + \frac{1}{2}}{x} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}}{x} \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\color{blue}{\frac{\frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374}{x \cdot x} + 0.5}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x}}{x \cdot x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 7.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.45:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x\_m \cdot x\_m}}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

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 1.45)
    (/
     x_m
     (fma
      x_m
      (*
       x_m
       (fma
        (* x_m x_m)
        (fma x_m (* x_m 0.015175085973910875) 0.17858401087518092)
        0.6665536072))
      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 <= 1.45) {
		tmp = x_m / fma(x_m, (x_m * fma((x_m * x_m), fma(x_m, (x_m * 0.015175085973910875), 0.17858401087518092), 0.6665536072)), 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 <= 1.45)
		tmp = Float64(x_m / fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.015175085973910875), 0.17858401087518092), 0.6665536072)), 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, 1.45], 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 * 0.015175085973910875), $MachinePrecision] + 0.17858401087518092), $MachinePrecision] + 0.6665536072), $MachinePrecision]), $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 < 1.44999999999999996

   1. Initial program 71.3%

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) + 1}} \cdot x \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)}} \cdot x \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}}, 1\right)} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x, \frac{833192009}{1250000000}\right)}, 1\right)} \cdot x \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2} + \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      14. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      15. associate-*l* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      17. lower-*.f64 71.2

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.015175085973910875}, 0.17858401087518092\right), 0.6665536072\right), 1\right)} \cdot x \]

   7. Applied rewrites 71.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \cdot x \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right) + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)\right)} + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right)} + 1} \cdot x \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \cdot x \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{x}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \]

      9. lower-/.f64 71.2

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right) + 1}} \]

   9. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

   if 1.44999999999999996 < x

   1. Initial program 3.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.1%

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

Alternative 8: 99.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.82:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m} \cdot \left(0.5 + \frac{0.2514179000665374}{x\_m \cdot x\_m}\right)\\ \end{array} \end{array} \]

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 1.82)
    (/
     x_m
     (fma
      x_m
      (*
       x_m
       (fma
        (* x_m x_m)
        (fma x_m (* x_m 0.015175085973910875) 0.17858401087518092)
        0.6665536072))
      1.0))
    (* (/ 1.0 x_m) (+ 0.5 (/ 0.2514179000665374 (* 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 <= 1.82) {
		tmp = x_m / fma(x_m, (x_m * fma((x_m * x_m), fma(x_m, (x_m * 0.015175085973910875), 0.17858401087518092), 0.6665536072)), 1.0);
	} else {
		tmp = (1.0 / x_m) * (0.5 + (0.2514179000665374 / (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 <= 1.82)
		tmp = Float64(x_m / fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * 0.015175085973910875), 0.17858401087518092), 0.6665536072)), 1.0));
	else
		tmp = Float64(Float64(1.0 / x_m) * Float64(0.5 + Float64(0.2514179000665374 / Float64(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, 1.82], 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 * 0.015175085973910875), $MachinePrecision] + 0.17858401087518092), $MachinePrecision] + 0.6665536072), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(0.5 + N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.82000000000000006

   1. Initial program 71.3%

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) + 1}} \cdot x \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)}} \cdot x \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}}, 1\right)} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x, \frac{833192009}{1250000000}\right)}, 1\right)} \cdot x \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2} + \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      14. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      15. associate-*l* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      17. lower-*.f64 71.2

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.015175085973910875}, 0.17858401087518092\right), 0.6665536072\right), 1\right)} \cdot x \]

   7. Applied rewrites 71.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \cdot x \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right) + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)\right)} + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right)} + 1} \cdot x \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \cdot x \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{x}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \]

      9. lower-/.f64 71.2

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right) + 1}} \]

   9. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

   if 1.82000000000000006 < x

   1. Initial program 3.7%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374, 0.5\right)}{x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{\color{blue}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \color{blue}{\frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\color{blue}{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{\color{blue}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\color{blue}{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \color{blue}{\left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right)} + \frac{1}{2}}{x} \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}{x} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}} \]

      11. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(0.5 + \frac{\frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374}{x \cdot x}\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.9

         \[\leadsto \frac{1}{x} \cdot \left(0.5 + \frac{0.2514179000665374}{\color{blue}{x \cdot x}}\right) \]

