Jmat.Real.erf

Percentage Accurate: 79.0% → 99.9%

Time: 28.6s

Alternatives: 13

Speedup: 279.5×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t_0 \cdot \left(0.254829592 + t_0 \cdot \left(-0.284496736 + t_0 \cdot \left(1.421413741 + t_0 \cdot \left(-1.453152027 + t_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (+
         -0.284496736
         (*
          t_0
          (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(fabs(x) * fabs(x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))
    code = 1.0d0 - ((t_0 * (0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (t_0 * 1.061405429d0))))))))) * exp(-(abs(x) * abs(x))))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * Math.abs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * Math.exp(-(Math.abs(x) * Math.abs(x))));
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + (0.3275911 * math.fabs(x)))
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * math.exp(-(math.fabs(x) * math.fabs(x))))

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * Float64(-1.453152027 + Float64(t_0 * 1.061405429))))))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / (1.0 + (0.3275911 * abs(x)));
	tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(abs(x) * abs(x))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(-1.453152027 + N[(t$95$0 * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t_0 \cdot \left(0.254829592 + t_0 \cdot \left(-0.284496736 + t_0 \cdot \left(1.421413741 + t_0 \cdot \left(-1.453152027 + t_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (+
         -0.284496736
         (*
          t_0
          (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(fabs(x) * fabs(x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))
    code = 1.0d0 - ((t_0 * (0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (t_0 * 1.061405429d0))))))))) * exp(-(abs(x) * abs(x))))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * Math.abs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * Math.exp(-(Math.abs(x) * Math.abs(x))));
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + (0.3275911 * math.fabs(x)))
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * math.exp(-(math.fabs(x) * math.fabs(x))))

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * Float64(-1.453152027 + Float64(t_0 * 1.061405429))))))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / (1.0 + (0.3275911 * abs(x)));
	tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(abs(x) * abs(x))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(-1.453152027 + N[(t$95$0 * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \frac{1}{1 + x \cdot 0.3275911}\\ t_1 := \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ t_2 := 1 + \left|x\right| \cdot 0.3275911\\ \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\frac{10^{-27} + {t_1}^{3}}{\mathsf{fma}\left(t_1, t_1 + -1 \cdot 10^{-9}, 10^{-18}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{t_2} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\left(-0.284496736 + t_0 \cdot \left(1.421413741 + t_0 \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2} \cdot 1.126581484710674 + \frac{1.5423834506201546}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}\right)\right) \cdot \frac{-1}{t_2} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* x 0.3275911))))
        (t_1
         (fma
          (pow x 3.0)
          -0.37545125292247583
          (* x (+ 1.128386358070218 (* x -0.00011824294398844343)))))
        (t_2 (+ 1.0 (* (fabs x) 0.3275911))))
   (if (<= (fabs x) 0.0001)
     (/ (+ 1e-27 (pow t_1 3.0)) (fma t_1 (+ t_1 -1e-9) 1e-18))
     (+
      1.0
      (*
       (/ 1.0 t_2)
       (*
        (exp (* x (- x)))
        (-
         (*
          (+
           -0.284496736
           (*
            t_0
            (+
             1.421413741
             (*
              t_0
              (/
               (+
                -3.0685496600615605
                (/ 1.1957597040827899 (pow (fma 0.3275911 x 1.0) 3.0)))
               (+
                2.111650813574209
                (+
                 (* (pow (fma 0.3275911 x 1.0) -2.0) 1.126581484710674)
                 (/ 1.5423834506201546 (fma 0.3275911 x 1.0)))))))))
          (/ -1.0 t_2))
         0.254829592)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 / (1.0 + (x * 0.3275911));
	double t_1 = fma(pow(x, 3.0), -0.37545125292247583, (x * (1.128386358070218 + (x * -0.00011824294398844343))));
	double t_2 = 1.0 + (fabs(x) * 0.3275911);
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = (1e-27 + pow(t_1, 3.0)) / fma(t_1, (t_1 + -1e-9), 1e-18);
	} else {
		tmp = 1.0 + ((1.0 / t_2) * (exp((x * -x)) * (((-0.284496736 + (t_0 * (1.421413741 + (t_0 * ((-3.0685496600615605 + (1.1957597040827899 / pow(fma(0.3275911, x, 1.0), 3.0))) / (2.111650813574209 + ((pow(fma(0.3275911, x, 1.0), -2.0) * 1.126581484710674) + (1.5423834506201546 / fma(0.3275911, x, 1.0))))))))) * (-1.0 / t_2)) - 0.254829592)));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(x * 0.3275911)))
	t_1 = fma((x ^ 3.0), -0.37545125292247583, Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343))))
	t_2 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64(Float64(1e-27 + (t_1 ^ 3.0)) / fma(t_1, Float64(t_1 + -1e-9), 1e-18));
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 / t_2) * Float64(exp(Float64(x * Float64(-x))) * Float64(Float64(Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * Float64(Float64(-3.0685496600615605 + Float64(1.1957597040827899 / (fma(0.3275911, x, 1.0) ^ 3.0))) / Float64(2.111650813574209 + Float64(Float64((fma(0.3275911, x, 1.0) ^ -2.0) * 1.126581484710674) + Float64(1.5423834506201546 / fma(0.3275911, x, 1.0))))))))) * Float64(-1.0 / t_2)) - 0.254829592))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(N[(1e-27 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t$95$1 + -1e-9), $MachinePrecision] + 1e-18), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(N[(-3.0685496600615605 + N[(1.1957597040827899 / N[Power[N[(0.3275911 * x + 1.0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.111650813574209 + N[(N[(N[Power[N[(0.3275911 * x + 1.0), $MachinePrecision], -2.0], $MachinePrecision] * 1.126581484710674), $MachinePrecision] + N[(1.5423834506201546 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   1. Initial program 58.4%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 58.4%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 58.4%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 56.5%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around 0 96.8%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. flip3-+ 96.8%

         \[\leadsto \color{blue}{\frac{{\left( 10^{-9} \right)}^{3} + {\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)}} \]

      2. metadata-eval 96.8%

         \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      3. fma-def 96.8%

         \[\leadsto \frac{10^{-27} + {\color{blue}{\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)}}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      4. pow2 96.8%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot \color{blue}{\left(x \cdot x\right)} + 1.128386358070218 \cdot x\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      5. *-commutative 96.8%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\left(x \cdot x\right) \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      6. *-commutative 96.8%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \left(x \cdot x\right) \cdot -0.00011824294398844343 + \color{blue}{x \cdot 1.128386358070218}\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      7. fma-udef 96.8%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)}\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      8. metadata-eval 96.8%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)}^{3}}{\color{blue}{10^{-18}} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

   10. Applied egg-rr 96.8%

       \[\leadsto \color{blue}{\frac{10^{-27} + {\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 96.8%

          \[\leadsto \frac{10^{-27} + {\color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)}}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]

       2. *-commutative 96.8%

          \[\leadsto \frac{10^{-27} + {\left(\color{blue}{{x}^{3} \cdot -0.37545125292247583} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]

       3. fma-def 96.8%

          \[\leadsto \frac{10^{-27} + {\color{blue}{\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)}}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]

       4. fma-def 96.8%

          \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{\left(x \cdot x\right) \cdot -0.00011824294398844343 + x \cdot 1.128386358070218}\right)\right)}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]

       5. +-commutative 96.8%

          \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{x \cdot 1.128386358070218 + \left(x \cdot x\right) \cdot -0.00011824294398844343}\right)\right)}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]

       6. associate-*l* 96.8%

          \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot 1.128386358070218 + \color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)}\right)\right)}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]

       7. distribute-lft-out 96.8%

          \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)}\right)\right)}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]

       8. +-commutative 96.8%

          \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\right)}^{3}}{\color{blue}{\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right) + 10^{-18}}} \]

   12. Simplified 96.8%

       \[\leadsto \color{blue}{\frac{10^{-27} + {\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right), \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right) + -1 \cdot 10^{-9}, 10^{-18}\right)}} \]

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 99.9%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]
   4. Step-by-step derivation

      1. pow1 99.9%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   5. Applied egg-rr 99.9%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   6. Step-by-step derivation

