Jmat.Real.erf

Average Error: 13.5 → 0.1

Time: 15.7s

Precision: binary64

Cost: 54532

Math

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

↓

\[\begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ \mathbf{if}\;\left|x\right| \leq 4 \cdot 10^{-5}:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{0.284496736 + \frac{1.453152027 \cdot \frac{1}{t_0} + \left(-1.421413741 + 1.061405429 \cdot \frac{-1}{{t_0}^{2}}\right)}{t_0}}{t_0}}{e^{x \cdot x}}}{t_0}\\ \end{array} \]

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 (* (fabs x) 0.3275911))))
   (if (<= (fabs x) 4e-5)
     (+
      1e-9
      (+
       (* -0.00011824294398844343 (pow x 2.0))
       (+
        (* -0.37545125292247583 (pow x 3.0))
        (sqrt (* x (* x 1.2732557730789702))))))
     (+
      1.0
      (/
       (/
        (+
         -0.254829592
         (/
          (+
           0.284496736
           (/
            (+
             (* 1.453152027 (/ 1.0 t_0))
             (+ -1.421413741 (* 1.061405429 (/ -1.0 (pow t_0 2.0)))))
            t_0))
          t_0))
        (exp (* x x)))
       t_0)))))

C

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

↓

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double tmp;
	if (fabs(x) <= 4e-5) {
		tmp = 1e-9 + ((-0.00011824294398844343 * pow(x, 2.0)) + ((-0.37545125292247583 * pow(x, 3.0)) + sqrt((x * (x * 1.2732557730789702)))));
	} else {
		tmp = 1.0 + (((-0.254829592 + ((0.284496736 + (((1.453152027 * (1.0 / t_0)) + (-1.421413741 + (1.061405429 * (-1.0 / pow(t_0, 2.0))))) / t_0)) / t_0)) / exp((x * x))) / t_0);
	}
	return tmp;
}

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) :: tmp
    t_0 = 1.0d0 + (abs(x) * 0.3275911d0)
    if (abs(x) <= 4d-5) then
        tmp = 1d-9 + (((-0.00011824294398844343d0) * (x ** 2.0d0)) + (((-0.37545125292247583d0) * (x ** 3.0d0)) + sqrt((x * (x * 1.2732557730789702d0)))))
    else
        tmp = 1.0d0 + ((((-0.254829592d0) + ((0.284496736d0 + (((1.453152027d0 * (1.0d0 / t_0)) + ((-1.421413741d0) + (1.061405429d0 * ((-1.0d0) / (t_0 ** 2.0d0))))) / t_0)) / t_0)) / exp((x * x))) / t_0)
    end if
    code = tmp
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 + (Math.abs(x) * 0.3275911);
	double tmp;
	if (Math.abs(x) <= 4e-5) {
		tmp = 1e-9 + ((-0.00011824294398844343 * Math.pow(x, 2.0)) + ((-0.37545125292247583 * Math.pow(x, 3.0)) + Math.sqrt((x * (x * 1.2732557730789702)))));
	} else {
		tmp = 1.0 + (((-0.254829592 + ((0.284496736 + (((1.453152027 * (1.0 / t_0)) + (-1.421413741 + (1.061405429 * (-1.0 / Math.pow(t_0, 2.0))))) / t_0)) / t_0)) / Math.exp((x * x))) / t_0);
	}
	return tmp;
}

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 + (math.fabs(x) * 0.3275911)
	tmp = 0
	if math.fabs(x) <= 4e-5:
		tmp = 1e-9 + ((-0.00011824294398844343 * math.pow(x, 2.0)) + ((-0.37545125292247583 * math.pow(x, 3.0)) + math.sqrt((x * (x * 1.2732557730789702)))))
	else:
		tmp = 1.0 + (((-0.254829592 + ((0.284496736 + (((1.453152027 * (1.0 / t_0)) + (-1.421413741 + (1.061405429 * (-1.0 / math.pow(t_0, 2.0))))) / t_0)) / t_0)) / math.exp((x * x))) / t_0)
	return tmp