   10. Applied rewrites 98.9%

       \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(0.5 + \frac{0.2514179000665374}{x \cdot x}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.6% accurate, 9.2× speedup

Math

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

FPCore

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

C

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.2)
		tmp = Float64(x_m * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), -0.0732490286039007, 0.265709700396151)), -0.6665536072), 1.0));
	else
		tmp = Float64(Float64(1.0 / x_m) * Float64(0.5 + Float64(0.2514179000665374 / Float64(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, 1.2], N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.0732490286039007 + 0.265709700396151), $MachinePrecision]), $MachinePrecision] + -0.6665536072), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(0.5 + N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 71.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right) + 1\right)} \cdot x \]

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-9156128575487588197208397249}{125000000000000000000000000000}, \frac{3321371254951887171}{12500000000000000000}\right)}, \frac{-833192009}{1250000000}\right), 1\right) \cdot x \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-9156128575487588197208397249}{125000000000000000000000000000}, \frac{3321371254951887171}{12500000000000000000}\right), \frac{-833192009}{1250000000}\right), 1\right) \cdot x \]

      15. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   if 1.19999999999999996 < x

   1. Initial program 3.7%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374, 0.5\right)}{x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{\color{blue}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \color{blue}{\frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\color{blue}{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{\color{blue}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\color{blue}{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \color{blue}{\left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right)} + \frac{1}{2}}{x} \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}{x} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}} \]

      11. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(0.5 + \frac{\frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374}{x \cdot x}\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.9

         \[\leadsto \frac{1}{x} \cdot \left(0.5 + \frac{0.2514179000665374}{\color{blue}{x \cdot x}}\right) \]

   10. Applied rewrites 98.9%

       \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(0.5 + \frac{0.2514179000665374}{x \cdot x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

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

Alternative 10: 99.6% accurate, 9.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.4:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.17858401087518092, 0.6665536072\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m} \cdot \left(0.5 + \frac{0.2514179000665374}{x\_m \cdot x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.4)
    (/
     x_m
     (fma (* x_m x_m) (fma (* x_m x_m) 0.17858401087518092 0.6665536072) 1.0))
    (* (/ 1.0 x_m) (+ 0.5 (/ 0.2514179000665374 (* x_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.4) {
		tmp = x_m / fma((x_m * x_m), fma((x_m * x_m), 0.17858401087518092, 0.6665536072), 1.0);
	} else {
		tmp = (1.0 / x_m) * (0.5 + (0.2514179000665374 / (x_m * x_m)));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 2.4)
		tmp = Float64(x_m / fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.17858401087518092, 0.6665536072), 1.0));
	else
		tmp = Float64(Float64(1.0 / x_m) * Float64(0.5 + Float64(0.2514179000665374 / Float64(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, 2.4], N[(x$95$m / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.17858401087518092 + 0.6665536072), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(0.5 + N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 71.3%

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) + 1}} \cdot x \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)}} \cdot x \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}}, 1\right)} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x, \frac{833192009}{1250000000}\right)}, 1\right)} \cdot x \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2} + \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      14. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      15. associate-*l* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      17. lower-*.f64 71.2

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.015175085973910875}, 0.17858401087518092\right), 0.6665536072\right), 1\right)} \cdot x \]

   7. Applied rewrites 71.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \cdot x \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right) + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)\right)} + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right)} + 1} \cdot x \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \cdot x \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{x}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \]

      9. lower-/.f64 71.2

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right) + 1}} \]

   9. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right) + 1}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}, 1\right)}} \]

       3. unpow2 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}, 1\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}, 1\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2} + \frac{833192009}{1250000000}}, 1\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{2232300135939761477}{12500000000000000000}} + \frac{833192009}{1250000000}, 1\right)} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2232300135939761477}{12500000000000000000}, \frac{833192009}{1250000000}\right)}, 1\right)} \]

       8. unpow2 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2232300135939761477}{12500000000000000000}, \frac{833192009}{1250000000}\right), 1\right)} \]

       9. lower-*.f64 71.2

          \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.17858401087518092, 0.6665536072\right), 1\right)} \]

