Jmat.Real.erf

Average Accuracy: 79.1% → 99.7%

Time: 30.6s

Precision: binary64

Cost: 200712

Math

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

↓

\[\begin{array}{l} t_0 := e^{x \cdot x}\\ t_1 := 1 + 0.3275911 \cdot \left|x\right|\\ t_2 := \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_1}}{t_1}}{t_1}}{t_1}}{t_0}}{t_1}\\ t_3 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_4 := {t_3}^{3}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{\left(0.284496736 + 1.453152027 \cdot \frac{1}{{t_1}^{2}}\right) + \left(1.061405429 \cdot \frac{-1}{{t_1}^{3}} + 1.421413741 \cdot \frac{-1}{t_1}\right)}{t_1}}{t_0}}{t_1}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{{\left(\left(\left(0.254829592 + \left(\frac{1.061405429}{{t_3}^{4}} + \frac{1.421413741}{{t_3}^{2}}\right)\right) + \frac{-0.284496736}{t_3}\right) + \frac{-1.453152027}{t_4}\right)}^{3}}{t_4 \cdot {\left({\left(e^{x}\right)}^{x}\right)}^{3}}}{1 + t_2 \cdot \left(1 + t_2\right)}\\ \end{array} \]

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 (+ 1.0 (* 0.3275911 (fabs x))))
        (t_2
         (/
          (/
           (+
            0.254829592
            (/
             (+
              -0.284496736
              (/
               (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_1)) t_1))
               t_1))
             t_1))
           t_0)
          t_1))
        (t_3 (fma 0.3275911 (fabs x) 1.0))
        (t_4 (pow t_3 3.0)))
   (if (<= x -1.7e-6)
     (+
      1.0
      (/
       (/
        (+
         -0.254829592
         (/
          (+
           (+ 0.284496736 (* 1.453152027 (/ 1.0 (pow t_1 2.0))))
           (+
            (* 1.061405429 (/ -1.0 (pow t_1 3.0)))
            (* 1.421413741 (/ -1.0 t_1))))
          t_1))
        t_0)
       t_1))
     (if (<= x 1.3e-6)
       (+ 1e-9 (sqrt (* (* x x) 1.2732557730789702)))
       (/
        (-
         1.0
         (/
          (pow
           (+
            (+
             (+
              0.254829592
              (+ (/ 1.061405429 (pow t_3 4.0)) (/ 1.421413741 (pow t_3 2.0))))
             (/ -0.284496736 t_3))
            (/ -1.453152027 t_4))
           3.0)
          (* t_4 (pow (pow (exp x) x) 3.0))))
        (+ 1.0 (* t_2 (+ 1.0 t_2))))))))

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 = 1.0 + (0.3275911 * fabs(x));
	double t_2 = ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_1)) / t_1)) / t_1)) / t_1)) / t_0) / t_1;
	double t_3 = fma(0.3275911, fabs(x), 1.0);
	double t_4 = pow(t_3, 3.0);
	double tmp;
	if (x <= -1.7e-6) {
		tmp = 1.0 + (((-0.254829592 + (((0.284496736 + (1.453152027 * (1.0 / pow(t_1, 2.0)))) + ((1.061405429 * (-1.0 / pow(t_1, 3.0))) + (1.421413741 * (-1.0 / t_1)))) / t_1)) / t_0) / t_1);
	} else if (x <= 1.3e-6) {
		tmp = 1e-9 + sqrt(((x * x) * 1.2732557730789702));
	} else {
		tmp = (1.0 - (pow((((0.254829592 + ((1.061405429 / pow(t_3, 4.0)) + (1.421413741 / pow(t_3, 2.0)))) + (-0.284496736 / t_3)) + (-1.453152027 / t_4)), 3.0) / (t_4 * pow(pow(exp(x), x), 3.0)))) / (1.0 + (t_2 * (1.0 + t_2)));
	}
	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(x * x))
	t_1 = Float64(1.0 + Float64(0.3275911 * abs(x)))
	t_2 = 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_0) / t_1)
	t_3 = fma(0.3275911, abs(x), 1.0)
	t_4 = t_3 ^ 3.0
	tmp = 0.0
	if (x <= -1.7e-6)
		tmp = Float64(1.0 + Float64(Float64(Float64(-0.254829592 + Float64(Float64(Float64(0.284496736 + Float64(1.453152027 * Float64(1.0 / (t_1 ^ 2.0)))) + Float64(Float64(1.061405429 * Float64(-1.0 / (t_1 ^ 3.0))) + Float64(1.421413741 * Float64(-1.0 / t_1)))) / t_1)) / t_0) / t_1));
	elseif (x <= 1.3e-6)
		tmp = Float64(1e-9 + sqrt(Float64(Float64(x * x) * 1.2732557730789702)));
	else
		tmp = Float64(Float64(1.0 - Float64((Float64(Float64(Float64(0.254829592 + Float64(Float64(1.061405429 / (t_3 ^ 4.0)) + Float64(1.421413741 / (t_3 ^ 2.0)))) + Float64(-0.284496736 / t_3)) + Float64(-1.453152027 / t_4)) ^ 3.0) / Float64(t_4 * ((exp(x) ^ x) ^ 3.0)))) / Float64(1.0 + Float64(t_2 * Float64(1.0 + t_2))));
	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[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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$0), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, 3.0], $MachinePrecision]}, If[LessEqual[x, -1.7e-6], N[(1.0 + N[(N[(N[(-0.254829592 + N[(N[(N[(0.284496736 + N[(1.453152027 * N[(1.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.061405429 * N[(-1.0 / N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.421413741 * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e-6], N[(1e-9 + N[Sqrt[N[(N[(x * x), $MachinePrecision] * 1.2732557730789702), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Power[N[(N[(N[(0.254829592 + N[(N[(1.061405429 / N[Power[t$95$3, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.421413741 / N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.284496736 / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(-1.453152027 / t$95$4), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(t$95$4 * N[Power[N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$2 * N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.70000000000000003e-6

