Jmat.Real.erf

Percentage Accurate: 78.9% → 99.0%

Time: 20.7s

Precision: binary64

Cost: 41988

Math

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

↓

\[\begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 - t_1 \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \]

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))) (t_1 (/ 1.0 t_0)))
   (if (<= x -2.5e-17)
     (-
      1.0
      (*
       t_1
       (*
        (exp (* x (- x)))
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+ 1.421413741 (* t_1 (+ -1.453152027 (/ 1.061405429 t_0)))))))))))
     (if (<= x 0.9)
       (+ (* x 1.128386358070218) 1e-9)
       (pow 1.0 0.3333333333333333)))))

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 t_1 = 1.0 / t_0;
	double tmp;
	if (x <= -2.5e-17) {
		tmp = 1.0 - (t_1 * (exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))));
	} else if (x <= 0.9) {
		tmp = (x * 1.128386358070218) + 1e-9;
	} else {
		tmp = pow(1.0, 0.3333333333333333);
	}
	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 + (abs(x) * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    if (x <= (-2.5d-17)) then
        tmp = 1.0d0 - (t_1 * (exp((x * -x)) * (0.254829592d0 + (t_1 * ((-0.284496736d0) + (t_1 * (1.421413741d0 + (t_1 * ((-1.453152027d0) + (1.061405429d0 / t_0))))))))))
    else if (x <= 0.9d0) then
        tmp = (x * 1.128386358070218d0) + 1d-9
    else
        tmp = 1.0d0 ** 0.3333333333333333d0
    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 t_1 = 1.0 / t_0;
	double tmp;
	if (x <= -2.5e-17) {
		tmp = 1.0 - (t_1 * (Math.exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))));
	} else if (x <= 0.9) {
		tmp = (x * 1.128386358070218) + 1e-9;
	} else {
		tmp = Math.pow(1.0, 0.3333333333333333);
	}
	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)
	t_1 = 1.0 / t_0
	tmp = 0
	if x <= -2.5e-17:
		tmp = 1.0 - (t_1 * (math.exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))))
	elif x <= 0.9:
		tmp = (x * 1.128386358070218) + 1e-9
	else:
		tmp = math.pow(1.0, 0.3333333333333333)
	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))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x <= -2.5e-17)
		tmp = Float64(1.0 - Float64(t_1 * Float64(exp(Float64(x * Float64(-x))) * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(t_1 * Float64(-1.453152027 + Float64(1.061405429 / t_0)))))))))));
	elseif (x <= 0.9)
		tmp = Float64(Float64(x * 1.128386358070218) + 1e-9);
	else
		tmp = 1.0 ^ 0.3333333333333333;
	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);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (x <= -2.5e-17)
		tmp = 1.0 - (t_1 * (exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))));
	elseif (x <= 0.9)
		tmp = (x * 1.128386358070218) + 1e-9;
	else
		tmp = 1.0 ^ 0.3333333333333333;
	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]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x, -2.5e-17], N[(1.0 - N[(t$95$1 * N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.9], N[(N[(x * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], N[Power[1.0, 0.3333333333333333], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4999999999999999e-17

   1. Initial program 98.9%

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

   2. Simplified 98.9%

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

      [Start] 98.9%

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

      associate-*l* [=>] 98.9%

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

   if -2.4999999999999999e-17 < x < 0.900000000000000022

   1. Initial program 57.8%

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

   2. Simplified 57.8%

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

      [Start] 57.8%

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

      associate-*l* [=>] 57.8%

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

   3. Applied egg-rr 57.8%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]
      Step-by-step derivation

      [Start] 57.8%

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

      add-cbrt-cube [=>] 57.8%

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

   4. Taylor expanded in x around 0 95.5%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right)}}^{3}\right)}^{0.3333333333333333} \]

   5. Simplified 95.5%

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + x \cdot 1.128386358070218\right)}}^{3}\right)}^{0.3333333333333333} \]
      Step-by-step derivation

      [Start] 95.5%	\[ {\left({\left(10^{-9} + 1.128386358070218 \cdot x\right)}^{3}\right)}^{0.3333333333333333} \]
      *-commutative [=>] 95.5%	\[ {\left({\left(10^{-9} + \color{blue}{x \cdot 1.128386358070218}\right)}^{3}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 99.9%

