Jmat.Real.erf

Average Error: 13.3 → 13.3

Time: 7.6s

Precision: binary64

Cost: 61248

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ t_1 := 1 + \left|x\right| \cdot 0.3275911\\ 1 - \left(0.254829592 + t_0 \cdot \left(-0.284496736 + t_0 \cdot \left(1.421413741 + \left(1.061405429 \cdot \frac{1}{{t_1}^{2}} + \frac{1}{t_1} \cdot -1.453152027\right)\right)\right)\right) \cdot \left(t_0 \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \end{array} \]

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 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))))
        (t_1 (+ 1.0 (* (fabs x) 0.3275911))))
   (-
    1.0
    (*
     (+
      0.254829592
      (*
       t_0
       (+
        -0.284496736
        (*
         t_0
         (+
          1.421413741
          (+
           (* 1.061405429 (/ 1.0 (pow t_1 2.0)))
           (* (/ 1.0 t_1) -1.453152027)))))))
     (* t_0 (exp (- (* (fabs x) (fabs x)))))))))

C

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

↓

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	double t_1 = 1.0 + (fabs(x) * 0.3275911);
	return 1.0 - ((0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + ((1.061405429 * (1.0 / pow(t_1, 2.0))) + ((1.0 / t_1) * -1.453152027))))))) * (t_0 * exp(-(fabs(x) * fabs(x)))));
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = 1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))
    t_1 = 1.0d0 + (abs(x) * 0.3275911d0)
    code = 1.0d0 - ((0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + ((1.061405429d0 * (1.0d0 / (t_1 ** 2.0d0))) + ((1.0d0 / t_1) * (-1.453152027d0)))))))) * (t_0 * exp(-(abs(x) * abs(x)))))
end function

Java

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

↓

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * Math.abs(x)));
	double t_1 = 1.0 + (Math.abs(x) * 0.3275911);
	return 1.0 - ((0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + ((1.061405429 * (1.0 / Math.pow(t_1, 2.0))) + ((1.0 / t_1) * -1.453152027))))))) * (t_0 * Math.exp(-(Math.abs(x) * Math.abs(x)))));
}

Python

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

↓

def code(x):
	t_0 = 1.0 / (1.0 + (0.3275911 * math.fabs(x)))
	t_1 = 1.0 + (math.fabs(x) * 0.3275911)
	return 1.0 - ((0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + ((1.061405429 * (1.0 / math.pow(t_1, 2.0))) + ((1.0 / t_1) * -1.453152027))))))) * (t_0 * math.exp(-(math.fabs(x) * math.fabs(x)))))

Julia

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

↓

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	t_1 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	return Float64(1.0 - Float64(Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(Float64(1.061405429 * Float64(1.0 / (t_1 ^ 2.0))) + Float64(Float64(1.0 / t_1) * -1.453152027))))))) * Float64(t_0 * exp(Float64(-Float64(abs(x) * abs(x)))))))
end

MATLAB

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

↓

function tmp = code(x)
	t_0 = 1.0 / (1.0 + (0.3275911 * abs(x)));
	t_1 = 1.0 + (abs(x) * 0.3275911);
	tmp = 1.0 - ((0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + ((1.061405429 * (1.0 / (t_1 ^ 2.0))) + ((1.0 / t_1) * -1.453152027))))))) * (t_0 * exp(-(abs(x) * abs(x)))));
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(N[(1.061405429 * N[(1.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / t$95$1), $MachinePrecision] * -1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 13.3

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

2. Simplified 13.3

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

   [Start] 13.3	\[ 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   rational.json-simplify-2 [=>] 13.3	\[ 1 - \color{blue}{e^{-\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right)} \]
   rational.json-simplify-2 [=>] 13.3	\[ 1 - e^{-\left|x\right| \cdot \left|x\right|} \cdot \color{blue}{\left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right) \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]

3. Taylor expanded in x around 0 13.3

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

4. Simplified 13.3

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

   [Start] 13.3	\[ 1 - \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right) - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right) \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \]
   rational.json-simplify-1 [=>] 13.3	\[ 1 - \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\color{blue}{\left(1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}} + 1.421413741\right)} - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right) \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \]
   rational.json-simplify-1 [<=] 13.3	\[ 1 - \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(1.061405429 \cdot \frac{1}{{\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)}}^{2}} + 1.421413741\right) - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right) \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \]
   rational.json-simplify-1 [<=] 13.3	\[ 1 - \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + 1.421413741\right) - 1.453152027 \cdot \frac{1}{\color{blue}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right) \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \]
   rational.json-simplify-34 [<=] 13.3	\[ 1 - \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(1.421413741 - \left(1.453152027 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right)}\right)\right) \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \]
   rational.json-simplify-35 [=>] 13.3	\[ 1 - \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(1.421413741 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} - 1.453152027 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)}\right)\right) \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \]
   rational.json-simplify-41 [=>] 13.3	\[ 1 - \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \color{blue}{\left(\left(-1.453152027 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + 1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}\right)\right)\right) \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \]
   rational.json-simplify-1 [=>] 13.3	\[ 1 - \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \color{blue}{\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + \left(-1.453152027 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)}\right)\right)\right) \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \]

5. Final simplification 13.3

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

Alternatives

Alternative 1
Error	13.3
Cost	54784

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

Alternative 2
Error	13.3
Cost	54784

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

Reproduce

herbie shell --seed 2023053 
(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)))))))