   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. Applied egg-rr 99.7%

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

      [Start] 99.7	\[ 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      sub-neg [=>] 99.7	\[ \color{blue}{1 + \left(-\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)\right)} \]
      +-commutative [=>] 99.7	\[ \color{blue}{\left(-\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)\right) + 1} \]

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in x around inf 99.7%

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

   if -1.70000000000000003e-6 < x < 1.30000000000000005e-6

   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%

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

      [Start] 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

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

   3. Applied egg-rr 57.7%

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

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

   4. Applied egg-rr 57.0%

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

      [Start] 57.7

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

      flip3-- [=>] 57.8

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

   5. Taylor expanded in x around 0 97.8%

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

   6. Simplified 97.8%

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

      [Start] 97.8	\[ 10^{-9} + 1.128386358070218 \cdot x \]
      *-commutative [=>] 97.8	\[ 10^{-9} + \color{blue}{x \cdot 1.128386358070218} \]

   7. Applied egg-rr 99.7%

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

      [Start] 97.8	\[ 10^{-9} + x \cdot 1.128386358070218 \]
      add-sqr-sqrt [=>] 50.6	\[ 10^{-9} + \color{blue}{\sqrt{x \cdot 1.128386358070218} \cdot \sqrt{x \cdot 1.128386358070218}} \]
      sqrt-unprod [=>] 99.7	\[ 10^{-9} + \color{blue}{\sqrt{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)}} \]
      swap-sqr [=>] 99.7	\[ 10^{-9} + \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(1.128386358070218 \cdot 1.128386358070218\right)}} \]
      metadata-eval [=>] 99.7	\[ 10^{-9} + \sqrt{\left(x \cdot x\right) \cdot \color{blue}{1.2732557730789702}} \]

   if 1.30000000000000005e-6 < x

   1. Initial program 99.6%

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

   2. Simplified 99.6%

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

      [Start] 99.6

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

      associate-*l* [=>] 99.6

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

   3. Applied egg-rr 99.6%

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

      [Start] 99.6

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

      flip3-- [=>] 99.6

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

   4. Taylor expanded in x around inf 99.6%

      \[\leadsto \frac{\color{blue}{1 - \frac{{\left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1} + 1.453152027 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3}}\right)\right)}^{3}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3} \cdot {\left(e^{{x}^{2}}\right)}^{3}}}}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]

   5. Simplified 99.6%

      \[\leadsto \frac{\color{blue}{1 - \frac{{\left(\left(\left(0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}}\right)\right) - \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}\right)}^{3}}{{\left({\left(e^{x}\right)}^{x}\right)}^{3} \cdot {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}}}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]
      Proof

      [Start] 99.6

      \[ \frac{1 - \frac{{\left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1} + 1.453152027 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3}}\right)\right)}^{3}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3} \cdot {\left(e^{{x}^{2}}\right)}^{3}}}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{\left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right) + \left(1.061405429 \cdot \frac{-1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{-1}{1 + 0.3275911 \cdot \left|x\right|}\right)}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{{\left(\left(\left(0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}}\right)\right) + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) + \frac{-1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}\right)}^{3}}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3} \cdot {\left({\left(e^{x}\right)}^{x}\right)}^{3}}}{1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1 + \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{e^{x \cdot x}}}{1 + 0.3275911 \cdot \left|x\right|}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	169864

