Jmat.Real.erf

Average Error: 20.61% → 0.44%

Time: 15.7s

Precision: binary64

Cost: 35460

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{1 + \left|x\right| \cdot 0.3275911}\\ t_1 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;\left|x\right| \leq 10^{-5}:\\ \;\;\;\;10^{-9} + \left(x \cdot \left(x \cdot -0.00011824294398844343\right) + \left(-0.37545125292247583 \cdot {x}^{3} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + t_0 \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(-0.254829592 + t_0 \cdot \left(0.284496736 + t_0 \cdot \left(-1.421413741 + \frac{1.453152027 + \frac{-1.061405429}{t_1}}{t_1}\right)\right)\right)\right)\\ \end{array} \]

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 (/ 1.0 (+ 1.0 (* (fabs x) 0.3275911))))
        (t_1 (+ 1.0 (* x 0.3275911))))
   (if (<= (fabs x) 1e-5)
     (+
      1e-9
      (+
       (* x (* x -0.00011824294398844343))
       (+
        (* -0.37545125292247583 (pow x 3.0))
        (sqrt (* x (* x 1.2732557730789702))))))
     (+
      1.0
      (*
       t_0
       (*
        (exp (* x (- x)))
        (+
         -0.254829592
         (*
          t_0
          (+
           0.284496736
           (*
            t_0
            (+
             -1.421413741
             (/ (+ 1.453152027 (/ -1.061405429 t_1)) t_1))))))))))))

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 + (fabs(x) * 0.3275911));
	double t_1 = 1.0 + (x * 0.3275911);
	double tmp;
	if (fabs(x) <= 1e-5) {
		tmp = 1e-9 + ((x * (x * -0.00011824294398844343)) + ((-0.37545125292247583 * pow(x, 3.0)) + sqrt((x * (x * 1.2732557730789702)))));
	} else {
		tmp = 1.0 + (t_0 * (exp((x * -x)) * (-0.254829592 + (t_0 * (0.284496736 + (t_0 * (-1.421413741 + ((1.453152027 + (-1.061405429 / t_1)) / t_1))))))));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (1.0d0 + (abs(x) * 0.3275911d0))
    t_1 = 1.0d0 + (x * 0.3275911d0)
    if (abs(x) <= 1d-5) then
        tmp = 1d-9 + ((x * (x * (-0.00011824294398844343d0))) + (((-0.37545125292247583d0) * (x ** 3.0d0)) + sqrt((x * (x * 1.2732557730789702d0)))))
    else
        tmp = 1.0d0 + (t_0 * (exp((x * -x)) * ((-0.254829592d0) + (t_0 * (0.284496736d0 + (t_0 * ((-1.421413741d0) + ((1.453152027d0 + ((-1.061405429d0) / t_1)) / t_1))))))))
    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 / (1.0 + (Math.abs(x) * 0.3275911));
	double t_1 = 1.0 + (x * 0.3275911);
	double tmp;
	if (Math.abs(x) <= 1e-5) {
		tmp = 1e-9 + ((x * (x * -0.00011824294398844343)) + ((-0.37545125292247583 * Math.pow(x, 3.0)) + Math.sqrt((x * (x * 1.2732557730789702)))));
	} else {
		tmp = 1.0 + (t_0 * (Math.exp((x * -x)) * (-0.254829592 + (t_0 * (0.284496736 + (t_0 * (-1.421413741 + ((1.453152027 + (-1.061405429 / t_1)) / t_1))))))));
	}
	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 / (1.0 + (math.fabs(x) * 0.3275911))
	t_1 = 1.0 + (x * 0.3275911)
	tmp = 0
	if math.fabs(x) <= 1e-5:
		tmp = 1e-9 + ((x * (x * -0.00011824294398844343)) + ((-0.37545125292247583 * math.pow(x, 3.0)) + math.sqrt((x * (x * 1.2732557730789702)))))
	else:
		tmp = 1.0 + (t_0 * (math.exp((x * -x)) * (-0.254829592 + (t_0 * (0.284496736 + (t_0 * (-1.421413741 + ((1.453152027 + (-1.061405429 / t_1)) / t_1))))))))
	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(1.0 + Float64(abs(x) * 0.3275911)))
	t_1 = Float64(1.0 + Float64(x * 0.3275911))
	tmp = 0.0
	if (abs(x) <= 1e-5)
		tmp = Float64(1e-9 + Float64(Float64(x * Float64(x * -0.00011824294398844343)) + Float64(Float64(-0.37545125292247583 * (x ^ 3.0)) + sqrt(Float64(x * Float64(x * 1.2732557730789702))))));
	else
		tmp = Float64(1.0 + Float64(t_0 * Float64(exp(Float64(x * Float64(-x))) * Float64(-0.254829592 + Float64(t_0 * Float64(0.284496736 + Float64(t_0 * Float64(-1.421413741 + Float64(Float64(1.453152027 + Float64(-1.061405429 / t_1)) / t_1)))))))));
	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 / (1.0 + (abs(x) * 0.3275911));
	t_1 = 1.0 + (x * 0.3275911);
	tmp = 0.0;
	if (abs(x) <= 1e-5)
		tmp = 1e-9 + ((x * (x * -0.00011824294398844343)) + ((-0.37545125292247583 * (x ^ 3.0)) + sqrt((x * (x * 1.2732557730789702)))));
	else
		tmp = 1.0 + (t_0 * (exp((x * -x)) * (-0.254829592 + (t_0 * (0.284496736 + (t_0 * (-1.421413741 + ((1.453152027 + (-1.061405429 / t_1)) / t_1))))))));
	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[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1e-5], N[(1e-9 + N[(N[(x * N[(x * -0.00011824294398844343), $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[(t$95$0 * N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(-0.254829592 + N[(t$95$0 * N[(0.284496736 + N[(t$95$0 * N[(-1.421413741 + N[(N[(1.453152027 + N[(-1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000008e-5