      1. unpow1 99.9%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. unpow1 99.9%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. sqr-pow 41.5%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. fabs-sqr 41.5%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. sqr-pow 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. unpow1 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   7. Simplified 99.4%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   8. Step-by-step derivation

      1. pow1 99.9%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   9. Applied egg-rr 99.4%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   10. Step-by-step derivation

       1. unpow1 99.9%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. unpow1 99.9%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. sqr-pow 41.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. fabs-sqr 41.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. sqr-pow 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. unpow1 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   11. Simplified 99.4%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   12. Step-by-step derivation

       1. pow1 99.9%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   13. Applied egg-rr 99.4%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   14. Step-by-step derivation

       1. unpow1 99.9%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. unpow1 99.9%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. sqr-pow 41.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. fabs-sqr 41.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. sqr-pow 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. unpow1 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   15. Simplified 99.3%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   16. Step-by-step derivation

       1. flip3-+ 99.3%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \color{blue}{\frac{{-1.453152027}^{3} + {\left(\frac{1.061405429}{1 + 0.3275911 \cdot x}\right)}^{3}}{-1.453152027 \cdot -1.453152027 + \left(\frac{1.061405429}{1 + 0.3275911 \cdot x} \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x} - -1.453152027 \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. metadata-eval 99.3%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{\color{blue}{-3.0685496600615605} + {\left(\frac{1.061405429}{1 + 0.3275911 \cdot x}\right)}^{3}}{-1.453152027 \cdot -1.453152027 + \left(\frac{1.061405429}{1 + 0.3275911 \cdot x} \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x} - -1.453152027 \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. +-commutative 99.3%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + {\left(\frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}}\right)}^{3}}{-1.453152027 \cdot -1.453152027 + \left(\frac{1.061405429}{1 + 0.3275911 \cdot x} \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x} - -1.453152027 \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. fma-udef 99.3%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + {\left(\frac{1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}^{3}}{-1.453152027 \cdot -1.453152027 + \left(\frac{1.061405429}{1 + 0.3275911 \cdot x} \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x} - -1.453152027 \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. cube-div 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \color{blue}{\frac{{1.061405429}^{3}}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}}{-1.453152027 \cdot -1.453152027 + \left(\frac{1.061405429}{1 + 0.3275911 \cdot x} \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x} - -1.453152027 \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. metadata-eval 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{\color{blue}{1.1957597040827899}}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{-1.453152027 \cdot -1.453152027 + \left(\frac{1.061405429}{1 + 0.3275911 \cdot x} \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x} - -1.453152027 \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       7. metadata-eval 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{\color{blue}{2.111650813574209} + \left(\frac{1.061405429}{1 + 0.3275911 \cdot x} \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x} - -1.453152027 \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       8. pow2 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left(\color{blue}{{\left(\frac{1.061405429}{1 + 0.3275911 \cdot x}\right)}^{2}} - -1.453152027 \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       9. +-commutative 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}}\right)}^{2} - -1.453152027 \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       10. fma-udef 99.4%

           \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\frac{1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}^{2} - -1.453152027 \cdot \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       11. +-commutative 99.4%

           \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2} - -1.453152027 \cdot \frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       12. fma-udef 99.4%

           \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2} - -1.453152027 \cdot \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   17. Applied egg-rr 99.4%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \color{blue}{\frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2} - -1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   18. Step-by-step derivation

       1. sub-neg 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \color{blue}{\left({\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2} + \left(--1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. unpow2 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left(\color{blue}{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}} + \left(--1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. metadata-eval 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left(\frac{\color{blue}{1.061405429 \cdot 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \left(--1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. associate-*r/ 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left(\color{blue}{\left(1.061405429 \cdot \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \left(--1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. *-commutative 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left(\color{blue}{\left(\frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot 1.061405429\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \left(--1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. metadata-eval 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left(\left(\frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot 1.061405429\right) \cdot \frac{\color{blue}{1.061405429 \cdot 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \left(--1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       7. associate-*r/ 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left(\left(\frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot 1.061405429\right) \cdot \color{blue}{\left(1.061405429 \cdot \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)} + \left(--1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       8. *-commutative 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left(\left(\frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot 1.061405429\right) \cdot \color{blue}{\left(\frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot 1.061405429\right)} + \left(--1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       9. swap-sqr 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left(\color{blue}{\left(\frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \left(1.061405429 \cdot 1.061405429\right)} + \left(--1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       10. unpow-1 99.4%

           \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left(\left(\color{blue}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-1}} \cdot \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \left(1.061405429 \cdot 1.061405429\right) + \left(--1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       11. unpow-1 99.4%

           \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left(\left({\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-1} \cdot \color{blue}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-1}}\right) \cdot \left(1.061405429 \cdot 1.061405429\right) + \left(--1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       12. pow-sqr 99.4%

           \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left(\color{blue}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{\left(2 \cdot -1\right)}} \cdot \left(1.061405429 \cdot 1.061405429\right) + \left(--1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       13. metadata-eval 99.4%

           \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{\color{blue}{-2}} \cdot \left(1.061405429 \cdot 1.061405429\right) + \left(--1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       14. metadata-eval 99.4%

           \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2} \cdot \color{blue}{1.126581484710674} + \left(--1.453152027 \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       15. associate-*r/ 99.4%

           \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2} \cdot 1.126581484710674 + \left(-\color{blue}{\frac{-1.453152027 \cdot 1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       16. distribute-neg-frac 99.4%

           \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2} \cdot 1.126581484710674 + \color{blue}{\frac{--1.453152027 \cdot 1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   19. Simplified 99.4%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \color{blue}{\frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2} \cdot 1.126581484710674 + \frac{1.5423834506201546}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\frac{10^{-27} + {\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right), \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right) + -1 \cdot 10^{-9}, 10^{-18}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}}{2.111650813574209 + \left({\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2} \cdot 1.126581484710674 + \frac{1.5423834506201546}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \frac{1}{1 + x \cdot 0.3275911}\\ t_1 := \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ t_2 := 1 + \left|x\right| \cdot 0.3275911\\ \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\frac{10^{-27} + {t_1}^{3}}{\mathsf{fma}\left(t_1, t_1 + -1 \cdot 10^{-9}, 10^{-18}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{t_2} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\left(-0.284496736 + t_0 \cdot \left(1.421413741 + t_0 \cdot \mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)\right)\right) \cdot \frac{-1}{t_2} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* x 0.3275911))))
        (t_1
         (fma
          (pow x 3.0)
          -0.37545125292247583
          (* x (+ 1.128386358070218 (* x -0.00011824294398844343)))))
        (t_2 (+ 1.0 (* (fabs x) 0.3275911))))
   (if (<= (fabs x) 0.0001)
     (/ (+ 1e-27 (pow t_1 3.0)) (fma t_1 (+ t_1 -1e-9) 1e-18))
     (+
      1.0
      (*
       (/ 1.0 t_2)
       (*
        (exp (* x (- x)))
        (-
         (*
          (+
           -0.284496736
           (*
            t_0
            (+
             1.421413741
             (*
              t_0
              (fma 1.061405429 (/ 1.0 (fma 0.3275911 x 1.0)) -1.453152027)))))
          (/ -1.0 t_2))
         0.254829592)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 / (1.0 + (x * 0.3275911));
	double t_1 = fma(pow(x, 3.0), -0.37545125292247583, (x * (1.128386358070218 + (x * -0.00011824294398844343))));
	double t_2 = 1.0 + (fabs(x) * 0.3275911);
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = (1e-27 + pow(t_1, 3.0)) / fma(t_1, (t_1 + -1e-9), 1e-18);
	} else {
		tmp = 1.0 + ((1.0 / t_2) * (exp((x * -x)) * (((-0.284496736 + (t_0 * (1.421413741 + (t_0 * fma(1.061405429, (1.0 / fma(0.3275911, x, 1.0)), -1.453152027))))) * (-1.0 / t_2)) - 0.254829592)));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(x * 0.3275911)))
	t_1 = fma((x ^ 3.0), -0.37545125292247583, Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343))))
	t_2 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64(Float64(1e-27 + (t_1 ^ 3.0)) / fma(t_1, Float64(t_1 + -1e-9), 1e-18));
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 / t_2) * Float64(exp(Float64(x * Float64(-x))) * Float64(Float64(Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * fma(1.061405429, Float64(1.0 / fma(0.3275911, x, 1.0)), -1.453152027))))) * Float64(-1.0 / t_2)) - 0.254829592))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(N[(1e-27 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t$95$1 + -1e-9), $MachinePrecision] + 1e-18), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(1.061405429 * N[(1.0 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   1. Initial program 58.4%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 58.4%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 58.4%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 56.5%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around 0 96.8%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. flip3-+ 96.8%

         \[\leadsto \color{blue}{\frac{{\left( 10^{-9} \right)}^{3} + {\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)}} \]

