Jmat.Real.erf

Average Accuracy: 79.2% → 99.7%

Time: 19.5s

Precision: binary64

Cost: 60868

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := 1 + 0.3275911 \cdot \left|x\right|\\ t_2 := \frac{1}{t_1}\\ t_3 := e^{x \cdot \left(-x\right)}\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;1 + \left(\left(0.254829592 + t_2 \cdot \left(-0.284496736 + t_2 \cdot \left(1.421413741 + \left(\frac{1.061405429}{{t_0}^{2}} + \frac{-1.453152027}{t_0}\right)\right)\right)\right) \cdot t_3\right) \cdot \frac{-1}{t_1}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-7}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 - t_2 \cdot \left(t_3 \cdot \left(0.254829592 + t_2 \cdot \left(-0.284496736 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(x \cdot 0.3275911\right)\right)} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{t_1}^{2}}\right) + t_2 \cdot -1.453152027\right)\right)\right)\right)\\ \end{array} \]

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 (fma 0.3275911 (fabs x) 1.0))
        (t_1 (+ 1.0 (* 0.3275911 (fabs x))))
        (t_2 (/ 1.0 t_1))
        (t_3 (exp (* x (- x)))))
   (if (<= x -1.2e-6)
     (+
      1.0
      (*
       (*
        (+
         0.254829592
         (*
          t_2
          (+
           -0.284496736
           (*
            t_2
            (+
             1.421413741
             (+ (/ 1.061405429 (pow t_0 2.0)) (/ -1.453152027 t_0)))))))
        t_3)
       (/ -1.0 t_1)))
     (if (<= x 9.6e-7)
       (+ 1e-9 (sqrt (* (* x x) 1.2732557730789702)))
       (-
        1.0
        (*
         t_2
         (*
          t_3
          (+
           0.254829592
           (*
            t_2
            (+
             -0.284496736
             (*
              (/ 1.0 (+ 1.0 (log (+ 1.0 (expm1 (* x 0.3275911))))))
              (+
               (+ 1.421413741 (* 1.061405429 (/ 1.0 (pow t_1 2.0))))
               (* t_2 -1.453152027)))))))))))))

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 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = 1.0 + (0.3275911 * fabs(x));
	double t_2 = 1.0 / t_1;
	double t_3 = exp((x * -x));
	double tmp;
	if (x <= -1.2e-6) {
		tmp = 1.0 + (((0.254829592 + (t_2 * (-0.284496736 + (t_2 * (1.421413741 + ((1.061405429 / pow(t_0, 2.0)) + (-1.453152027 / t_0))))))) * t_3) * (-1.0 / t_1));
	} else if (x <= 9.6e-7) {
		tmp = 1e-9 + sqrt(((x * x) * 1.2732557730789702));
	} else {
		tmp = 1.0 - (t_2 * (t_3 * (0.254829592 + (t_2 * (-0.284496736 + ((1.0 / (1.0 + log((1.0 + expm1((x * 0.3275911)))))) * ((1.421413741 + (1.061405429 * (1.0 / pow(t_1, 2.0)))) + (t_2 * -1.453152027))))))));
	}
	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 = fma(0.3275911, abs(x), 1.0)
	t_1 = Float64(1.0 + Float64(0.3275911 * abs(x)))
	t_2 = Float64(1.0 / t_1)
	t_3 = exp(Float64(x * Float64(-x)))
	tmp = 0.0
	if (x <= -1.2e-6)
		tmp = Float64(1.0 + Float64(Float64(Float64(0.254829592 + Float64(t_2 * Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(Float64(1.061405429 / (t_0 ^ 2.0)) + Float64(-1.453152027 / t_0))))))) * t_3) * Float64(-1.0 / t_1)));
	elseif (x <= 9.6e-7)
		tmp = Float64(1e-9 + sqrt(Float64(Float64(x * x) * 1.2732557730789702)));
	else
		tmp = Float64(1.0 - Float64(t_2 * Float64(t_3 * Float64(0.254829592 + Float64(t_2 * Float64(-0.284496736 + Float64(Float64(1.0 / Float64(1.0 + log(Float64(1.0 + expm1(Float64(x * 0.3275911)))))) * Float64(Float64(1.421413741 + Float64(1.061405429 * Float64(1.0 / (t_1 ^ 2.0)))) + Float64(t_2 * -1.453152027)))))))));
	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[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.2e-6], N[(1.0 + N[(N[(N[(0.254829592 + N[(t$95$2 * N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(N[(1.061405429 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.453152027 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.6e-7], N[(1e-9 + N[Sqrt[N[(N[(x * x), $MachinePrecision] * 1.2732557730789702), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(t$95$2 * N[(t$95$3 * N[(0.254829592 + N[(t$95$2 * N[(-0.284496736 + N[(N[(1.0 / N[(1.0 + N[Log[N[(1.0 + N[(Exp[N[(x * 0.3275911), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.421413741 + N[(1.061405429 * N[(1.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * -1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1999999999999999e-6