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := 1 + 0.3275911 \cdot \left|x\right|\\ t_2 := {\left(e^{x}\right)}^{x}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{\left(0.284496736 + 1.453152027 \cdot \frac{1}{{t_1}^{2}}\right) + \left(1.061405429 \cdot \frac{-1}{{t_1}^{3}} + 1.421413741 \cdot \frac{-1}{t_1}\right)}{t_1}}{e^{x \cdot x}}}{t_1}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{0.284496736}{{t_0}^{2} \cdot t_2} + \left(\left(1 + \frac{1.453152027}{{t_0}^{4} \cdot t_2}\right) - \left(\frac{1.421413741}{{t_0}^{3} \cdot t_2} + \left(\frac{\frac{0.254829592}{t_2}}{t_0} + \frac{1.061405429}{t_2 \cdot {t_0}^{5}}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	102152

\[\begin{array}{l} t_0 := e^{x \cdot x}\\ t_1 := 0.3275911 \cdot \left|x\right|\\ t_2 := 1 + t_1\\ t_3 := \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_2}}{t_2}}{t_2}}{t_2}}{t_0}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{\left(0.284496736 + 1.453152027 \cdot \frac{1}{{t_2}^{2}}\right) + \left(1.061405429 \cdot \frac{-1}{{t_2}^{3}} + 1.421413741 \cdot \frac{-1}{t_2}\right)}{t_2}}{t_0}}{t_2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\frac{t_3}{1 + \log \left(e^{t_1}\right)}\right)}^{2}}{1 + \frac{t_3}{t_2}}\\ \end{array} \]

Alternative 3
Accuracy	99.7%
Cost	67144

\[\begin{array}{l} t_0 := e^{x \cdot x}\\ t_1 := 0.3275911 \cdot \left|x\right|\\ t_2 := 1 + t_1\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{\left(0.284496736 + 1.453152027 \cdot \frac{1}{{t_2}^{2}}\right) + \left(1.061405429 \cdot \frac{-1}{{t_2}^{3}} + 1.421413741 \cdot \frac{-1}{t_2}\right)}{t_2}}{t_0}}{t_2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(1 + \frac{\frac{-0.254829592 + \frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_2}}{t_2}}{t_2}}{t_2}}{t_0}}{1 + \log \left(e^{t_1}\right)}\right)}\\ \end{array} \]

Alternative 4
Accuracy	99.7%
Cost	54724

\[\begin{array}{l} t_0 := e^{x \cdot x}\\ t_1 := 1 + 0.3275911 \cdot \left|x\right|\\ t_2 := \frac{1}{{t_1}^{2}}\\ t_3 := \frac{-1}{t_1}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{\left(0.284496736 + 1.453152027 \cdot t_2\right) + \left(1.061405429 \cdot \frac{-1}{{t_1}^{3}} + 1.421413741 \cdot t_3\right)}{t_1}}{t_0}}{t_1}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{0.284496736 + \frac{-1.453152027 \cdot t_3 - \left(1.421413741 + 1.061405429 \cdot t_2\right)}{t_1}}{t_1}}{t_0}}{t_1}\\ \end{array} \]

Alternative 5
Accuracy	99.7%
Cost	48264

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -7 \cdot 10^{-7}:\\ \;\;\;\;1 - t_1 \cdot \left(\left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + \left(-1.453152027 + \frac{1.061405429}{t_0}\right) \cdot t_1\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{0.284496736 + \frac{-1.453152027 \cdot \frac{-1}{t_0} - \left(1.421413741 + 1.061405429 \cdot \frac{1}{{t_0}^{2}}\right)}{t_0}}{t_0}}{e^{x \cdot x}}}{t_0}\\ \end{array} \]

Alternative 6
Accuracy	99.7%
Cost	42121

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -7 \cdot 10^{-7} \lor \neg \left(x \leq 8.2 \cdot 10^{-7}\right):\\ \;\;\;\;1 - t_1 \cdot \left(\left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + \left(-1.453152027 + \frac{1.061405429}{t_0}\right) \cdot t_1\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \end{array} \]

Alternative 7
Accuracy	99.7%
Cost	41412

\[\begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := 1 + 0.3275911 \cdot \left|x\right|\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{-6}:\\ \;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_1}}{t_1}}{t_1}}{t_1}}{e^{x \cdot x}}}{t_1}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_0}}{t_0}}{t_0}}{t_0}}{{\left(e^{x}\right)}^{x} \cdot t_0}\right)}\\ \end{array} \]

Alternative 8
Accuracy	99.4%
Cost	28680

\[\begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_0}}{t_0}}{t_0}}{t_0}}{{\left(e^{x}\right)}^{x} \cdot t_0}\right)}\\ \end{array} \]

Alternative 9
Accuracy	99.2%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 0.88\right):\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \end{array} \]

Alternative 10
Accuracy	53.0%
Cost	64

\[10^{-9} \]

Reproduce

herbie shell --seed 2023130 
(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)))))))