      2. metadata-eval 96.8%

         \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      3. fma-def 96.8%

         \[\leadsto \frac{10^{-27} + {\color{blue}{\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)}}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      4. pow2 96.8%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot \color{blue}{\left(x \cdot x\right)} + 1.128386358070218 \cdot x\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      5. *-commutative 96.8%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\left(x \cdot x\right) \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      6. *-commutative 96.8%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \left(x \cdot x\right) \cdot -0.00011824294398844343 + \color{blue}{x \cdot 1.128386358070218}\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      7. fma-udef 96.8%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)}\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      8. metadata-eval 96.8%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)}^{3}}{\color{blue}{10^{-18}} + \left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

   10. Applied egg-rr 96.8%

       \[\leadsto \color{blue}{\frac{10^{-27} + {\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 96.8%

          \[\leadsto \frac{10^{-27} + {\color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)}}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]

       2. *-commutative 96.8%

          \[\leadsto \frac{10^{-27} + {\left(\color{blue}{{x}^{3} \cdot -0.37545125292247583} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]

       3. fma-def 96.8%

          \[\leadsto \frac{10^{-27} + {\color{blue}{\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)}}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]

       4. fma-def 96.8%

          \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{\left(x \cdot x\right) \cdot -0.00011824294398844343 + x \cdot 1.128386358070218}\right)\right)}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]

       5. +-commutative 96.8%

          \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{x \cdot 1.128386358070218 + \left(x \cdot x\right) \cdot -0.00011824294398844343}\right)\right)}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]

       6. associate-*l* 96.8%

          \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot 1.128386358070218 + \color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)}\right)\right)}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]

       7. distribute-lft-out 96.8%

          \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)}\right)\right)}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]

       8. +-commutative 96.8%

          \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\right)}^{3}}{\color{blue}{\left(\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right) + 10^{-18}}} \]

   12. Simplified 96.8%

       \[\leadsto \color{blue}{\frac{10^{-27} + {\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right), \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right) + -1 \cdot 10^{-9}, 10^{-18}\right)}} \]

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 99.9%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]
   4. Step-by-step derivation

      1. pow1 99.9%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   5. Applied egg-rr 99.9%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   6. Step-by-step derivation

      1. unpow1 99.9%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. unpow1 99.9%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. sqr-pow 41.5%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. fabs-sqr 41.5%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. sqr-pow 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. unpow1 99.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   7. Simplified 99.4%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   8. Step-by-step derivation

      1. pow1 99.9%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   9. Applied egg-rr 99.4%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   10. Step-by-step derivation

       1. unpow1 99.9%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. unpow1 99.9%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. sqr-pow 41.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. fabs-sqr 41.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. sqr-pow 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. unpow1 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   11. Simplified 99.4%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   12. Step-by-step derivation

       1. pow1 99.9%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   13. Applied egg-rr 99.4%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   14. Step-by-step derivation

       1. unpow1 99.9%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. unpow1 99.9%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. sqr-pow 41.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. fabs-sqr 41.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. sqr-pow 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. unpow1 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   15. Simplified 99.3%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   16. Step-by-step derivation

       1. +-commutative 99.3%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \color{blue}{\left(\frac{1.061405429}{1 + 0.3275911 \cdot x} + -1.453152027\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. +-commutative 99.3%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(\frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}} + -1.453152027\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. fma-udef 99.3%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(\frac{1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, x, 1\right)}} + -1.453152027\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. div-inv 99.3%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(\color{blue}{1.061405429 \cdot \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}} + -1.453152027\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. fma-def 99.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \color{blue}{\mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   17. Applied egg-rr 99.4%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \color{blue}{\mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\frac{10^{-27} + {\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right), \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right) + -1 \cdot 10^{-9}, 10^{-18}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \frac{1}{1 + x \cdot 0.3275911}\\ t_1 := 1 + \left|x\right| \cdot 0.3275911\\ \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right), {x}^{3} \cdot -0.37545125292247583\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{t_1} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\left(-0.284496736 + t_0 \cdot \left(1.421413741 + t_0 \cdot \mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)\right)\right) \cdot \frac{-1}{t_1} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* x 0.3275911))))
        (t_1 (+ 1.0 (* (fabs x) 0.3275911))))
   (if (<= x 0.0005)
     (+
      1e-9
      (fma
       x
       (fma x -0.00011824294398844343 1.128386358070218)
       (* (pow x 3.0) -0.37545125292247583)))
     (+
      1.0
      (*
       (/ 1.0 t_1)
       (*
        (exp (* x (- x)))
        (-
         (*
          (+
           -0.284496736
           (*
            t_0
            (+
             1.421413741
             (*
              t_0
              (fma 1.061405429 (/ 1.0 (fma 0.3275911 x 1.0)) -1.453152027)))))
          (/ -1.0 t_1))
         0.254829592)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 / (1.0 + (x * 0.3275911));
	double t_1 = 1.0 + (fabs(x) * 0.3275911);
	double tmp;
	if (x <= 0.0005) {
		tmp = 1e-9 + fma(x, fma(x, -0.00011824294398844343, 1.128386358070218), (pow(x, 3.0) * -0.37545125292247583));
	} else {
		tmp = 1.0 + ((1.0 / t_1) * (exp((x * -x)) * (((-0.284496736 + (t_0 * (1.421413741 + (t_0 * fma(1.061405429, (1.0 / fma(0.3275911, x, 1.0)), -1.453152027))))) * (-1.0 / t_1)) - 0.254829592)));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(x * 0.3275911)))
	t_1 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	tmp = 0.0
	if (x <= 0.0005)
		tmp = Float64(1e-9 + fma(x, fma(x, -0.00011824294398844343, 1.128386358070218), Float64((x ^ 3.0) * -0.37545125292247583)));
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 / t_1) * Float64(exp(Float64(x * Float64(-x))) * Float64(Float64(Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * fma(1.061405429, Float64(1.0 / fma(0.3275911, x, 1.0)), -1.453152027))))) * Float64(-1.0 / t_1)) - 0.254829592))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.0005], N[(1e-9 + N[(x * N[(x * -0.00011824294398844343 + 1.128386358070218), $MachinePrecision] + N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(1.061405429 * N[(1.0 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-4

   1. Initial program 74.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 74.1%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 74.0%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 72.4%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around 0 61.5%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 61.5%

         \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      2. fma-def 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)}\right) \]

      3. pow2 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot \color{blue}{\left(x \cdot x\right)} + 1.128386358070218 \cdot x\right)\right) \]

      4. *-commutative 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\left(x \cdot x\right) \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right)\right) \]

      5. *-commutative 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \left(x \cdot x\right) \cdot -0.00011824294398844343 + \color{blue}{x \cdot 1.128386358070218}\right)\right) \]

      6. fma-udef 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)}\right)\right) \]

   10. Applied egg-rr 61.5%

       \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 61.5%

          \[\leadsto \color{blue}{10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)} \]

       2. fma-udef 61.5%

          \[\leadsto 10^{-9} + \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)} \]

       3. *-commutative 61.5%

          \[\leadsto 10^{-9} + \left(\color{blue}{{x}^{3} \cdot -0.37545125292247583} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \]

       4. fma-def 61.5%

          \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)} \]

       5. fma-def 61.5%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{\left(x \cdot x\right) \cdot -0.00011824294398844343 + x \cdot 1.128386358070218}\right) \]

