Jmat.Real.erf

Average Error: 13.5 → 0.2

Time: 28.8s

Precision: binary64

Cost: 195332

Math

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

↓

\[\begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := 1 + 0.3275911 \cdot \left|x\right|\\ t_2 := \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_0}}{t_0}}{t_0}}{t_0}}{{\left(e^{x}\right)}^{x} \cdot t_0}\\ t_3 := e^{x \cdot x}\\ t_4 := \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_1}}{t_1}}{t_1}}{t_1}}{t_3}}{t_1}\\ t_5 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_6 := 1 + t_2\\ t_7 := \frac{1}{t_6}\\ t_8 := \frac{{t_2}^{2}}{t_6}\\ \mathbf{if}\;x \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log \left(e^{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{-3.0685496600615605 + \frac{1.1957597040827899}{{t_5}^{3}}}{\mathsf{fma}\left({t_5}^{-2}, 1.126581484710674, 2.111650813574209\right) + \frac{1.5423834506201546}{t_5}}}{t_1}}{t_1}}{t_1}}{t_1 \cdot t_3}\right)}^{3}}\right)}{1 + t_4 \cdot \left(1 + t_4\right)}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left({\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_7 \cdot t_7 - t_8 \cdot t_8}{t_7 + t_8}\\ \end{array} \]

FPCore

(FPCore (x)
 :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)))))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x 0.3275911)))
        (t_1 (+ 1.0 (* 0.3275911 (fabs x))))
        (t_2
         (/
          (+
           0.254829592
           (/
            (+
             -0.284496736
             (/
              (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_0)) t_0))
              t_0))
            t_0))
          (* (pow (exp x) x) t_0)))
        (t_3 (exp (* x x)))
        (t_4
         (/
          (/
           (+
            0.254829592
            (/
             (+
              -0.284496736
              (/
               (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_1)) t_1))
               t_1))
             t_1))
           t_3)
          t_1))
        (t_5 (fma 0.3275911 (fabs x) 1.0))
        (t_6 (+ 1.0 t_2))
        (t_7 (/ 1.0 t_6))
        (t_8 (/ (pow t_2 2.0) t_6)))
   (if (<= x -9.6e-7)
     (/
      (log
       (exp
        (-
         1.0
         (pow
          (/
           (+
            0.254829592
            (/
             (+
              -0.284496736
              (/
               (+
                1.421413741
                (/
                 (/
                  (+ -3.0685496600615605 (/ 1.1957597040827899 (pow t_5 3.0)))
                  (+
                   (fma (pow t_5 -2.0) 1.126581484710674 2.111650813574209)
                   (/ 1.5423834506201546 t_5)))
                 t_1))
               t_1))
             t_1))
           (* t_1 t_3))
          3.0))))
      (+ 1.0 (* t_4 (+ 1.0 t_4))))
     (if (<= x 1.3e-6)
       (+ 1e-9 (sqrt (pow (pow (cbrt (* x 1.128386358070218)) 2.0) 3.0)))
       (/ (- (* t_7 t_7) (* t_8 t_8)) (+ t_7 t_8))))))

C

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

↓

double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = 1.0 + (0.3275911 * fabs(x));
	double t_2 = (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0)) / (pow(exp(x), x) * t_0);
	double t_3 = exp((x * x));
	double t_4 = ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) / t_3) / t_1;
	double t_5 = fma(0.3275911, fabs(x), 1.0);
	double t_6 = 1.0 + t_2;
	double t_7 = 1.0 / t_6;
	double t_8 = pow(t_2, 2.0) / t_6;
	double tmp;
	if (x <= -9.6e-7) {
		tmp = log(exp((1.0 - pow(((0.254829592 + ((-0.284496736 + ((1.421413741 + (((-3.0685496600615605 + (1.1957597040827899 / pow(t_5, 3.0))) / (fma(pow(t_5, -2.0), 1.126581484710674, 2.111650813574209) + (1.5423834506201546 / t_5))) / t_1)) / t_1)) / t_1)) / (t_1 * t_3)), 3.0)))) / (1.0 + (t_4 * (1.0 + t_4)));
	} else if (x <= 1.3e-6) {
		tmp = 1e-9 + sqrt(pow(pow(cbrt((x * 1.128386358070218)), 2.0), 3.0));
	} else {
		tmp = ((t_7 * t_7) - (t_8 * t_8)) / (t_7 + t_8);
	}
	return tmp;
}

