Jmat.Real.erf

Average Error: 13.3 → 0.2

Time: 37.8s

Precision: binary64

Cost: 227652

Math

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

↓

\[\begin{array}{l} t_0 := e^{-x \cdot x}\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := 1 + 0.3275911 \cdot \left|x\right|\\ t_3 := \frac{1.061405429}{t_2}\\ t_4 := \frac{1}{t_2}\\ t_5 := {\left(t_4 \cdot \left(\left(0.254829592 + t_4 \cdot \left(-0.284496736 + t_4 \cdot \left(1.421413741 + t_4 \cdot \frac{1.126581484710674 \cdot {t_2}^{-2} + -2.111650813574209}{t_3 + 1.453152027}\right)\right)\right) \cdot t_0\right)\right)}^{2}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1 - t_5 \cdot t_5}{1 + t_5} \cdot \frac{1}{1 + t_4 \cdot \left(t_0 \cdot \left(0.254829592 + t_4 \cdot \left(-0.284496736 + t_4 \cdot \left(1.421413741 + t_4 \cdot \left(t_3 + -1.453152027\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{t_1}, -0.254829592 + \frac{0.284496736 + \frac{t_4 \cdot 1.453152027 + \left(-1.421413741 + \frac{1}{{t_2}^{2}} \cdot -1.061405429\right)}{t_1}}{t_1}, 1\right)\right)}\\ \end{array} \]

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 (exp (- (* x x))))
        (t_1 (fma 0.3275911 (fabs x) 1.0))
        (t_2 (+ 1.0 (* 0.3275911 (fabs x))))
        (t_3 (/ 1.061405429 t_2))
        (t_4 (/ 1.0 t_2))
        (t_5
         (pow
          (*
           t_4
           (*
            (+
             0.254829592
             (*
              t_4
              (+
               -0.284496736
               (*
                t_4
                (+
                 1.421413741
                 (*
                  t_4
                  (/
                   (+ (* 1.126581484710674 (pow t_2 -2.0)) -2.111650813574209)
                   (+ t_3 1.453152027))))))))
            t_0))
          2.0)))
   (if (<= x -8e-7)
     (*
      (/ (- 1.0 (* t_5 t_5)) (+ 1.0 t_5))
      (/
       1.0
       (+
        1.0
        (*
         t_4
         (*
          t_0
          (+
           0.254829592
           (*
            t_4
            (+
             -0.284496736
             (* t_4 (+ 1.421413741 (* t_4 (+ t_3 -1.453152027))))))))))))
     (if (<= x 1.85e-6)
       (+ 1e-9 (sqrt (* x (* x 1.2732557730789702))))
       (exp
        (log
         (fma
          (/ (pow (exp x) (- x)) t_1)
          (+
           -0.254829592
           (/
            (+
             0.284496736
             (/
              (+
               (* t_4 1.453152027)
               (+ -1.421413741 (* (/ 1.0 (pow t_2 2.0)) -1.061405429)))
              t_1))
            t_1))
          1.0)))))))