       6. +-commutative 61.5%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{x \cdot 1.128386358070218 + \left(x \cdot x\right) \cdot -0.00011824294398844343}\right) \]

       7. associate-*l* 61.5%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot 1.128386358070218 + \color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)}\right) \]

       8. distribute-lft-out 61.5%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)}\right) \]

   12. Simplified 61.5%

       \[\leadsto \color{blue}{10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)} \]

   13. Taylor expanded in x around 0 61.5%

       \[\leadsto 10^{-9} + \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]
   14. Step-by-step derivation

       1. +-commutative 61.5%

          \[\leadsto 10^{-9} + \color{blue}{\left(\left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right) + -0.37545125292247583 \cdot {x}^{3}\right)} \]

       2. *-commutative 61.5%

          \[\leadsto 10^{-9} + \left(\left(\color{blue}{{x}^{2} \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right) + -0.37545125292247583 \cdot {x}^{3}\right) \]

       3. unpow2 61.5%

          \[\leadsto 10^{-9} + \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot -0.00011824294398844343 + 1.128386358070218 \cdot x\right) + -0.37545125292247583 \cdot {x}^{3}\right) \]

       4. associate-*r* 61.5%

          \[\leadsto 10^{-9} + \left(\left(\color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)} + 1.128386358070218 \cdot x\right) + -0.37545125292247583 \cdot {x}^{3}\right) \]

       5. *-commutative 61.5%

          \[\leadsto 10^{-9} + \left(\left(x \cdot \left(x \cdot -0.00011824294398844343\right) + \color{blue}{x \cdot 1.128386358070218}\right) + -0.37545125292247583 \cdot {x}^{3}\right) \]

       6. distribute-lft-in 61.5%

          \[\leadsto 10^{-9} + \left(\color{blue}{x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)} + -0.37545125292247583 \cdot {x}^{3}\right) \]

       7. fma-def 62.0%

          \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left(x, x \cdot -0.00011824294398844343 + 1.128386358070218, -0.37545125292247583 \cdot {x}^{3}\right)} \]

       8. fma-def 62.0%

          \[\leadsto 10^{-9} + \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right)}, -0.37545125292247583 \cdot {x}^{3}\right) \]

       9. *-commutative 62.0%

          \[\leadsto 10^{-9} + \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right), \color{blue}{{x}^{3} \cdot -0.37545125292247583}\right) \]

   15. Simplified 62.0%

       \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right), {x}^{3} \cdot -0.37545125292247583\right)} \]

   if 5.0000000000000001e-4 < x

   1. Initial program 99.8%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]
   4. Step-by-step derivation

      1. pow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   5. Applied egg-rr 99.8%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   6. Step-by-step derivation

      1. unpow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. unpow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. sqr-pow 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. fabs-sqr 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. sqr-pow 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. unpow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   7. Simplified 99.8%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   8. Step-by-step derivation

      1. pow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   9. Applied egg-rr 99.8%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   10. Step-by-step derivation

       1. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. fabs-sqr 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   11. Simplified 99.8%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   12. Step-by-step derivation

       1. pow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   13. Applied egg-rr 99.8%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   14. Step-by-step derivation

       1. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. fabs-sqr 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   15. Simplified 99.8%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   16. Step-by-step derivation

       1. +-commutative 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \color{blue}{\left(\frac{1.061405429}{1 + 0.3275911 \cdot x} + -1.453152027\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. +-commutative 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(\frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}} + -1.453152027\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. fma-udef 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(\frac{1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, x, 1\right)}} + -1.453152027\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. div-inv 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(\color{blue}{1.061405429 \cdot \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}} + -1.453152027\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. fma-def 99.9%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \color{blue}{\mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   17. Applied egg-rr 99.9%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \color{blue}{\mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right), {x}^{3} \cdot -0.37545125292247583\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]

Alternative 4: 99.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq 0.0006:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right), {x}^{3} \cdot -0.37545125292247583\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= x 0.0006)
     (+
      1e-9
      (fma
       x
       (fma x -0.00011824294398844343 1.128386358070218)
       (* (pow x 3.0) -0.37545125292247583)))
     (+
      1.0
      (*
       (*
        (exp (* x (- x)))
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+ 1.421413741 (* t_1 (+ -1.453152027 (/ 1.061405429 t_0)))))))))
       (/ -1.0 (+ 1.0 (* (fabs x) 0.3275911))))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x <= 0.0006) {
		tmp = 1e-9 + fma(x, fma(x, -0.00011824294398844343, 1.128386358070218), (pow(x, 3.0) * -0.37545125292247583));
	} else {
		tmp = 1.0 + ((exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))) * (-1.0 / (1.0 + (fabs(x) * 0.3275911))));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 + Float64(x * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x <= 0.0006)
		tmp = Float64(1e-9 + fma(x, fma(x, -0.00011824294398844343, 1.128386358070218), Float64((x ^ 3.0) * -0.37545125292247583)));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(x * Float64(-x))) * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(t_1 * Float64(-1.453152027 + Float64(1.061405429 / t_0))))))))) * Float64(-1.0 / Float64(1.0 + Float64(abs(x) * 0.3275911)))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x, 0.0006], N[(1e-9 + N[(x * N[(x * -0.00011824294398844343 + 1.128386358070218), $MachinePrecision] + N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999947e-4

   1. Initial program 74.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 74.1%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 74.0%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 72.4%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around 0 61.5%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 61.5%

         \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      2. fma-def 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)}\right) \]

      3. pow2 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot \color{blue}{\left(x \cdot x\right)} + 1.128386358070218 \cdot x\right)\right) \]

      4. *-commutative 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\left(x \cdot x\right) \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right)\right) \]

      5. *-commutative 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \left(x \cdot x\right) \cdot -0.00011824294398844343 + \color{blue}{x \cdot 1.128386358070218}\right)\right) \]

      6. fma-udef 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)}\right)\right) \]

   10. Applied egg-rr 61.5%

       \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 61.5%

          \[\leadsto \color{blue}{10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)} \]

       2. fma-udef 61.5%

          \[\leadsto 10^{-9} + \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)} \]

       3. *-commutative 61.5%

          \[\leadsto 10^{-9} + \left(\color{blue}{{x}^{3} \cdot -0.37545125292247583} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \]

       4. fma-def 61.5%

          \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)} \]

       5. fma-def 61.5%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{\left(x \cdot x\right) \cdot -0.00011824294398844343 + x \cdot 1.128386358070218}\right) \]

       6. +-commutative 61.5%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{x \cdot 1.128386358070218 + \left(x \cdot x\right) \cdot -0.00011824294398844343}\right) \]

       7. associate-*l* 61.5%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot 1.128386358070218 + \color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)}\right) \]

       8. distribute-lft-out 61.5%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)}\right) \]

   12. Simplified 61.5%

       \[\leadsto \color{blue}{10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)} \]

   13. Taylor expanded in x around 0 61.5%

       \[\leadsto 10^{-9} + \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]
   14. Step-by-step derivation

       1. +-commutative 61.5%

          \[\leadsto 10^{-9} + \color{blue}{\left(\left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right) + -0.37545125292247583 \cdot {x}^{3}\right)} \]

       2. *-commutative 61.5%

          \[\leadsto 10^{-9} + \left(\left(\color{blue}{{x}^{2} \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right) + -0.37545125292247583 \cdot {x}^{3}\right) \]

       3. unpow2 61.5%

          \[\leadsto 10^{-9} + \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot -0.00011824294398844343 + 1.128386358070218 \cdot x\right) + -0.37545125292247583 \cdot {x}^{3}\right) \]

       4. associate-*r* 61.5%

          \[\leadsto 10^{-9} + \left(\left(\color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)} + 1.128386358070218 \cdot x\right) + -0.37545125292247583 \cdot {x}^{3}\right) \]

       5. *-commutative 61.5%

          \[\leadsto 10^{-9} + \left(\left(x \cdot \left(x \cdot -0.00011824294398844343\right) + \color{blue}{x \cdot 1.128386358070218}\right) + -0.37545125292247583 \cdot {x}^{3}\right) \]

       6. distribute-lft-in 61.5%

          \[\leadsto 10^{-9} + \left(\color{blue}{x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)} + -0.37545125292247583 \cdot {x}^{3}\right) \]