Julia

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

↓

function code(x)
	t_0 = Float64(1.0 + Float64(x * 0.3275911))
	t_1 = Float64(1.0 + Float64(0.3275911 * abs(x)))
	t_2 = Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_0)) / t_0)) / t_0)) / t_0)) / Float64((exp(x) ^ x) * t_0))
	t_3 = exp(Float64(x * x))
	t_4 = Float64(Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) / t_3) / t_1)
	t_5 = fma(0.3275911, abs(x), 1.0)
	t_6 = Float64(1.0 + t_2)
	t_7 = Float64(1.0 / t_6)
	t_8 = Float64((t_2 ^ 2.0) / t_6)
	tmp = 0.0
	if (x <= -9.6e-7)
		tmp = Float64(log(exp(Float64(1.0 - (Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(Float64(-3.0685496600615605 + Float64(1.1957597040827899 / (t_5 ^ 3.0))) / Float64(fma((t_5 ^ -2.0), 1.126581484710674, 2.111650813574209) + Float64(1.5423834506201546 / t_5))) / t_1)) / t_1)) / t_1)) / Float64(t_1 * t_3)) ^ 3.0)))) / Float64(1.0 + Float64(t_4 * Float64(1.0 + t_4))));
	elseif (x <= 1.3e-6)
		tmp = Float64(1e-9 + sqrt(((cbrt(Float64(x * 1.128386358070218)) ^ 2.0) ^ 3.0)));
	else
		tmp = Float64(Float64(Float64(t_7 * t_7) - Float64(t_8 * t_8)) / Float64(t_7 + t_8));
	end
	return tmp
end

Wolfram

code[x_] := N[(1.0 - N[(N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.254829592 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.284496736 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.421413741 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.453152027 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]

↓

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$6 = N[(1.0 + t$95$2), $MachinePrecision]}, Block[{t$95$7 = N[(1.0 / t$95$6), $MachinePrecision]}, Block[{t$95$8 = N[(N[Power[t$95$2, 2.0], $MachinePrecision] / t$95$6), $MachinePrecision]}, If[LessEqual[x, -9.6e-7], N[(N[Log[N[Exp[N[(1.0 - N[Power[N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(N[(-3.0685496600615605 + N[(1.1957597040827899 / N[Power[t$95$5, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[t$95$5, -2.0], $MachinePrecision] * 1.126581484710674 + 2.111650813574209), $MachinePrecision] + N[(1.5423834506201546 / t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(t$95$4 * N[(1.0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e-6], N[(1e-9 + N[Sqrt[N[Power[N[Power[N[Power[N[(x * 1.128386358070218), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$7 * t$95$7), $MachinePrecision] - N[(t$95$8 * t$95$8), $MachinePrecision]), $MachinePrecision] / N[(t$95$7 + t$95$8), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.59999999999999914e-7

   1. Initial program 0.2

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

   2. Simplified 0.2

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

      [Start] 0.2

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

      associate-*l* [=>] 0.2

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

   3. Applied egg-rr 0.2

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

   4. Applied egg-rr 0.2

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

   5. Applied egg-rr 0.2

      \[\leadsto \frac{\log \left(e^{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\color{blue}{\left(-3.0685496600615605 + {\left(\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}\right) \cdot \frac{1}{2.111650813574209 + \left(1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} - \frac{-1.5423834506201546}{1 + 0.3275911 \cdot \left|x\right|}\right)}}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}\right)}^{3}}\right)}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]

   6. Simplified 0.2

      \[\leadsto \frac{\log \left(e^{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\color{blue}{\frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}}{\mathsf{fma}\left({\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, 1.126581484710674, 2.111650813574209\right) + \frac{1.5423834506201546}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}\right)}^{3}}\right)}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]
      Proof