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 = exp(-(x * x));
	double t_1 = fma(0.3275911, fabs(x), 1.0);
	double t_2 = 1.0 + (0.3275911 * fabs(x));
	double t_3 = 1.061405429 / t_2;
	double t_4 = 1.0 / t_2;
	double t_5 = pow((t_4 * ((0.254829592 + (t_4 * (-0.284496736 + (t_4 * (1.421413741 + (t_4 * (((1.126581484710674 * pow(t_2, -2.0)) + -2.111650813574209) / (t_3 + 1.453152027)))))))) * t_0)), 2.0);
	double tmp;
	if (x <= -8e-7) {
		tmp = ((1.0 - (t_5 * t_5)) / (1.0 + t_5)) * (1.0 / (1.0 + (t_4 * (t_0 * (0.254829592 + (t_4 * (-0.284496736 + (t_4 * (1.421413741 + (t_4 * (t_3 + -1.453152027)))))))))));
	} else if (x <= 1.85e-6) {
		tmp = 1e-9 + sqrt((x * (x * 1.2732557730789702)));
	} else {
		tmp = exp(log(fma((pow(exp(x), -x) / t_1), (-0.254829592 + ((0.284496736 + (((t_4 * 1.453152027) + (-1.421413741 + ((1.0 / pow(t_2, 2.0)) * -1.061405429))) / t_1)) / t_1)), 1.0)));
	}
	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 = exp(Float64(-Float64(x * x)))
	t_1 = fma(0.3275911, abs(x), 1.0)
	t_2 = Float64(1.0 + Float64(0.3275911 * abs(x)))
	t_3 = Float64(1.061405429 / t_2)
	t_4 = Float64(1.0 / t_2)
	t_5 = Float64(t_4 * Float64(Float64(0.254829592 + Float64(t_4 * Float64(-0.284496736 + Float64(t_4 * Float64(1.421413741 + Float64(t_4 * Float64(Float64(Float64(1.126581484710674 * (t_2 ^ -2.0)) + -2.111650813574209) / Float64(t_3 + 1.453152027)))))))) * t_0)) ^ 2.0
	tmp = 0.0
	if (x <= -8e-7)
		tmp = Float64(Float64(Float64(1.0 - Float64(t_5 * t_5)) / Float64(1.0 + t_5)) * Float64(1.0 / Float64(1.0 + Float64(t_4 * Float64(t_0 * Float64(0.254829592 + Float64(t_4 * Float64(-0.284496736 + Float64(t_4 * Float64(1.421413741 + Float64(t_4 * Float64(t_3 + -1.453152027))))))))))));
	elseif (x <= 1.85e-6)
		tmp = Float64(1e-9 + sqrt(Float64(x * Float64(x * 1.2732557730789702))));
	else
		tmp = exp(log(fma(Float64((exp(x) ^ Float64(-x)) / t_1), Float64(-0.254829592 + Float64(Float64(0.284496736 + Float64(Float64(Float64(t_4 * 1.453152027) + Float64(-1.421413741 + Float64(Float64(1.0 / (t_2 ^ 2.0)) * -1.061405429))) / t_1)) / t_1)), 1.0)));
	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[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.061405429 / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(t$95$4 * N[(N[(0.254829592 + N[(t$95$4 * N[(-0.284496736 + N[(t$95$4 * N[(1.421413741 + N[(t$95$4 * N[(N[(N[(1.126581484710674 * N[Power[t$95$2, -2.0], $MachinePrecision]), $MachinePrecision] + -2.111650813574209), $MachinePrecision] / N[(t$95$3 + 1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -8e-7], N[(N[(N[(1.0 - N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$5), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(t$95$4 * N[(t$95$0 * N[(0.254829592 + N[(t$95$4 * N[(-0.284496736 + N[(t$95$4 * N[(1.421413741 + N[(t$95$4 * N[(t$95$3 + -1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.85e-6], N[(1e-9 + N[Sqrt[N[(x * N[(x * 1.2732557730789702), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[N[(N[(N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(-0.254829592 + N[(N[(0.284496736 + N[(N[(N[(t$95$4 * 1.453152027), $MachinePrecision] + N[(-1.421413741 + N[(N[(1.0 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * -1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.9999999999999996e-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

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

   4. Applied egg-rr 0.2

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

   5. Applied egg-rr 0.2

      \[\leadsto \color{blue}{\frac{1 - {\left(\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 \frac{1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} - 2.111650813574209}{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} - -1.453152027}\right)\right)\right) \cdot e^{-x \cdot x}\right)\right)}^{2} \cdot {\left(\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 \frac{1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} - 2.111650813574209}{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} - -1.453152027}\right)\right)\right) \cdot e^{-x \cdot x}\right)\right)}^{2}}{1 + {\left(\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 \frac{1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} - 2.111650813574209}{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} - -1.453152027}\right)\right)\right) \cdot e^{-x \cdot x}\right)\right)}^{2}}} \cdot \frac{1}{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)} \]