       7. fma-def 62.0%

          \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left(x, x \cdot -0.00011824294398844343 + 1.128386358070218, -0.37545125292247583 \cdot {x}^{3}\right)} \]

       8. fma-def 62.0%

          \[\leadsto 10^{-9} + \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right)}, -0.37545125292247583 \cdot {x}^{3}\right) \]

       9. *-commutative 62.0%

          \[\leadsto 10^{-9} + \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right), \color{blue}{{x}^{3} \cdot -0.37545125292247583}\right) \]

   15. Simplified 62.0%

       \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right), {x}^{3} \cdot -0.37545125292247583\right)} \]

   if 5.99999999999999947e-4 < x

   1. Initial program 99.8%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]
   4. Step-by-step derivation

      1. pow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   5. Applied egg-rr 99.8%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   6. Step-by-step derivation

      1. unpow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. unpow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. sqr-pow 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. fabs-sqr 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. sqr-pow 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. unpow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   7. Simplified 99.8%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   8. Step-by-step derivation

      1. pow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   9. Applied egg-rr 99.8%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   10. Step-by-step derivation

       1. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. fabs-sqr 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   11. Simplified 99.8%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   12. Step-by-step derivation

       1. pow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   13. Applied egg-rr 99.8%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   14. Step-by-step derivation

       1. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. fabs-sqr 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   15. Simplified 99.8%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   16. Step-by-step derivation

       1. pow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   17. Applied egg-rr 99.8%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   18. Step-by-step derivation

       1. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. fabs-sqr 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   19. Simplified 99.8%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0006:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right), {x}^{3} \cdot -0.37545125292247583\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 5: 99.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq 0.0006:\\ \;\;\;\;10^{-9} + \left({x}^{3} \cdot -0.37545125292247583 + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= x 0.0006)
     (+
      1e-9
      (+
       (* (pow x 3.0) -0.37545125292247583)
       (* x (+ 1.128386358070218 (* x -0.00011824294398844343)))))
     (+
      1.0
      (*
       (*
        (exp (* x (- x)))
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+ 1.421413741 (* t_1 (+ -1.453152027 (/ 1.061405429 t_0)))))))))
       (/ -1.0 (+ 1.0 (* (fabs x) 0.3275911))))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x <= 0.0006) {
		tmp = 1e-9 + ((pow(x, 3.0) * -0.37545125292247583) + (x * (1.128386358070218 + (x * -0.00011824294398844343))));
	} else {
		tmp = 1.0 + ((exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))) * (-1.0 / (1.0 + (fabs(x) * 0.3275911))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    if (x <= 0.0006d0) then
        tmp = 1d-9 + (((x ** 3.0d0) * (-0.37545125292247583d0)) + (x * (1.128386358070218d0 + (x * (-0.00011824294398844343d0)))))
    else
        tmp = 1.0d0 + ((exp((x * -x)) * (0.254829592d0 + (t_1 * ((-0.284496736d0) + (t_1 * (1.421413741d0 + (t_1 * ((-1.453152027d0) + (1.061405429d0 / t_0))))))))) * ((-1.0d0) / (1.0d0 + (abs(x) * 0.3275911d0))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x <= 0.0006) {
		tmp = 1e-9 + ((Math.pow(x, 3.0) * -0.37545125292247583) + (x * (1.128386358070218 + (x * -0.00011824294398844343))));
	} else {
		tmp = 1.0 + ((Math.exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))) * (-1.0 / (1.0 + (Math.abs(x) * 0.3275911))));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = 1.0 + (x * 0.3275911)
	t_1 = 1.0 / t_0
	tmp = 0
	if x <= 0.0006:
		tmp = 1e-9 + ((math.pow(x, 3.0) * -0.37545125292247583) + (x * (1.128386358070218 + (x * -0.00011824294398844343))))
	else:
		tmp = 1.0 + ((math.exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))) * (-1.0 / (1.0 + (math.fabs(x) * 0.3275911))))
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 + Float64(x * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x <= 0.0006)
		tmp = Float64(1e-9 + Float64(Float64((x ^ 3.0) * -0.37545125292247583) + Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343)))));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(x * Float64(-x))) * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(t_1 * Float64(-1.453152027 + Float64(1.061405429 / t_0))))))))) * Float64(-1.0 / Float64(1.0 + Float64(abs(x) * 0.3275911)))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = 1.0 + (x * 0.3275911);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (x <= 0.0006)
		tmp = 1e-9 + (((x ^ 3.0) * -0.37545125292247583) + (x * (1.128386358070218 + (x * -0.00011824294398844343))));
	else
		tmp = 1.0 + ((exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))) * (-1.0 / (1.0 + (abs(x) * 0.3275911))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x, 0.0006], N[(1e-9 + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision] + N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999947e-4

   1. Initial program 74.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 74.1%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 74.0%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 72.4%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around 0 61.5%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 61.5%

         \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      2. fma-def 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)}\right) \]

      3. pow2 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot \color{blue}{\left(x \cdot x\right)} + 1.128386358070218 \cdot x\right)\right) \]

      4. *-commutative 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\left(x \cdot x\right) \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right)\right) \]

      5. *-commutative 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \left(x \cdot x\right) \cdot -0.00011824294398844343 + \color{blue}{x \cdot 1.128386358070218}\right)\right) \]

      6. fma-udef 61.5%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)}\right)\right) \]

   10. Applied egg-rr 61.5%

       \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 61.5%

          \[\leadsto \color{blue}{10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)} \]

       2. fma-udef 61.5%

          \[\leadsto 10^{-9} + \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)} \]

       3. *-commutative 61.5%

          \[\leadsto 10^{-9} + \left(\color{blue}{{x}^{3} \cdot -0.37545125292247583} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \]

       4. fma-def 61.5%

          \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)} \]

       5. fma-def 61.5%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{\left(x \cdot x\right) \cdot -0.00011824294398844343 + x \cdot 1.128386358070218}\right) \]

       6. +-commutative 61.5%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{x \cdot 1.128386358070218 + \left(x \cdot x\right) \cdot -0.00011824294398844343}\right) \]

       7. associate-*l* 61.5%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot 1.128386358070218 + \color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)}\right) \]

       8. distribute-lft-out 61.5%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)}\right) \]

   12. Simplified 61.5%

       \[\leadsto \color{blue}{10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)} \]
   13. Step-by-step derivation

       1. fma-udef 61.5%

          \[\leadsto 10^{-9} + \color{blue}{\left({x}^{3} \cdot -0.37545125292247583 + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)} \]

       2. +-commutative 61.5%

          \[\leadsto 10^{-9} + \left({x}^{3} \cdot -0.37545125292247583 + x \cdot \color{blue}{\left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)}\right) \]

   14. Applied egg-rr 61.5%

       \[\leadsto 10^{-9} + \color{blue}{\left({x}^{3} \cdot -0.37545125292247583 + x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)\right)} \]

   if 5.99999999999999947e-4 < x

   1. Initial program 99.8%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]
   4. Step-by-step derivation

      1. pow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   5. Applied egg-rr 99.8%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   6. Step-by-step derivation

      1. unpow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. unpow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. sqr-pow 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. fabs-sqr 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. sqr-pow 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. unpow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   7. Simplified 99.8%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   8. Step-by-step derivation

      1. pow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   9. Applied egg-rr 99.8%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   10. Step-by-step derivation

       1. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. fabs-sqr 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   11. Simplified 99.8%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   12. Step-by-step derivation

       1. pow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   13. Applied egg-rr 99.8%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   14. Step-by-step derivation

       1. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. fabs-sqr 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   15. Simplified 99.8%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   16. Step-by-step derivation

       1. pow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   17. Applied egg-rr 99.8%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   18. Step-by-step derivation

