Jmat.Real.dawson

Average Error: 28.9 → 0.0

Time: 12.0s

Precision: binary64

Cost: 88456

Math

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

↓

\[\begin{array}{l} t_0 := {\left(x \cdot x\right)}^{5}\\ t_1 := {\left(x \cdot x\right)}^{2}\\ t_2 := {\left(x \cdot x\right)}^{3}\\ \mathbf{if}\;x \leq -30540458040.676582:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 434.0116910138749:\\ \;\;\;\;x \cdot \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \mathsf{fma}\left(0.0424060604, t_1, 0.0072644182 \cdot t_2\right)\right) + \left({x}^{8} \cdot 0.0005064034 + 0.0001789971 \cdot t_0\right)}{\left(\left(1 + \mathsf{fma}\left(x \cdot x, 0.7715471019, t_1 \cdot 0.2909738639\right)\right) + \mathsf{fma}\left(t_2, 0.0694555761, \left(x \cdot x\right) \cdot \left(t_2 \cdot 0.0140005442\right)\right)\right) + \mathsf{fma}\left(t_0, 0.0008327945, 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \end{array} \]

FPCore

(FPCore (x)
 :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))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (* x x) 5.0))
        (t_1 (pow (* x x) 2.0))
        (t_2 (pow (* x x) 3.0)))
   (if (<= x -30540458040.676582)
     (/ 0.5 x)
     (if (<= x 434.0116910138749)
       (*
        x
        (/
         (+
          (+
           (+ 1.0 (* 0.1049934947 (* x x)))
           (fma 0.0424060604 t_1 (* 0.0072644182 t_2)))
          (+ (* (pow x 8.0) 0.0005064034) (* 0.0001789971 t_0)))
         (+
          (+
           (+ 1.0 (fma (* x x) 0.7715471019 (* t_1 0.2909738639)))
           (fma t_2 0.0694555761 (* (* x x) (* t_2 0.0140005442))))
          (fma t_0 0.0008327945 (* 0.0003579942 (* (* x x) t_0))))))
       (+ (/ 0.5 x) (/ 0.2514179000665374 (pow x 3.0)))))))

C

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

↓

double code(double x) {
	double t_0 = pow((x * x), 5.0);
	double t_1 = pow((x * x), 2.0);
	double t_2 = pow((x * x), 3.0);
	double tmp;
	if (x <= -30540458040.676582) {
		tmp = 0.5 / x;
	} else if (x <= 434.0116910138749) {
		tmp = x * ((((1.0 + (0.1049934947 * (x * x))) + fma(0.0424060604, t_1, (0.0072644182 * t_2))) + ((pow(x, 8.0) * 0.0005064034) + (0.0001789971 * t_0))) / (((1.0 + fma((x * x), 0.7715471019, (t_1 * 0.2909738639))) + fma(t_2, 0.0694555761, ((x * x) * (t_2 * 0.0140005442)))) + fma(t_0, 0.0008327945, (0.0003579942 * ((x * x) * t_0)))));
	} else {
		tmp = (0.5 / x) + (0.2514179000665374 / pow(x, 3.0));
	}
	return tmp;
}

Julia

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

↓

function code(x)
	t_0 = Float64(x * x) ^ 5.0
	t_1 = Float64(x * x) ^ 2.0
	t_2 = Float64(x * x) ^ 3.0
	tmp = 0.0
	if (x <= -30540458040.676582)
		tmp = Float64(0.5 / x);
	elseif (x <= 434.0116910138749)
		tmp = Float64(x * Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + fma(0.0424060604, t_1, Float64(0.0072644182 * t_2))) + Float64(Float64((x ^ 8.0) * 0.0005064034) + Float64(0.0001789971 * t_0))) / Float64(Float64(Float64(1.0 + fma(Float64(x * x), 0.7715471019, Float64(t_1 * 0.2909738639))) + fma(t_2, 0.0694555761, Float64(Float64(x * x) * Float64(t_2 * 0.0140005442)))) + fma(t_0, 0.0008327945, Float64(0.0003579942 * Float64(Float64(x * x) * t_0))))));
	else
		tmp = Float64(Float64(0.5 / x) + Float64(0.2514179000665374 / (x ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 + N[(0.7715471019 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[Power[N[(x * x), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x * x), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(x * x), $MachinePrecision], 3.0], $MachinePrecision]}, If[LessEqual[x, -30540458040.676582], N[(0.5 / x), $MachinePrecision], If[LessEqual[x, 434.0116910138749], N[(x * N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$1 + N[(0.0072644182 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 8.0], $MachinePrecision] * 0.0005064034), $MachinePrecision] + N[(0.0001789971 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.7715471019 + N[(t$95$1 * 0.2909738639), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * 0.0694555761 + N[(N[(x * x), $MachinePrecision] * N[(t$95$2 * 0.0140005442), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.0008327945 + N[(0.0003579942 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] + N[(0.2514179000665374 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -30540458040.676582