   12. Applied rewrites 71.2%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.17858401087518092, 0.6665536072\right), 1\right)}} \]

   if 2.39999999999999991 < x

   1. Initial program 3.7%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374, 0.5\right)}{x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot x}} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{\color{blue}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \color{blue}{\frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\color{blue}{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}}{x \cdot x} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{\color{blue}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \left(\color{blue}{\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x}} + \frac{600041}{2386628}\right) + \frac{1}{2}}{x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x \cdot x} \cdot \color{blue}{\left(\frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}\right)} + \frac{1}{2}}{x} \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}{x} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x}{\mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)}}} \]

      11. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \mathsf{fma}\left(\frac{1}{x \cdot x}, \frac{\frac{1307076337763}{8543989815576} + \frac{\frac{344398180852034095277}{30586987988352776592}}{x \cdot x}}{x \cdot x} + \frac{600041}{2386628}, \frac{1}{2}\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(0.5 + \frac{\frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot x} + 0.2514179000665374}{x \cdot x}\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.9

         \[\leadsto \frac{1}{x} \cdot \left(0.5 + \frac{0.2514179000665374}{\color{blue}{x \cdot x}}\right) \]

   10. Applied rewrites 98.9%

       \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(0.5 + \frac{0.2514179000665374}{x \cdot x}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 99.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.4:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.17858401087518092, 0.6665536072\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.4)
    (/
     x_m
     (fma (* x_m x_m) (fma (* x_m x_m) 0.17858401087518092 0.6665536072) 1.0))
    (/ (+ 0.5 (/ 0.2514179000665374 (* x_m x_m))) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.4) {
		tmp = x_m / fma((x_m * x_m), fma((x_m * x_m), 0.17858401087518092, 0.6665536072), 1.0);
	} else {
		tmp = (0.5 + (0.2514179000665374 / (x_m * x_m))) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 2.4)
		tmp = Float64(x_m / fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.17858401087518092, 0.6665536072), 1.0));
	else
		tmp = Float64(Float64(0.5 + Float64(0.2514179000665374 / 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, 2.4], N[(x$95$m / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.17858401087518092 + 0.6665536072), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 71.3%

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) + 1}} \cdot x \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)}} \cdot x \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}}, 1\right)} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x, \frac{833192009}{1250000000}\right)}, 1\right)} \cdot x \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2} + \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      14. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      15. associate-*l* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      17. lower-*.f64 71.2

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.015175085973910875}, 0.17858401087518092\right), 0.6665536072\right), 1\right)} \cdot x \]

   7. Applied rewrites 71.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \cdot x \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right) + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)\right)} + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right)} + 1} \cdot x \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \cdot x \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{x}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \]

      9. lower-/.f64 71.2

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right) + 1}} \]

   9. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right) + 1}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}, 1\right)}} \]

       3. unpow2 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}, 1\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}, 1\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2} + \frac{833192009}{1250000000}}, 1\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{2232300135939761477}{12500000000000000000}} + \frac{833192009}{1250000000}, 1\right)} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2232300135939761477}{12500000000000000000}, \frac{833192009}{1250000000}\right)}, 1\right)} \]

       8. unpow2 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2232300135939761477}{12500000000000000000}, \frac{833192009}{1250000000}\right), 1\right)} \]

       9. lower-*.f64 71.2

          \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.17858401087518092, 0.6665536072\right), 1\right)} \]

   12. Applied rewrites 71.2%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.17858401087518092, 0.6665536072\right), 1\right)}} \]

   if 2.39999999999999991 < x

   1. Initial program 3.7%

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 12: 99.5% accurate, 11.2× speedup

Math

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

FPCore

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

C

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = fma(Float64(x_m * x_m), Float64(x_m * fma(x_m, Float64(x_m * 0.265709700396151), -0.6665536072)), x_m);
	else
		tmp = Float64(Float64(0.5 + Float64(0.2514179000665374 / 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, 1.1], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.265709700396151), $MachinePrecision] + -0.6665536072), $MachinePrecision]), $MachinePrecision] + x$95$m), $MachinePrecision], N[(N[(0.5 + N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 71.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right) + x \cdot 1} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right) \cdot x} + x \cdot 1 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) \cdot x\right)} + x \cdot 1 \]