       1. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. fabs-sqr 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. sqr-pow 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. unpow1 99.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \color{blue}{x}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   19. Simplified 99.8%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0006:\\ \;\;\;\;10^{-9} + \left({x}^{3} \cdot -0.37545125292247583 + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 6: 99.7% accurate, 7.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.02:\\ \;\;\;\;10^{-9} + \left({x}^{3} \cdot -0.37545125292247583 + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.02)
   (+
    1e-9
    (+
     (* (pow x 3.0) -0.37545125292247583)
     (* x (+ 1.128386358070218 (* x -0.00011824294398844343)))))
   (- 1.0 (/ 0.7778892405807117 (* x (exp (* x x)))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.02) {
		tmp = 1e-9 + ((pow(x, 3.0) * -0.37545125292247583) + (x * (1.128386358070218 + (x * -0.00011824294398844343))));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x * exp((x * x))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.02d0) then
        tmp = 1d-9 + (((x ** 3.0d0) * (-0.37545125292247583d0)) + (x * (1.128386358070218d0 + (x * (-0.00011824294398844343d0)))))
    else
        tmp = 1.0d0 - (0.7778892405807117d0 / (x * exp((x * x))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.02) {
		tmp = 1e-9 + ((Math.pow(x, 3.0) * -0.37545125292247583) + (x * (1.128386358070218 + (x * -0.00011824294398844343))));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x * Math.exp((x * x))));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.02:
		tmp = 1e-9 + ((math.pow(x, 3.0) * -0.37545125292247583) + (x * (1.128386358070218 + (x * -0.00011824294398844343))))
	else:
		tmp = 1.0 - (0.7778892405807117 / (x * math.exp((x * x))))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.02)
		tmp = Float64(1e-9 + Float64(Float64((x ^ 3.0) * -0.37545125292247583) + Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343)))));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x * exp(Float64(x * x)))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.02)
		tmp = 1e-9 + (((x ^ 3.0) * -0.37545125292247583) + (x * (1.128386358070218 + (x * -0.00011824294398844343))));
	else
		tmp = 1.0 - (0.7778892405807117 / (x * exp((x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.02], N[(1e-9 + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision] + N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.02

   1. Initial program 74.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 74.2%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 74.1%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 74.1%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 72.5%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around 0 61.4%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 61.4%

         \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      2. fma-def 61.4%

         \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)}\right) \]

      3. pow2 61.4%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot \color{blue}{\left(x \cdot x\right)} + 1.128386358070218 \cdot x\right)\right) \]

      4. *-commutative 61.4%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\left(x \cdot x\right) \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right)\right) \]

      5. *-commutative 61.4%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \left(x \cdot x\right) \cdot -0.00011824294398844343 + \color{blue}{x \cdot 1.128386358070218}\right)\right) \]

      6. fma-udef 61.4%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)}\right)\right) \]

   10. Applied egg-rr 61.4%

       \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 61.4%

          \[\leadsto \color{blue}{10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)} \]

       2. fma-udef 61.4%

          \[\leadsto 10^{-9} + \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)} \]

       3. *-commutative 61.4%

          \[\leadsto 10^{-9} + \left(\color{blue}{{x}^{3} \cdot -0.37545125292247583} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right) \]

       4. fma-def 61.4%

          \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\right)} \]

       5. fma-def 61.4%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{\left(x \cdot x\right) \cdot -0.00011824294398844343 + x \cdot 1.128386358070218}\right) \]

       6. +-commutative 61.4%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{x \cdot 1.128386358070218 + \left(x \cdot x\right) \cdot -0.00011824294398844343}\right) \]

       7. associate-*l* 61.4%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot 1.128386358070218 + \color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)}\right) \]

       8. distribute-lft-out 61.5%

          \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)}\right) \]

   12. Simplified 61.5%

       \[\leadsto \color{blue}{10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)} \]
   13. Step-by-step derivation

       1. fma-udef 61.5%

          \[\leadsto 10^{-9} + \color{blue}{\left({x}^{3} \cdot -0.37545125292247583 + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)} \]

       2. +-commutative 61.5%

          \[\leadsto 10^{-9} + \left({x}^{3} \cdot -0.37545125292247583 + x \cdot \color{blue}{\left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)}\right) \]

   14. Applied egg-rr 61.5%

       \[\leadsto 10^{-9} + \color{blue}{\left({x}^{3} \cdot -0.37545125292247583 + x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)\right)} \]

   if 1.02 < x

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 100.0%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - 0.7778892405807117 \cdot \frac{1}{x \cdot e^{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto 1 - \color{blue}{\frac{0.7778892405807117 \cdot 1}{x \cdot e^{{x}^{2}}}} \]

      2. metadata-eval 100.0%

         \[\leadsto 1 - \frac{\color{blue}{0.7778892405807117}}{x \cdot e^{{x}^{2}}} \]

      3. unpow2 100.0%

         \[\leadsto 1 - \frac{0.7778892405807117}{x \cdot e^{\color{blue}{x \cdot x}}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02:\\ \;\;\;\;10^{-9} + \left({x}^{3} \cdot -0.37545125292247583 + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 7: 99.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{10^{-18} - t_0 \cdot t_0}{10^{-9} - t_0}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (+ 1.128386358070218 (* x -0.00011824294398844343)))))
   (if (<= x 0.88)
     (/ (- 1e-18 (* t_0 t_0)) (- 1e-9 t_0))
     (- 1.0 (/ 0.7778892405807117 (* x (exp (* x x))))))))

C

x = abs(x);
double code(double x) {
	double t_0 = x * (1.128386358070218 + (x * -0.00011824294398844343));
	double tmp;
	if (x <= 0.88) {
		tmp = (1e-18 - (t_0 * t_0)) / (1e-9 - t_0);
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x * exp((x * x))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.128386358070218d0 + (x * (-0.00011824294398844343d0)))
    if (x <= 0.88d0) then
        tmp = (1d-18 - (t_0 * t_0)) / (1d-9 - t_0)
    else
        tmp = 1.0d0 - (0.7778892405807117d0 / (x * exp((x * x))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = x * (1.128386358070218 + (x * -0.00011824294398844343));
	double tmp;
	if (x <= 0.88) {
		tmp = (1e-18 - (t_0 * t_0)) / (1e-9 - t_0);
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x * Math.exp((x * x))));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = x * (1.128386358070218 + (x * -0.00011824294398844343))
	tmp = 0
	if x <= 0.88:
		tmp = (1e-18 - (t_0 * t_0)) / (1e-9 - t_0)
	else:
		tmp = 1.0 - (0.7778892405807117 / (x * math.exp((x * x))))
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343)))
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(Float64(1e-18 - Float64(t_0 * t_0)) / Float64(1e-9 - t_0));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x * exp(Float64(x * x)))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = x * (1.128386358070218 + (x * -0.00011824294398844343));
	tmp = 0.0;
	if (x <= 0.88)
		tmp = (1e-18 - (t_0 * t_0)) / (1e-9 - t_0);
	else
		tmp = 1.0 - (0.7778892405807117 / (x * exp((x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.88], N[(N[(1e-18 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1e-9 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 74.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 74.2%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 74.1%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 74.1%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 72.5%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around 0 60.4%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   9. Step-by-step derivation

      1. *-commutative 60.4%

         \[\leadsto 10^{-9} + \left(\color{blue}{{x}^{2} \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right) \]

      2. fma-def 60.4%

         \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left({x}^{2}, -0.00011824294398844343, 1.128386358070218 \cdot x\right)} \]

      3. unpow2 60.4%

         \[\leadsto 10^{-9} + \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.00011824294398844343, 1.128386358070218 \cdot x\right) \]

      4. *-commutative 60.4%

         \[\leadsto 10^{-9} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, \color{blue}{x \cdot 1.128386358070218}\right) \]

   10. Simplified 60.4%

       \[\leadsto \color{blue}{10^{-9} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)} \]

   11. Taylor expanded in x around 0 60.4%

       \[\leadsto 10^{-9} + \color{blue}{\left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   12. Step-by-step derivation

       1. +-commutative 60.4%

          \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

       2. *-commutative 60.4%

          \[\leadsto 10^{-9} + \left(\color{blue}{x \cdot 1.128386358070218} + -0.00011824294398844343 \cdot {x}^{2}\right) \]

       3. *-commutative 60.4%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) \]

       4. unpow2 60.4%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{\left(x \cdot x\right)} \cdot -0.00011824294398844343\right) \]

       5. associate-*l* 60.4%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)}\right) \]

       6. distribute-lft-out 60.4%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]