   1. Initial program 99.8%

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

   2. Simplified 99.8%

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

      [Start] 99.8

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

      associate-*l* [=>] 99.8

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

   3. Taylor expanded in x around 0 99.8%

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

   4. Simplified 99.8%

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

      [Start] 99.8	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right) - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      associate--l+ [=>] 99.8	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(1.421413741 + \left(1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}} - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)}\right)\right) \cdot e^{-x \cdot x}\right) \]
      associate-*r/ [=>] 99.8	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \left(\color{blue}{\frac{1.061405429 \cdot 1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}} - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      +-commutative [=>] 99.8	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \left(\frac{1.061405429 \cdot 1}{{\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)}}^{2}} - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      metadata-eval [=>] 99.8	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \left(\frac{\color{blue}{1.061405429}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      +-commutative [<=] 99.8	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \left(\frac{1.061405429}{{\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}}^{2}} - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      fma-def [=>] 99.8	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \left(\frac{1.061405429}{{\color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}}^{2}} - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      associate-*r/ [=>] 99.8	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} - \color{blue}{\frac{1.453152027 \cdot 1}{0.3275911 \cdot \left|x\right| + 1}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      metadata-eval [=>] 99.8	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} - \frac{\color{blue}{1.453152027}}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      fma-def [=>] 99.8	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} - \frac{1.453152027}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   if -1.1999999999999999e-6 < x < 9.59999999999999914e-7

   1. Initial program 57.7%

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

   2. Simplified 57.7%

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

      [Start] 57.7	\[ 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
      associate-*l* [=>] 57.7	\[ 1 - \color{blue}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)} \]
      associate-*l/ [=>] 57.7	\[ 1 - \color{blue}{\frac{1 \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   3. Applied egg-rr 54.9%

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

      [Start] 57.7	\[ 1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]
      add-log-exp [=>] 55.4	\[ \color{blue}{\log \left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)} \]
      associate-/l/ [=>] 55.4	\[ \log \left(e^{1 - \color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}}}\right) \]

   4. Taylor expanded in x around 0 98.3%

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

   5. Applied egg-rr 99.7%

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

      [Start] 98.3	\[ 10^{-9} + 1.128386358070218 \cdot x \]
      add-sqr-sqrt [=>] 50.3	\[ 10^{-9} + \color{blue}{\sqrt{1.128386358070218 \cdot x} \cdot \sqrt{1.128386358070218 \cdot x}} \]
      sqrt-unprod [=>] 99.7	\[ 10^{-9} + \color{blue}{\sqrt{\left(1.128386358070218 \cdot x\right) \cdot \left(1.128386358070218 \cdot x\right)}} \]
      swap-sqr [=>] 99.7	\[ 10^{-9} + \sqrt{\color{blue}{\left(1.128386358070218 \cdot 1.128386358070218\right) \cdot \left(x \cdot x\right)}} \]
      metadata-eval [=>] 99.7	\[ 10^{-9} + \sqrt{\color{blue}{1.2732557730789702} \cdot \left(x \cdot x\right)} \]

   if 9.59999999999999914e-7 < x

   1. Initial program 99.7%

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

   2. Simplified 99.7%

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

      [Start] 99.7

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

      associate-*l* [=>] 99.7

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

   3. Taylor expanded in x around 0 99.7%

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

   4. Applied egg-rr 99.7%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot x\right)\right)}} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right) - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      Proof

      [Start] 99.7	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right) - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      log1p-expm1-u [=>] 99.7	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right) - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      log1p-udef [=>] 99.7	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right) - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      add-sqr-sqrt [=>] 99.7	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right)} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right) - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      fabs-sqr [=>] 99.7	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)\right)} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right) - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      add-sqr-sqrt [<=] 99.7	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(0.3275911 \cdot \color{blue}{x}\right)\right)} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right) - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;1 + \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\right) \cdot \frac{-1}{1 + 0.3275911 \cdot \left|x\right|}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-7}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(x \cdot 0.3275911\right)\right)} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right) + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot -1.453152027\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	55240

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ t_2 := 1.061405429 \cdot \frac{1}{{t_0}^{2}}\\ t_3 := e^{x \cdot \left(-x\right)}\\ t_4 := t_1 \cdot -1.453152027\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;1 - t_1 \cdot \left(t_3 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + \left(t_2 + t_4\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-7}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 - t_1 \cdot \left(t_3 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + \frac{1}{1 + \log \left(1 + \mathsf{expm1}\left(x \cdot 0.3275911\right)\right)} \cdot \left(\left(1.421413741 + t_2\right) + t_4\right)\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	48580

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ t_2 := e^{x \cdot \left(-x\right)}\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;1 - t_1 \cdot \left(t_2 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + \left(1.061405429 \cdot \frac{1}{{t_0}^{2}} + t_1 \cdot -1.453152027\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 + t_1 \cdot \left(t_2 \cdot \left(-0.254829592 - t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \log \left(1 + \mathsf{expm1}\left(x \cdot 0.3275911\right)\right)}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	99.7%
Cost	42248

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ t_2 := e^{x \cdot \left(-x\right)}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;1 - t_1 \cdot \left(t_2 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 - \frac{-1.061405429}{t_0}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 + t_1 \cdot \left(t_2 \cdot \left(-0.254829592 - t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \log \left(1 + \mathsf{expm1}\left(x \cdot 0.3275911\right)\right)}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	99.7%
Cost	41988

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;1 - t_1 \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 - \frac{-1.061405429}{t_0}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 5
Accuracy	99.6%
Cost	40904

\[\begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 6
Accuracy	99.6%
Cost	29320

\[\begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ t_1 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 + t_0 \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(-0.254829592 - t_0 \cdot \left(-0.284496736 + t_0 \cdot \left(1.421413741 + \frac{1}{t_1} \cdot \left(-1.453152027 + \frac{1.061405429}{t_1}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	99.4%
Cost	7112

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

Alternative 8
Accuracy	98.8%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -0.37545125292247583 + -0.00011824294398844343\right) + \left(10^{-9} + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Accuracy	98.6%
Cost	1096

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

Alternative 10
Accuracy	98.6%
Cost	584

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

Alternative 11
Accuracy	97.8%
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 12
Accuracy	53.2%
Cost	64

\[10^{-9} \]

Reproduce

herbie shell --seed 2023136 
(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)))))))