      5. *-commutative N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)} + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto {x}^{2} \cdot \left(x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right), x\right)} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right), x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right), x\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)}, x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right)\right)}, x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{3321371254951887171}{12500000000000000000}} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right)\right), x\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{3321371254951887171}{12500000000000000000} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right)\right), x\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{3321371254951887171}{12500000000000000000}\right)} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(x \cdot \frac{3321371254951887171}{12500000000000000000}\right) + \color{blue}{\frac{-833192009}{1250000000}}\right), x\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{3321371254951887171}{12500000000000000000}, \frac{-833192009}{1250000000}\right)}, x\right) \]

      17. lower-*.f64 70.0

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.265709700396151}, -0.6665536072\right), x\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.265709700396151, -0.6665536072\right), x\right)} \]

   if 1.1000000000000001 < x

   1. Initial program 3.7%

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 13: 99.3% accurate, 12.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.9:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.265709700396151, -0.6665536072\right), x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.9)
    (fma
     (* x_m x_m)
     (* x_m (fma x_m (* x_m 0.265709700396151) -0.6665536072))
     x_m)
    (/ 0.5 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = fma((x_m * x_m), (x_m * fma(x_m, (x_m * 0.265709700396151), -0.6665536072)), x_m);
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.9)
		tmp = fma(Float64(x_m * x_m), Float64(x_m * fma(x_m, Float64(x_m * 0.265709700396151), -0.6665536072)), x_m);
	else
		tmp = Float64(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_] := N[(x$95$s * If[LessEqual[x$95$m, 0.9], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.265709700396151), $MachinePrecision] + -0.6665536072), $MachinePrecision]), $MachinePrecision] + x$95$m), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 71.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right) + x \cdot 1} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right) \cdot x} + x \cdot 1 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) \cdot x\right)} + x \cdot 1 \]

      5. *-commutative N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)} + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto {x}^{2} \cdot \left(x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right), x\right)} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right), x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right), x\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)}, x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right)\right)}, x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{3321371254951887171}{12500000000000000000}} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right)\right), x\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{3321371254951887171}{12500000000000000000} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right)\right), x\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{3321371254951887171}{12500000000000000000}\right)} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(x \cdot \frac{3321371254951887171}{12500000000000000000}\right) + \color{blue}{\frac{-833192009}{1250000000}}\right), x\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{3321371254951887171}{12500000000000000000}, \frac{-833192009}{1250000000}\right)}, x\right) \]

      17. lower-*.f64 70.0

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.265709700396151}, -0.6665536072\right), x\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.265709700396151, -0.6665536072\right), x\right)} \]

   if 0.900000000000000022 < x

   1. Initial program 3.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 14: 99.2% accurate, 14.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.86:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(x\_m, x\_m \cdot 0.6665536072, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.86) (/ x_m (fma x_m (* x_m 0.6665536072) 1.0)) (/ 0.5 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.86) {
		tmp = x_m / fma(x_m, (x_m * 0.6665536072), 1.0);
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.86)
		tmp = Float64(x_m / fma(x_m, Float64(x_m * 0.6665536072), 1.0));
	else
		tmp = Float64(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_] := N[(x$95$s * If[LessEqual[x$95$m, 0.86], N[(x$95$m / N[(x$95$m * N[(x$95$m * 0.6665536072), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.859999999999999987

   1. Initial program 71.3%

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) + 1}} \cdot x \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)}} \cdot x \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), 1\right)} \cdot x \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}}, 1\right)} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x\right)} + \frac{833192009}{1250000000}, 1\right)} \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot x, \frac{833192009}{1250000000}\right)}, 1\right)} \cdot x \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2} + \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      14. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      15. associate-*l* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}, \frac{833192009}{1250000000}\right), 1\right)} \cdot x \]

      17. lower-*.f64 71.2

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.015175085973910875}, 0.17858401087518092\right), 0.6665536072\right), 1\right)} \cdot x \]