   13. Simplified 60.4%

       \[\leadsto 10^{-9} + \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]
   14. Step-by-step derivation

       1. flip-+ 60.3%

          \[\leadsto \color{blue}{\frac{10^{-9} \cdot 10^{-9} - \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right) \cdot \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)}{10^{-9} - x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)}} \]

       2. metadata-eval 60.3%

          \[\leadsto \frac{\color{blue}{10^{-18}} - \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right) \cdot \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)}{10^{-9} - x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]

       3. +-commutative 60.3%

          \[\leadsto \frac{10^{-18} - \left(x \cdot \color{blue}{\left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)}\right) \cdot \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)}{10^{-9} - x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]

       4. +-commutative 60.3%

          \[\leadsto \frac{10^{-18} - \left(x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)}\right)}{10^{-9} - x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]

       5. +-commutative 60.3%

          \[\leadsto \frac{10^{-18} - \left(x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)\right) \cdot \left(x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)\right)}{10^{-9} - x \cdot \color{blue}{\left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)}} \]

   15. Applied egg-rr 60.3%

       \[\leadsto \color{blue}{\frac{10^{-18} - \left(x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)\right) \cdot \left(x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)\right)}{10^{-9} - x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)}} \]

   if 0.880000000000000004 < x

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 100.0%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 - 0.7778892405807117 \cdot \frac{1}{x \cdot e^{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto 1 - \color{blue}{\frac{0.7778892405807117 \cdot 1}{x \cdot e^{{x}^{2}}}} \]

      2. metadata-eval 100.0%

         \[\leadsto 1 - \frac{\color{blue}{0.7778892405807117}}{x \cdot e^{{x}^{2}}} \]

      3. unpow2 100.0%

         \[\leadsto 1 - \frac{0.7778892405807117}{x \cdot e^{\color{blue}{x \cdot x}}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{10^{-18} - \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right) \cdot \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)}{10^{-9} - x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 8: 99.5% accurate, 29.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;\frac{10^{-18} - t_0 \cdot t_0}{10^{-9} - t_0}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (+ 1.128386358070218 (* x -0.00011824294398844343)))))
   (if (<= x 0.9) (/ (- 1e-18 (* t_0 t_0)) (- 1e-9 t_0)) 1.0)))

C

x = abs(x);
double code(double x) {
	double t_0 = x * (1.128386358070218 + (x * -0.00011824294398844343));
	double tmp;
	if (x <= 0.9) {
		tmp = (1e-18 - (t_0 * t_0)) / (1e-9 - t_0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.128386358070218d0 + (x * (-0.00011824294398844343d0)))
    if (x <= 0.9d0) then
        tmp = (1d-18 - (t_0 * t_0)) / (1d-9 - t_0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = x * (1.128386358070218 + (x * -0.00011824294398844343));
	double tmp;
	if (x <= 0.9) {
		tmp = (1e-18 - (t_0 * t_0)) / (1e-9 - t_0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = x * (1.128386358070218 + (x * -0.00011824294398844343))
	tmp = 0
	if x <= 0.9:
		tmp = (1e-18 - (t_0 * t_0)) / (1e-9 - t_0)
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343)))
	tmp = 0.0
	if (x <= 0.9)
		tmp = Float64(Float64(1e-18 - Float64(t_0 * t_0)) / Float64(1e-9 - t_0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = x * (1.128386358070218 + (x * -0.00011824294398844343));
	tmp = 0.0;
	if (x <= 0.9)
		tmp = (1e-18 - (t_0 * t_0)) / (1e-9 - t_0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.9], N[(N[(1e-18 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1e-9 - t$95$0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 74.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 74.2%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 74.1%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 74.1%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 72.5%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around 0 60.4%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   9. Step-by-step derivation

      1. *-commutative 60.4%

         \[\leadsto 10^{-9} + \left(\color{blue}{{x}^{2} \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right) \]

      2. fma-def 60.4%

         \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left({x}^{2}, -0.00011824294398844343, 1.128386358070218 \cdot x\right)} \]

      3. unpow2 60.4%

         \[\leadsto 10^{-9} + \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.00011824294398844343, 1.128386358070218 \cdot x\right) \]

      4. *-commutative 60.4%

         \[\leadsto 10^{-9} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, \color{blue}{x \cdot 1.128386358070218}\right) \]

   10. Simplified 60.4%

       \[\leadsto \color{blue}{10^{-9} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)} \]

   11. Taylor expanded in x around 0 60.4%

       \[\leadsto 10^{-9} + \color{blue}{\left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   12. Step-by-step derivation

       1. +-commutative 60.4%

          \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

       2. *-commutative 60.4%

          \[\leadsto 10^{-9} + \left(\color{blue}{x \cdot 1.128386358070218} + -0.00011824294398844343 \cdot {x}^{2}\right) \]

       3. *-commutative 60.4%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) \]

       4. unpow2 60.4%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{\left(x \cdot x\right)} \cdot -0.00011824294398844343\right) \]

       5. associate-*l* 60.4%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)}\right) \]

       6. distribute-lft-out 60.4%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]

   13. Simplified 60.4%

       \[\leadsto 10^{-9} + \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]
   14. Step-by-step derivation

       1. flip-+ 60.3%

          \[\leadsto \color{blue}{\frac{10^{-9} \cdot 10^{-9} - \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right) \cdot \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)}{10^{-9} - x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)}} \]

       2. metadata-eval 60.3%

          \[\leadsto \frac{\color{blue}{10^{-18}} - \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right) \cdot \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)}{10^{-9} - x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]

       3. +-commutative 60.3%

          \[\leadsto \frac{10^{-18} - \left(x \cdot \color{blue}{\left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)}\right) \cdot \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)}{10^{-9} - x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]

       4. +-commutative 60.3%

          \[\leadsto \frac{10^{-18} - \left(x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)}\right)}{10^{-9} - x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]

       5. +-commutative 60.3%

          \[\leadsto \frac{10^{-18} - \left(x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)\right) \cdot \left(x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)\right)}{10^{-9} - x \cdot \color{blue}{\left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)}} \]

   15. Applied egg-rr 60.3%

       \[\leadsto \color{blue}{\frac{10^{-18} - \left(x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)\right) \cdot \left(x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)\right)}{10^{-9} - x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)}} \]

   if 0.900000000000000022 < x

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 100.0%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;\frac{10^{-18} - \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right) \cdot \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)}{10^{-9} - x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 99.5% accurate, 44.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + x \cdot \frac{1.2732557730789702 - \left(x \cdot x\right) \cdot 1.3981393803054172 \cdot 10^{-8}}{1.128386358070218 - x \cdot -0.00011824294398844343}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.9)
   (+
    1e-9
    (*
     x
     (/
      (- 1.2732557730789702 (* (* x x) 1.3981393803054172e-8))
      (- 1.128386358070218 (* x -0.00011824294398844343)))))
   1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.9) {
		tmp = 1e-9 + (x * ((1.2732557730789702 - ((x * x) * 1.3981393803054172e-8)) / (1.128386358070218 - (x * -0.00011824294398844343))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.9d0) then
        tmp = 1d-9 + (x * ((1.2732557730789702d0 - ((x * x) * 1.3981393803054172d-8)) / (1.128386358070218d0 - (x * (-0.00011824294398844343d0)))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.9) {
		tmp = 1e-9 + (x * ((1.2732557730789702 - ((x * x) * 1.3981393803054172e-8)) / (1.128386358070218 - (x * -0.00011824294398844343))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.9:
		tmp = 1e-9 + (x * ((1.2732557730789702 - ((x * x) * 1.3981393803054172e-8)) / (1.128386358070218 - (x * -0.00011824294398844343))))
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.9)
		tmp = Float64(1e-9 + Float64(x * Float64(Float64(1.2732557730789702 - Float64(Float64(x * x) * 1.3981393803054172e-8)) / Float64(1.128386358070218 - Float64(x * -0.00011824294398844343)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.9)
		tmp = 1e-9 + (x * ((1.2732557730789702 - ((x * x) * 1.3981393803054172e-8)) / (1.128386358070218 - (x * -0.00011824294398844343))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.9], N[(1e-9 + N[(x * N[(N[(1.2732557730789702 - N[(N[(x * x), $MachinePrecision] * 1.3981393803054172e-8), $MachinePrecision]), $MachinePrecision] / N[(1.128386358070218 - N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 74.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 74.2%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 74.1%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 74.1%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 72.5%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around 0 60.4%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   9. Step-by-step derivation