   if -7.9999999999999996e-7 < x < 1.8500000000000001e-6

   1. Initial program 27.0

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

   2. Simplified 27.0

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

      [Start] 27.0

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

      associate-*l* [=>] 27.0

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

   3. Applied egg-rr 27.0

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

   4. Applied egg-rr 27.0

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

   5. Simplified 27.4

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

      [Start] 27.0	\[ 1 \cdot \left(\frac{-\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}}{1 + 0.3275911 \cdot \left|x\right|} + 1\right) \]
      *-lft-identity [=>] 27.0	\[ \color{blue}{\frac{-\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}}{1 + 0.3275911 \cdot \left|x\right|} + 1} \]
      +-commutative [=>] 27.0	\[ \color{blue}{1 + \frac{-\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}}{1 + 0.3275911 \cdot \left|x\right|}} \]
      metadata-eval [<=] 27.0	\[ \color{blue}{\left(1 - 0\right)} + \frac{-\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}}{1 + 0.3275911 \cdot \left|x\right|} \]
      associate--r- [<=] 27.0	\[ \color{blue}{1 - \left(0 - \frac{-\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]

   6. Taylor expanded in x around 0 1.1

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

   7. Applied egg-rr 0.2

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

   8. 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	\[ 10^{-9} + \sqrt{\color{blue}{x \cdot \left(x \cdot 1.2732557730789702\right)}} \]

   if 1.8500000000000001e-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

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

      [Start] 0.1	\[ 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
      cancel-sign-sub-inv [=>] 0.1	\[ \color{blue}{1 + \left(-\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \]
      +-commutative [=>] 0.1	\[ \color{blue}{\left(-\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} + 1} \]

   3. Taylor expanded in x around 0 0.1

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

   4. Applied egg-rr 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1 - {\left(\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 \frac{1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} + -2.111650813574209}{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027}\right)\right)\right) \cdot e^{-x \cdot x}\right)\right)}^{2} \cdot {\left(\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 \frac{1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} + -2.111650813574209}{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027}\right)\right)\right) \cdot e^{-x \cdot x}\right)\right)}^{2}}{1 + {\left(\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 \frac{1.126581484710674 \cdot {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-2} + -2.111650813574209}{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027}\right)\right)\right) \cdot e^{-x \cdot x}\right)\right)}^{2}} \cdot \frac{1}{1 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(e^{-x \cdot x} \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(\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} + -1.453152027\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\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{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.453152027 + \left(-1.421413741 + \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} \cdot -1.061405429\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	116804

\[\begin{array}{l} t_0 := e^{-x \cdot x}\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := 1 + 0.3275911 \cdot \left|x\right|\\ t_3 := \frac{1}{t_2}\\ t_4 := \frac{1.061405429}{t_2}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{1 + t_3 \cdot \left(t_0 \cdot \left(0.254829592 + t_3 \cdot \left(-0.284496736 + t_3 \cdot \left(1.421413741 + t_3 \cdot \left(t_4 + -1.453152027\right)\right)\right)\right)\right)} \cdot e^{\log \left(1 - {\left(t_3 \cdot \left(\left(0.254829592 + t_3 \cdot \left(-0.284496736 + t_3 \cdot \left(1.421413741 + t_3 \cdot \frac{1.126581484710674 \cdot {t_2}^{-2} + -2.111650813574209}{t_4 + 1.453152027}\right)\right)\right) \cdot t_0\right)\right)}^{2}\right)}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{t_1}, -0.254829592 + \frac{0.284496736 + \frac{t_3 \cdot 1.453152027 + \left(-1.421413741 + \frac{1}{{t_2}^{2}} \cdot -1.061405429\right)}{t_1}}{t_1}, 1\right)\right)}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	104004

\[\begin{array}{l} t_0 := e^{-x \cdot x}\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := 1 + 0.3275911 \cdot \left|x\right|\\ t_3 := \frac{1}{t_2}\\ t_4 := \frac{1.061405429}{t_2}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{1 + t_3 \cdot \left(t_0 \cdot \left(0.254829592 + t_3 \cdot \left(-0.284496736 + t_3 \cdot \left(1.421413741 + t_3 \cdot \left(t_4 + -1.453152027\right)\right)\right)\right)\right)} \cdot \left(1 - {\left(t_3 \cdot \left(\left(0.254829592 + t_3 \cdot \left(-0.284496736 + t_3 \cdot \left(1.421413741 + t_3 \cdot \frac{1.126581484710674 \cdot {t_2}^{-2} + -2.111650813574209}{t_4 + 1.453152027}\right)\right)\right) \cdot t_0\right)\right)}^{2}\right)\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{t_1}, -0.254829592 + \frac{0.284496736 + \frac{t_3 \cdot 1.453152027 + \left(-1.421413741 + \frac{1}{{t_2}^{2}} \cdot -1.061405429\right)}{t_1}}{t_1}, 1\right)\right)}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	92552