   7. Applied rewrites 71.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \cdot x \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right) + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}\right)} + \frac{2232300135939761477}{12500000000000000000}\right)\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)}\right) + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right)\right)} + \frac{833192009}{1250000000}\right) + 1} \cdot x \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right)} + 1} \cdot x \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \cdot x \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)}} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{x}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right), 1\right)} \]

      9. lower-/.f64 71.2

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1896885746738859378453363281}{125000000000000000000000000000}, \frac{2232300135939761477}{12500000000000000000}\right), \frac{833192009}{1250000000}\right) + 1}} \]

   9. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.015175085973910875, 0.17858401087518092\right), 0.6665536072\right), 1\right)}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{x}{\color{blue}{1 + \frac{833192009}{1250000000} \cdot {x}^{2}}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{833192009}{1250000000} \cdot {x}^{2} + 1}} \]

       2. unpow2 N/A

          \[\leadsto \frac{x}{\frac{833192009}{1250000000} \cdot \color{blue}{\left(x \cdot x\right)} + 1} \]

       3. associate-*r* N/A

          \[\leadsto \frac{x}{\color{blue}{\left(\frac{833192009}{1250000000} \cdot x\right) \cdot x} + 1} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{x \cdot \left(\frac{833192009}{1250000000} \cdot x\right)} + 1} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, \frac{833192009}{1250000000} \cdot x, 1\right)}} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{833192009}{1250000000}}, 1\right)} \]

       7. lower-*.f64 72.8

          \[\leadsto \frac{x}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.6665536072}, 1\right)} \]

   12. Applied rewrites 72.8%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, x \cdot 0.6665536072, 1\right)}} \]

   if 0.859999999999999987 < x

   1. Initial program 3.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 15: 99.2% accurate, 18.0× speedup

Math

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

FPCore

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

C

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.78)
		tmp = fma(x_m, Float64(x_m * Float64(x_m * -0.6665536072)), x_m);
	else
		tmp = Float64(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_] := N[(x$95$s * If[LessEqual[x$95$m, 0.78], N[(x$95$m * N[(x$95$m * N[(x$95$m * -0.6665536072), $MachinePrecision]), $MachinePrecision] + x$95$m), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.78000000000000003

   1. Initial program 71.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-833192009}{1250000000} \cdot {x}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-833192009}{1250000000} \cdot {x}^{2}\right) + x \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto x \cdot \left(\frac{-833192009}{1250000000} \cdot {x}^{2}\right) + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-833192009}{1250000000} \cdot {x}^{2}, x\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-833192009}{1250000000}}, x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-833192009}{1250000000}, x\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{-833192009}{1250000000}\right)}, x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{-833192009}{1250000000}\right)}, x\right) \]

      9. lower-*.f64 69.5

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot -0.6665536072\right)}, x\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot -0.6665536072\right), x\right)} \]

   if 0.78000000000000003 < x

   1. Initial program 3.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 16: 99.2% accurate, 18.0× speedup

Math

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

FPCore

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

C

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.78)
		tmp = Float64(x_m * fma(x_m, Float64(x_m * -0.6665536072), 1.0));
	else
		tmp = Float64(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_] := N[(x$95$s * If[LessEqual[x$95$m, 0.78], N[(x$95$m * N[(x$95$m * N[(x$95$m * -0.6665536072), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.78000000000000003

   1. Initial program 71.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-833192009}{1250000000} \cdot {x}^{2} + 1\right)} \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \frac{-833192009}{1250000000}} + 1\right) \cdot x \]

      3. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-833192009}{1250000000} + 1\right) \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(x \cdot \frac{-833192009}{1250000000}\right)} + 1\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-833192009}{1250000000}, 1\right)} \cdot x \]

      6. lower-*.f64 69.5

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

   5. Applied rewrites 69.5%

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

   if 0.78000000000000003 < x

   1. Initial program 3.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 75.5%

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

Alternative 17: 99.0% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 71.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 69.8%

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

      1. *-lft-identity 69.8

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

   7. Applied rewrites 69.8%

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

   if 0.69999999999999996 < x

   1. Initial program 3.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 18: 50.4% accurate, 415.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 55.8%

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

   1. *-lft-identity 55.8

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

7. Applied rewrites 55.8%

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

Reproduce

herbie shell --seed 2024216 
(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))