      1. *-commutative 60.4%

         \[\leadsto 10^{-9} + \left(\color{blue}{{x}^{2} \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right) \]

      2. fma-def 60.4%

         \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left({x}^{2}, -0.00011824294398844343, 1.128386358070218 \cdot x\right)} \]

      3. unpow2 60.4%

         \[\leadsto 10^{-9} + \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.00011824294398844343, 1.128386358070218 \cdot x\right) \]

      4. *-commutative 60.4%

         \[\leadsto 10^{-9} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, \color{blue}{x \cdot 1.128386358070218}\right) \]

   10. Simplified 60.4%

       \[\leadsto \color{blue}{10^{-9} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)} \]

   11. Taylor expanded in x around 0 60.4%

       \[\leadsto 10^{-9} + \color{blue}{\left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   12. Step-by-step derivation

       1. +-commutative 60.4%

          \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

       2. *-commutative 60.4%

          \[\leadsto 10^{-9} + \left(\color{blue}{x \cdot 1.128386358070218} + -0.00011824294398844343 \cdot {x}^{2}\right) \]

       3. *-commutative 60.4%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) \]

       4. unpow2 60.4%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{\left(x \cdot x\right)} \cdot -0.00011824294398844343\right) \]

       5. associate-*l* 60.4%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)}\right) \]

       6. distribute-lft-out 60.4%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]

   13. Simplified 60.4%

       \[\leadsto 10^{-9} + \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]
   14. Step-by-step derivation

       1. flip-+ 60.4%

          \[\leadsto 10^{-9} + x \cdot \color{blue}{\frac{1.128386358070218 \cdot 1.128386358070218 - \left(x \cdot -0.00011824294398844343\right) \cdot \left(x \cdot -0.00011824294398844343\right)}{1.128386358070218 - x \cdot -0.00011824294398844343}} \]

       2. metadata-eval 60.4%

          \[\leadsto 10^{-9} + x \cdot \frac{\color{blue}{1.2732557730789702} - \left(x \cdot -0.00011824294398844343\right) \cdot \left(x \cdot -0.00011824294398844343\right)}{1.128386358070218 - x \cdot -0.00011824294398844343} \]

   15. Applied egg-rr 60.4%

       \[\leadsto 10^{-9} + x \cdot \color{blue}{\frac{1.2732557730789702 - \left(x \cdot -0.00011824294398844343\right) \cdot \left(x \cdot -0.00011824294398844343\right)}{1.128386358070218 - x \cdot -0.00011824294398844343}} \]
   16. Step-by-step derivation

       1. swap-sqr 60.4%

          \[\leadsto 10^{-9} + x \cdot \frac{1.2732557730789702 - \color{blue}{\left(x \cdot x\right) \cdot \left(-0.00011824294398844343 \cdot -0.00011824294398844343\right)}}{1.128386358070218 - x \cdot -0.00011824294398844343} \]

       2. metadata-eval 60.4%

          \[\leadsto 10^{-9} + x \cdot \frac{1.2732557730789702 - \left(x \cdot x\right) \cdot \color{blue}{1.3981393803054172 \cdot 10^{-8}}}{1.128386358070218 - x \cdot -0.00011824294398844343} \]

   17. Simplified 60.4%

       \[\leadsto 10^{-9} + x \cdot \color{blue}{\frac{1.2732557730789702 - \left(x \cdot x\right) \cdot 1.3981393803054172 \cdot 10^{-8}}{1.128386358070218 - x \cdot -0.00011824294398844343}} \]

   if 0.900000000000000022 < x

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 100.0%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + x \cdot \frac{1.2732557730789702 - \left(x \cdot x\right) \cdot 1.3981393803054172 \cdot 10^{-8}}{1.128386358070218 - x \cdot -0.00011824294398844343}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 99.5% accurate, 77.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.9)
   (+ 1e-9 (* x (+ 1.128386358070218 (* x -0.00011824294398844343))))
   1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.9) {
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.9d0) then
        tmp = 1d-9 + (x * (1.128386358070218d0 + (x * (-0.00011824294398844343d0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.9) {
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.9:
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)))
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.9)
		tmp = Float64(1e-9 + Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.9)
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.9], N[(1e-9 + N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 74.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 74.2%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 74.1%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 74.1%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 72.5%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around 0 60.4%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   9. Step-by-step derivation

      1. *-commutative 60.4%

         \[\leadsto 10^{-9} + \left(\color{blue}{{x}^{2} \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right) \]

      2. fma-def 60.4%

         \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left({x}^{2}, -0.00011824294398844343, 1.128386358070218 \cdot x\right)} \]

      3. unpow2 60.4%

         \[\leadsto 10^{-9} + \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.00011824294398844343, 1.128386358070218 \cdot x\right) \]

      4. *-commutative 60.4%

         \[\leadsto 10^{-9} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, \color{blue}{x \cdot 1.128386358070218}\right) \]

   10. Simplified 60.4%

       \[\leadsto \color{blue}{10^{-9} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)} \]

   11. Taylor expanded in x around 0 60.4%

       \[\leadsto 10^{-9} + \color{blue}{\left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   12. Step-by-step derivation

       1. +-commutative 60.4%

          \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

       2. *-commutative 60.4%

          \[\leadsto 10^{-9} + \left(\color{blue}{x \cdot 1.128386358070218} + -0.00011824294398844343 \cdot {x}^{2}\right) \]

       3. *-commutative 60.4%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) \]

       4. unpow2 60.4%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{\left(x \cdot x\right)} \cdot -0.00011824294398844343\right) \]

       5. associate-*l* 60.4%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)}\right) \]

       6. distribute-lft-out 60.4%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]

   13. Simplified 60.4%

       \[\leadsto 10^{-9} + \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]

   if 0.900000000000000022 < x

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 100.0%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 99.4% accurate, 121.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.9) (+ 1e-9 (* x 1.128386358070218)) 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.9) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.9d0) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.9) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.9:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.9)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.9)
		tmp = 1e-9 + (x * 1.128386358070218);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.9], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 74.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 74.2%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 74.1%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 74.1%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 72.5%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around 0 60.3%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   9. Step-by-step derivation

      1. *-commutative 60.3%

         \[\leadsto 10^{-9} + \color{blue}{x \cdot 1.128386358070218} \]

   10. Simplified 60.3%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]

   if 0.900000000000000022 < x

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 100.0%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 97.7% accurate, 279.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 2.8e-5) 1e-9 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.8e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.8d-5) then
        tmp = 1d-9
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.8e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.8e-5:
		tmp = 1e-9
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.8e-5)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.8e-5)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.8e-5], 1e-9, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 74.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 74.1%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 74.0%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 72.4%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around 0 63.0%

      \[\leadsto \color{blue}{10^{-9}} \]

   if 2.79999999999999996e-5 < x

   1. Initial program 99.8%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
   6. Step-by-step derivation

      1. distribute-frac-neg 99.8%

         \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   8. Taylor expanded in x around inf 98.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 53.4% accurate, 856.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 10^{-9} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 1e-9)

C

x = abs(x);
double code(double x) {
	return 1e-9;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 1d-9
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 1e-9;
}

Python

x = abs(x)
def code(x):
	return 1e-9

Julia

x = abs(x)
function code(x)
	return 1e-9
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = 1e-9;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 1e-9

Derivation

1. Initial program 79.5%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

3. Simplified 79.5%

   \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]

5. Applied egg-rr 79.5%

   \[\leadsto \color{blue}{1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
6. Step-by-step derivation

   1. distribute-frac-neg 79.5%

      \[\leadsto 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

7. Simplified 78.2%

   \[\leadsto \color{blue}{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

8. Taylor expanded in x around 0 52.1%

   \[\leadsto \color{blue}{10^{-9}} \]

9. Final simplification 52.1%

   \[\leadsto 10^{-9} \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1.0 (* (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))