   1. Initial program 60.9

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

   2. Taylor expanded in x around inf 0

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

   if -30540458040.676582 < x < 434.011691013874895

   1. Initial program 0.0

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

   2. Applied egg-rr 57.8

      \[\leadsto \color{blue}{\log \left(e^{x \cdot \frac{\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot {\left(x \cdot x\right)}^{2} + 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right)\right) + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)\right)}}\right)} \]

   3. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \mathsf{fma}\left(0.0424060604, {\left(x \cdot x\right)}^{2}, 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \left(0.0005064034 \cdot \left(x \cdot \left(x \cdot {\left(x \cdot x\right)}^{3}\right)\right) + 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)}{\left(\left(1 + \mathsf{fma}\left(x \cdot x, 0.7715471019, {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 0.0694555761, \left(x \cdot x\right) \cdot \left({\left(x \cdot x\right)}^{3} \cdot 0.0140005442\right)\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{5}\right)\right)} \cdot x} \]

   4. Taylor expanded in x around 0 0.0

      \[\leadsto \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \mathsf{fma}\left(0.0424060604, {\left(x \cdot x\right)}^{2}, 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \left(\color{blue}{0.0005064034 \cdot {x}^{8}} + 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)}{\left(\left(1 + \mathsf{fma}\left(x \cdot x, 0.7715471019, {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 0.0694555761, \left(x \cdot x\right) \cdot \left({\left(x \cdot x\right)}^{3} \cdot 0.0140005442\right)\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{5}\right)\right)} \cdot x \]

   5. Simplified 0.0

      \[\leadsto \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \mathsf{fma}\left(0.0424060604, {\left(x \cdot x\right)}^{2}, 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \left(\color{blue}{{x}^{8} \cdot 0.0005064034} + 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)}{\left(\left(1 + \mathsf{fma}\left(x \cdot x, 0.7715471019, {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 0.0694555761, \left(x \cdot x\right) \cdot \left({\left(x \cdot x\right)}^{3} \cdot 0.0140005442\right)\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{5}\right)\right)} \cdot x \]
      Proof
      (*.f64 (pow.f64 x 8) 2532017/5000000000): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 2532017/5000000000 (pow.f64 x 8))): 0 points increase in error, 0 points decrease in error

   if 434.011691013874895 < x

   1. Initial program 59.4

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

   2. Applied egg-rr 59.4

      \[\leadsto \color{blue}{\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot {\left(x \cdot x\right)}^{2} + 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right)\right) + 0.0001789971 \cdot \left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)\right) \cdot x}{\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \left({\left(x \cdot x\right)}^{3} \cdot 0.0694555761 + \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 0.0140005442\right)\right) + \left(\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{2}\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{3}\right) \cdot {\left(x \cdot x\right)}^{2}\right)\right)}} \]