      [Start] 0.2	\[ \frac{\log \left(e^{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\left(-3.0685496600615605 + {\left(\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}\right) \cdot \frac{1}{2.111650813574209 + \left(1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} - \frac{-1.5423834506201546}{1 + 0.3275911 \cdot \left|x\right|}\right)}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}\right)}^{3}}\right)}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]
      associate-*r/ [=>] 0.2	\[ \frac{\log \left(e^{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\color{blue}{\frac{\left(-3.0685496600615605 + {\left(\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}\right) \cdot 1}{2.111650813574209 + \left(1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} - \frac{-1.5423834506201546}{1 + 0.3275911 \cdot \left|x\right|}\right)}}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}\right)}^{3}}\right)}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]
      *-rgt-identity [=>] 0.2	\[ \frac{\log \left(e^{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{\color{blue}{-3.0685496600615605 + {\left(\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}}{2.111650813574209 + \left(1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} - \frac{-1.5423834506201546}{1 + 0.3275911 \cdot \left|x\right|}\right)}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}\right)}^{3}}\right)}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]
      cube-div [=>] 0.2	\[ \frac{\log \left(e^{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{-3.0685496600615605 + \color{blue}{\frac{{1.061405429}^{3}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}}}{2.111650813574209 + \left(1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} - \frac{-1.5423834506201546}{1 + 0.3275911 \cdot \left|x\right|}\right)}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}\right)}^{3}}\right)}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]
      metadata-eval [=>] 0.2	\[ \frac{\log \left(e^{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{-3.0685496600615605 + \frac{\color{blue}{1.1957597040827899}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}}{2.111650813574209 + \left(1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} - \frac{-1.5423834506201546}{1 + 0.3275911 \cdot \left|x\right|}\right)}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}\right)}^{3}}\right)}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]
      +-commutative [<=] 0.2	\[ \frac{\log \left(e^{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}}^{3}}}{2.111650813574209 + \left(1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} - \frac{-1.5423834506201546}{1 + 0.3275911 \cdot \left|x\right|}\right)}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}\right)}^{3}}\right)}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]
      fma-def [=>] 0.2	\[ \frac{\log \left(e^{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}}^{3}}}{2.111650813574209 + \left(1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} - \frac{-1.5423834506201546}{1 + 0.3275911 \cdot \left|x\right|}\right)}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}\right)}^{3}}\right)}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]
      sub-neg [=>] 0.2	\[ \frac{\log \left(e^{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}}{2.111650813574209 + \color{blue}{\left(1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} + \left(-\frac{-1.5423834506201546}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)}}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}\right)}^{3}}\right)}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]
      associate-+r+ [=>] 0.2	\[ \frac{\log \left(e^{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}}{\color{blue}{\left(2.111650813574209 + 1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2}\right) + \left(-\frac{-1.5423834506201546}{1 + 0.3275911 \cdot \left|x\right|}\right)}}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}\right)}^{3}}\right)}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]
      +-commutative [<=] 0.2	\[ \frac{\log \left(e^{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}}{\color{blue}{\left(1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} + 2.111650813574209\right)} + \left(-\frac{-1.5423834506201546}{1 + 0.3275911 \cdot \left|x\right|}\right)}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}\right)}^{3}}\right)}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]

   if -9.59999999999999914e-7 < x < 1.30000000000000005e-6

   1. Initial program 27.1

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

   2. Simplified 27.1

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

      [Start] 27.1	\[ 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
      cancel-sign-sub-inv [=>] 27.1	\[ \color{blue}{1 + \left(-\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \]
      +-commutative [=>] 27.1	\[ \color{blue}{\left(-\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} + 1} \]

   3. Applied egg-rr 27.4

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

   4. Taylor expanded in x around 0 1.0

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

   5. Applied egg-rr 0.2

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

   6. Simplified 0.2

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

      [Start] 0.2	\[ 10^{-9} + \sqrt{1.2732557730789702 \cdot \left(x \cdot x\right)} \]
      *-commutative [=>] 0.2	\[ 10^{-9} + \sqrt{\color{blue}{\left(x \cdot x\right) \cdot 1.2732557730789702}} \]
      associate-*l* [=>] 0.2	\[ 10^{-9} + \sqrt{\color{blue}{x \cdot \left(x \cdot 1.2732557730789702\right)}} \]

   7. Applied egg-rr 0.2

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

   if 1.30000000000000005e-6 < x

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1

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

      associate-*l* [=>] 0.1

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

   3. Applied egg-rr 0.1

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

   4. Applied egg-rr 0.1

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

   5. Applied egg-rr 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log \left(e^{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{-3.0685496600615605 + \frac{1.1957597040827899}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}}{\mathsf{fma}\left({\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, 1.126581484710674, 2.111650813574209\right) + \frac{1.5423834506201546}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}\right)}^{3}}\right)}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left({\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{{\left(e^{x}\right)}^{x} \cdot \left(1 + x \cdot 0.3275911\right)}} \cdot \frac{1}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{{\left(e^{x}\right)}^{x} \cdot \left(1 + x \cdot 0.3275911\right)}} - \frac{{\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{{\left(e^{x}\right)}^{x} \cdot \left(1 + x \cdot 0.3275911\right)}\right)}^{2}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{{\left(e^{x}\right)}^{x} \cdot \left(1 + x \cdot 0.3275911\right)}} \cdot \frac{{\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{{\left(e^{x}\right)}^{x} \cdot \left(1 + x \cdot 0.3275911\right)}\right)}^{2}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{{\left(e^{x}\right)}^{x} \cdot \left(1 + x \cdot 0.3275911\right)}}}{\frac{1}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{{\left(e^{x}\right)}^{x} \cdot \left(1 + x \cdot 0.3275911\right)}} + \frac{{\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{{\left(e^{x}\right)}^{x} \cdot \left(1 + x \cdot 0.3275911\right)}\right)}^{2}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{1 + x \cdot 0.3275911}}{{\left(e^{x}\right)}^{x} \cdot \left(1 + x \cdot 0.3275911\right)}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	160712