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-7}:\\ \;\;\;\;1 + \frac{e^{-x \cdot x} \cdot \left(-0.254829592 + t_2 \cdot \left(0.284496736 - t_2 \cdot \left(1.421413741 + t_2 \cdot \frac{1.126581484710674 \cdot {t_0}^{-2} + -2.111650813574209}{\frac{1.061405429}{t_0} + 1.453152027}\right)\right)\right)}{t_0}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{t_1}, -0.254829592 + \frac{0.284496736 + \frac{t_2 \cdot 1.453152027 + \left(-1.421413741 + \frac{1}{{t_0}^{2}} \cdot -1.061405429\right)}{t_1}}{t_1}, 1\right)\right)}\\ \end{array} \]

Alternative 4
Error	0.2
Cost	73480

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-7}:\\ \;\;\;\;1 + \frac{e^{-x \cdot x} \cdot \left(-0.254829592 + t_2 \cdot \left(0.284496736 - t_2 \cdot \left(1.421413741 + t_2 \cdot \frac{1.126581484710674 \cdot {t_0}^{-2} + -2.111650813574209}{\frac{1.061405429}{t_0} + 1.453152027}\right)\right)\right)}{t_0}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{t_1}, -0.254829592 + \frac{0.284496736 + \frac{t_2 \cdot 1.453152027 + \left(-1.421413741 + \frac{1}{{t_0}^{2}} \cdot -1.061405429\right)}{t_1}}{t_0}, 1\right)\\ \end{array} \]

Alternative 5
Error	0.2
Cost	60104

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-7}:\\ \;\;\;\;1 + \frac{e^{-x \cdot x} \cdot \left(-0.254829592 + t_1 \cdot \left(0.284496736 - t_1 \cdot \left(1.421413741 + t_1 \cdot \frac{1.126581484710674 \cdot {t_0}^{-2} + -2.111650813574209}{\frac{1.061405429}{t_0} + 1.453152027}\right)\right)\right)}{t_0}\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}}\right)}^{3}\\ \end{array} \]

Alternative 6
Error	0.2
Cost	55364

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-7}:\\ \;\;\;\;1 + \frac{e^{-x \cdot x} \cdot \left(-0.254829592 + t_1 \cdot \left(0.284496736 - t_1 \cdot \left(1.421413741 + t_1 \cdot \frac{1.126581484710674 \cdot {t_0}^{-2} + -2.111650813574209}{\frac{1.061405429}{t_0} + 1.453152027}\right)\right)\right)}{t_0}\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 7
Error	0.2
Cost	54660

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-7}:\\ \;\;\;\;e^{\log \left(1 + \frac{e^{-x \cdot x} \cdot \left(-0.254829592 + \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(\frac{1.061405429}{t_0} + -1.453152027\right)\right)\right) \cdot \frac{-1}{t_0}\right)}{t_0}\right)}\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 8
Error	0.2
Cost	41860

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-7}:\\ \;\;\;\;1 + \frac{e^{-x \cdot x} \cdot \left(-0.254829592 + \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(\frac{1.061405429}{t_0} + -1.453152027\right)\right)\right) \cdot \frac{-1}{t_0}\right)}{t_0}\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 9
Error	0.3
Cost	40904

\[\begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 10
Error	0.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	0.9
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{1}{\frac{10^{-9} + x \cdot -1.128386358070218}{10^{-18} + \left(x \cdot x\right) \cdot -1.2732557730789702}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	0.9
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 1.2732557730789702 + -1 \cdot 10^{-18}}{x \cdot 1.128386358070218 + -1 \cdot 10^{-9}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	0.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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	30.0
Cost	64

\[10^{-9} \]

Reproduce

herbie shell --seed 2023011 
(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)))))))