   3. Taylor expanded in x around inf 0.0

      \[\leadsto \color{blue}{0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}} \]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\frac{0.2514179000665374}{{x}^{3}} + \frac{0.5}{x}} \]
      Proof
      (+.f64 (/.f64 600041/2386628 (pow.f64 x 3)) (/.f64 1/2 x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 600041/2386628 1)) (pow.f64 x 3)) (/.f64 1/2 x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 600041/2386628 (/.f64 1 (pow.f64 x 3)))) (/.f64 1/2 x)): 6 points increase in error, 4 points decrease in error
      (+.f64 (*.f64 600041/2386628 (/.f64 1 (pow.f64 x 3))) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 600041/2386628 (/.f64 1 (pow.f64 x 3))) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -30540458040.676582:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 434.0116910138749:\\ \;\;\;\;x \cdot \frac{\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \mathsf{fma}\left(0.0424060604, {\left(x \cdot x\right)}^{2}, 0.0072644182 \cdot {\left(x \cdot x\right)}^{3}\right)\right) + \left({x}^{8} \cdot 0.0005064034 + 0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)}{\left(\left(1 + \mathsf{fma}\left(x \cdot x, 0.7715471019, {\left(x \cdot x\right)}^{2} \cdot 0.2909738639\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 0.0694555761, \left(x \cdot x\right) \cdot \left({\left(x \cdot x\right)}^{3} \cdot 0.0140005442\right)\right)\right) + \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot {\left(x \cdot x\right)}^{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	77064

\[\begin{array}{l} t_0 := {\left(x \cdot x\right)}^{3}\\ t_1 := {\left(x \cdot x\right)}^{2}\\ t_2 := \left(x \cdot x\right) \cdot t_0\\ \mathbf{if}\;x \leq -30540458040.676582:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 434.0116910138749:\\ \;\;\;\;\frac{x \cdot \left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0072644182 \cdot t_0 + 0.0424060604 \cdot t_1\right)\right) + 0.0005064034 \cdot t_2\right) + 0.0001789971 \cdot \left(t_1 \cdot t_0\right)\right)}{\left(\left(1 + \left(t_1 \cdot 0.2909738639 + \left(x \cdot x\right) \cdot 0.7715471019\right)\right) + \left(\left(x \cdot x\right) \cdot \left(t_0 \cdot 0.0140005442\right) + t_0 \cdot 0.0694555761\right)\right) + \left(0.0008327945 \cdot {x}^{10} + 0.0003579942 \cdot \left(t_1 \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \end{array} \]

Alternative 2
Error	0.0
Cost	76680

\[\begin{array}{l} t_0 := {\left(x \cdot x\right)}^{2}\\ t_1 := {\left(x \cdot x\right)}^{3}\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ \mathbf{if}\;x \leq -30540458040.676582:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 434.0116910138749:\\ \;\;\;\;\frac{x \cdot \left(0.0001789971 \cdot \left(t_0 \cdot t_1\right) + \left({x}^{8} \cdot 0.0005064034 + \left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0072644182 \cdot t_1 + 0.0424060604 \cdot t_0\right)\right)\right)\right)}{\left(0.0008327945 \cdot {x}^{10} + 0.0003579942 \cdot \left(t_0 \cdot t_2\right)\right) + \left(\left(1 + \left(t_0 \cdot 0.2909738639 + \left(x \cdot x\right) \cdot 0.7715471019\right)\right) + \left(t_1 \cdot 0.0694555761 + 0.0140005442 \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \end{array} \]

Alternative 3
Error	0.0
Cost	11208

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ t_3 := \left(x \cdot x\right) \cdot t_2\\ \mathbf{if}\;x \leq -30540458040.676582:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 434.0116910138749:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot t_1\right) + 0.0005064034 \cdot t_2\right) + 0.0001789971 \cdot t_3}{\left(\left(\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + 0.2909738639 \cdot t_0\right) + 0.0694555761 \cdot t_1\right) + 0.0140005442 \cdot t_2\right) + 0.0008327945 \cdot t_3\right) + 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \end{array} \]

Alternative 4
Error	0.4
Cost	7176

\[\begin{array}{l} t_0 := \frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \mathbf{if}\;x \leq -3.018783031278212:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0029421815227809534:\\ \;\;\;\;x \cdot \left(2 + \left(x \cdot x\right) \cdot -0.6665536072\right) - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.5
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -3.018783031278212:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.0029421815227809534:\\ \;\;\;\;x \cdot \left(2 + \left(x \cdot x\right) \cdot -0.6665536072\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 6
Error	0.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -3.018783031278212:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.0029421815227809534:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot -0.6665536072\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 7
Error	0.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -3.018783031278212:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.0029421815227809534:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 8
Error	30.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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))