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := 1 + x \cdot 0.3275911\\ t_2 := \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_1}}{t_1}}{t_1}}{t_1}}{{\left(e^{x}\right)}^{x} \cdot t_1}\\ t_3 := 1 + t_2\\ t_4 := \frac{{t_2}^{2}}{t_3}\\ t_5 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_6 := \frac{1}{t_3}\\ \mathbf{if}\;x \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;e^{\log \left(1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{-3.0685496600615605 + \frac{1.1957597040827899}{{t_5}^{3}}}{\frac{1.5423834506201546}{t_5} + \left(2.111650813574209 + 1.126581484710674 \cdot \frac{1}{{t_0}^{2}}\right)}}{t_0}}{t_0}}{t_0}}{e^{x \cdot x}}}{t_0}\right)}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left({\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_6 \cdot t_6 - t_4 \cdot t_4}{t_6 + t_4}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	93764

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := 1 + x \cdot 0.3275911\\ t_2 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_3 := e^{x \cdot x}\\ t_4 := \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_1}}{t_1}}{t_1}}{t_1}}{t_3 \cdot t_1}\\ t_5 := {t_4}^{2}\\ \mathbf{if}\;x \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;e^{\log \left(1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{-3.0685496600615605 + \frac{1.1957597040827899}{{t_2}^{3}}}{\frac{1.5423834506201546}{t_2} + \left(2.111650813574209 + 1.126581484710674 \cdot \frac{1}{{t_0}^{2}}\right)}}{t_0}}{t_0}}{t_0}}{t_3}}{t_0}\right)}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left({\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {\left(-t_5\right)}^{2}}{1 + t_5}}{1 + t_4}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	67588

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := 1 + x \cdot 0.3275911\\ t_2 := \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_1}}{t_1}}{t_1}}{t_1}}{e^{x \cdot x} \cdot t_1}\\ t_3 := {t_2}^{2}\\ \mathbf{if}\;x \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;1 + \frac{e^{-{x}^{2}} \cdot \left(\left(-1.453152027 \cdot \frac{-1}{{t_0}^{3}} + -0.284496736 \cdot \frac{-1}{t_0}\right) + \left(-0.254829592 + \left(1.061405429 \cdot \frac{-1}{{t_0}^{4}} + {t_0}^{-2} \cdot -1.421413741\right)\right)\right)}{t_0}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left({\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {\left(-t_3\right)}^{2}}{1 + t_3}}{1 + t_2}\\ \end{array} \]

Alternative 4
Error	0.5
Cost	53956

\[\begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_0}}{t_0}}{t_0}}{t_0}}{e^{x \cdot x} \cdot t_0}\\ t_2 := {t_1}^{2}\\ \mathbf{if}\;\left|x\right| \leq 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left({\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {\left(-t_2\right)}^{2}}{1 + t_2}}{1 + t_1}\\ \end{array} \]

Alternative 5
Error	0.3
Cost	38024

\[\begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_0}}{t_0}}{t_0}}{t_0}}{{\left(e^{x}\right)}^{x} \cdot t_0}\\ \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left({\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t_1}^{2}}{1 + t_1}\\ \end{array} \]

Alternative 6
Error	0.3
Cost	26312

\[\begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left({\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{1 + \frac{-0.254829592 + \frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_0}}{t_0}}{t_0}}{t_0}}{{\left(e^{x}\right)}^{x} \cdot t_0}}}\\ \end{array} \]

Alternative 7
Error	0.3
Cost	16136

\[\begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{1 + \frac{-0.254829592 + \frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_0}}{t_0}}{t_0}}{t_0}}{{\left(e^{x}\right)}^{x} \cdot t_0}}}\\ \end{array} \]

Alternative 8
Error	0.3
Cost	15880

\[\begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592 + \frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_0}}{t_0}}{t_0}}{t_0}}{{\left(e^{x}\right)}^{x} \cdot t_0}\\ \end{array} \]

Alternative 9
Error	0.4
Cost	14856

\[\begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143}{t_0}}{t_0}}{{\left(e^{x}\right)}^{x} \cdot t_0}\\ \end{array} \]

Alternative 10
Error	0.4
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.7778892405807117}{e^{x \cdot x}}}{x}\\ \end{array} \]

Alternative 11
Error	0.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	0.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	0.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	1.4
Cost	328

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

Alternative 15
Error	29.5
Cost	64

\[10^{-9} \]

Reproduce

herbie shell --seed 2023039 
(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)))))))