Julia

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

↓

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	tmp = 0.0
	if (abs(x) <= 4e-5)
		tmp = Float64(1e-9 + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(Float64(-0.37545125292247583 * (x ^ 3.0)) + sqrt(Float64(x * Float64(x * 1.2732557730789702))))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(-0.254829592 + Float64(Float64(0.284496736 + Float64(Float64(Float64(1.453152027 * Float64(1.0 / t_0)) + Float64(-1.421413741 + Float64(1.061405429 * Float64(-1.0 / (t_0 ^ 2.0))))) / t_0)) / t_0)) / exp(Float64(x * x))) / t_0));
	end
	return tmp
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_2 = code(x)
	t_0 = 1.0 + (abs(x) * 0.3275911);
	tmp = 0.0;
	if (abs(x) <= 4e-5)
		tmp = 1e-9 + ((-0.00011824294398844343 * (x ^ 2.0)) + ((-0.37545125292247583 * (x ^ 3.0)) + sqrt((x * (x * 1.2732557730789702)))));
	else
		tmp = 1.0 + (((-0.254829592 + ((0.284496736 + (((1.453152027 * (1.0 / t_0)) + (-1.421413741 + (1.061405429 * (-1.0 / (t_0 ^ 2.0))))) / t_0)) / t_0)) / exp((x * x))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 4e-5], N[(1e-9 + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.37545125292247583 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(x * N[(x * 1.2732557730789702), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(-0.254829592 + N[(N[(0.284496736 + N[(N[(N[(1.453152027 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-1.421413741 + N[(1.061405429 * N[(-1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.00000000000000033e-5

   1. Initial program 27.0

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

   2. Simplified 27.0

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

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

   3. Applied egg-rr 27.6

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

   4. Simplified 27.6

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

      [Start] 27.6	\[ 1 + \left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right) \]
      distribute-neg-frac [=>] 27.6	\[ 1 + \color{blue}{\frac{-\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]
      neg-sub0 [=>] 27.6	\[ 1 + \frac{\color{blue}{0 - \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]
      associate--r+ [=>] 27.6	\[ 1 + \frac{\color{blue}{\left(0 - 0.254829592\right) - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]
      metadata-eval [=>] 27.6	\[ 1 + \frac{\color{blue}{-0.254829592} - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]
      *-commutative [=>] 27.6	\[ 1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\color{blue}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}} \]
      exp-prod [<=] 27.6	\[ 1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\color{blue}{e^{x \cdot x}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)} \]

   5. Taylor expanded in x around 0 1.4

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)} \]

   6. Applied egg-rr 0.1

      \[\leadsto 10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + \color{blue}{\sqrt{1.2732557730789702 \cdot \left(x \cdot x\right)}}\right)\right) \]

   7. Simplified 0.1

      \[\leadsto 10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + \color{blue}{\sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}}\right)\right) \]
      Proof

      [Start] 0.1	\[ 10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + \sqrt{1.2732557730789702 \cdot \left(x \cdot x\right)}\right)\right) \]
      *-commutative [=>] 0.1	\[ 10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + \sqrt{\color{blue}{\left(x \cdot x\right) \cdot 1.2732557730789702}}\right)\right) \]
      associate-*l* [=>] 0.1	\[ 10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + \sqrt{\color{blue}{x \cdot \left(x \cdot 1.2732557730789702\right)}}\right)\right) \]

   if 4.00000000000000033e-5 < (fabs.f64 x)

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1

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

      associate-*l* [=>] 0.1

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

   3. Applied egg-rr 0.1

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

   4. Taylor expanded in x around 0 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4 \cdot 10^{-5}:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{0.284496736 + \frac{1.453152027 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911} + \left(-1.421413741 + 1.061405429 \cdot \frac{-1}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{2}}\right)}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{x \cdot x}}}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	47748

\[\begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ \mathbf{if}\;x \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;1 + \frac{-0.254829592 + \frac{0.284496736 + \frac{-1.421413741 - \frac{-1.453152027 + \frac{1.061405429}{t_0}}{t_0}}{t_0}}{t_0}}{t_0 \cdot {\left(e^{x}\right)}^{x}}\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	0.2
Cost	41412

\[\begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{-6}:\\ \;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{0.284496736 + \frac{-1.421413741 - \frac{-1.453152027 + \frac{1.061405429}{t_0}}{t_0}}{t_0}}{t_0}}{e^{x \cdot x}}}{t_0}\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	0.3
Cost	35076

\[\begin{array}{l} t_0 := \frac{1}{1 + \left|x\right| \cdot 0.3275911}\\ \mathbf{if}\;x \leq -0.52:\\ \;\;\;\;1 - t_0 \cdot \left(\left(0.254829592 + t_0 \cdot \left(-0.284496736 + t_0 \cdot \left(1.421413741 + t_0 \cdot -0.391746598\right)\right)\right) \cdot e^{-x \cdot x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	0.3
Cost	20552

\[\begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	0.3
Cost	20232

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;10^{-9} + \left|\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	0.4
Cost	19848

\[\begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + \sqrt{{\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{6}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	0.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	0.9
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -8.9 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;10^{-9} + \left(x \cdot 1.128386358070218 + \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	1.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -8.9 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	1.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8.9 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	1.0
Cost	584

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

Alternative 12
Error	1.5
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	29.7
Cost	64

\[10^{-9} \]

Reproduce

herbie shell --seed 2023237 
(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)))))))