   1. Initial program 42.23

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

   2. Simplified 42.22

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

      [Start] 42.23	\[ 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
      associate-*l/ [=>] 42.23	\[ 1 - \color{blue}{\frac{1 \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
      associate-*l/ [=>] 42.23	\[ 1 - \color{blue}{\frac{\left(1 \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}} \]

   3. Applied egg-rr 42.91

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

   4. Simplified 42.91

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

      [Start] 42.91	\[ 1 \cdot \left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right) \]
      *-lft-identity [=>] 42.91	\[ \color{blue}{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]
      *-commutative [=>] 42.91	\[ 1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\color{blue}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}} \]
      exp-prod [<=] 42.91	\[ 1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\color{blue}{e^{x \cdot x}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)} \]

   5. Taylor expanded in x around 0 1.89

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

   6. Applied egg-rr 2.09

      \[\leadsto 10^{-9} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(-0.00011824294398844343 \cdot \left(x \cdot x\right)\right)} - 1\right)} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]

   7. Simplified 1.89

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

      [Start] 2.09	\[ 10^{-9} + \left(\left(e^{\mathsf{log1p}\left(-0.00011824294398844343 \cdot \left(x \cdot x\right)\right)} - 1\right) + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]
      expm1-def [=>] 1.89	\[ 10^{-9} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.00011824294398844343 \cdot \left(x \cdot x\right)\right)\right)} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]
      expm1-log1p [=>] 1.89	\[ 10^{-9} + \left(\color{blue}{-0.00011824294398844343 \cdot \left(x \cdot x\right)} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]
      *-commutative [=>] 1.89	\[ 10^{-9} + \left(\color{blue}{\left(x \cdot x\right) \cdot -0.00011824294398844343} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]
      associate-*l* [=>] 1.89	\[ 10^{-9} + \left(\color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]

   8. Applied egg-rr 0.1

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

   9. Simplified 0.1

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

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

   if 1.00000000000000008e-5 < (fabs.f64 x)

   1. Initial program 0.18

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

   2. Simplified 0.18

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

      [Start] 0.18

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

      associate-*l* [=>] 0.18

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

   3. Applied egg-rr 0.18

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

   4. Applied egg-rr 0.77

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

   5. Simplified 0.77

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

      [Start] 0.77

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

      *-lft-identity [=>] 0.77

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

3. Recombined 2 regimes into one program.

4. Final simplification 0.44

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

Alternatives

Alternative 1
Error	0.31%
Cost	28680

\[\begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.00064:\\ \;\;\;\;10^{-9} + \left(x \cdot \left(x \cdot -0.00011824294398844343\right) + \left(-0.37545125292247583 \cdot {x}^{3} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(1 + \frac{-0.254829592 + \frac{0.284496736 + \frac{-1.421413741 + \frac{1.453152027 + \frac{-1.061405429}{t_0}}{t_0}}{t_0}}{t_0}}{t_0 \cdot {\left(e^{x}\right)}^{x}}\right)}\\ \end{array} \]

Alternative 2
Error	0.44%
Cost	27336

\[\begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.78:\\ \;\;\;\;10^{-9} + \left(x \cdot \left(x \cdot -0.00011824294398844343\right) + \left(-0.37545125292247583 \cdot {x}^{3} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592 + \frac{0.284496736 + \frac{-1.029667143}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 3
Error	0.44%
Cost	14216

\[\begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;10^{-9} + \left(x \cdot \left(x \cdot -0.00011824294398844343\right) + \left(-0.37545125292247583 \cdot {x}^{3} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 4
Error	0.68%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 5
Error	0.68%
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 6
Error	1.52%
Cost	840

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

Alternative 7
Error	1.6%
Cost	584

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

Alternative 8
Error	1.56%
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 9
Error	2.41%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.82 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	47.41%
Cost	64

\[10^{-9} \]

Reproduce

herbie shell --seed 2023090 
(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)))))))