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

      [Start] 95.5%	\[ {\left({\left(10^{-9} + x \cdot 1.128386358070218\right)}^{3}\right)}^{0.3333333333333333} \]
      pow-pow [=>] 99.9%	\[ \color{blue}{{\left(10^{-9} + x \cdot 1.128386358070218\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]
      metadata-eval [=>] 99.9%	\[ {\left(10^{-9} + x \cdot 1.128386358070218\right)}^{\color{blue}{1}} \]
      pow1 [<=] 99.9%	\[ \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]
      +-commutative [=>] 99.9%	\[ \color{blue}{x \cdot 1.128386358070218 + 10^{-9}} \]

   if 0.900000000000000022 < x

   1. Initial program 100.0%

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

   2. Simplified 100.0%

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

      [Start] 100.0%

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

      associate-*l* [=>] 100.0%

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

   3. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]
      Step-by-step derivation

      [Start] 100.0%

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

      add-cbrt-cube [=>] 100.0%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto {\left({\color{blue}{1}}^{3}\right)}^{0.3333333333333333} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	41988

\[\begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 - t_1 \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \]

Alternative 2
Accuracy	98.5%
Cost	250500

\[\begin{array}{l} t_0 := e^{x \cdot \left(-x\right)}\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := \frac{\left(0.254829592 + \frac{-0.284496736 - \frac{\frac{1.453152027}{t_1} - \left(1.421413741 + \frac{1.061405429}{{t_1}^{2}}\right)}{t_1}}{t_1}\right) \cdot t_0}{t_1}\\ t_3 := {t_1}^{3}\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t_2 \cdot t_2}{1 + \frac{\left(0.254829592 + \left(\frac{\frac{1.421413741}{t_1} + \left(\frac{1.061405429}{t_3} - 0.284496736\right)}{t_1} - \frac{1.453152027}{t_3}\right)\right) \cdot t_0}{t_1}}\\ \end{array} \]

Alternative 3
Accuracy	98.5%
Cost	244036

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \frac{\left(0.254829592 + \frac{-0.284496736 - \frac{\frac{1.453152027}{t_0} - \left(1.421413741 + \frac{1.061405429}{{t_0}^{2}}\right)}{t_0}}{t_0}\right) \cdot e^{x \cdot \left(-x\right)}}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t_1 \cdot t_1}{1 + t_1}\\ \end{array} \]

Alternative 4
Accuracy	98.5%
Cost	225796

\[\begin{array}{l} t_0 := e^{x \cdot \left(-x\right)}\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := 1 + \left|x\right| \cdot 0.3275911\\ t_3 := \frac{\left(0.254829592 + \frac{-0.284496736 - \frac{\frac{1.453152027}{t_1} - \left(1.421413741 + \frac{1.061405429}{{t_1}^{2}}\right)}{t_1}}{t_1}\right) \cdot t_0}{t_1}\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t_3 \cdot \frac{\left(0.254829592 + \frac{\left(1.061405429 \cdot \frac{1}{{t_2}^{3}} + 1.421413741 \cdot \frac{1}{t_2}\right) + \left(1.453152027 \cdot \frac{-1}{{t_2}^{2}} - 0.284496736\right)}{t_2}\right) \cdot t_0}{t_1}}{1 + t_3}\\ \end{array} \]

Alternative 5
Accuracy	98.5%
Cost	98628

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{e^{x \cdot \left(-x\right)}}{\frac{t_0}{0.254829592 + \left(\left(\frac{1.421413741}{{t_0}^{2}} + \frac{1.061405429}{{t_0}^{4}}\right) - \left(\frac{1.453152027}{{t_0}^{3}} + \frac{0.284496736}{t_0}\right)\right)}}\\ \end{array} \]

Alternative 6
Accuracy	98.5%
Cost	98500

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 - \frac{\frac{1.453152027}{t_0} - \left(1.421413741 + \frac{1.061405429}{{t_0}^{2}}\right)}{t_0}}{t_0}\right) \cdot e^{x \cdot \left(-x\right)}}{t_0}\right)}\\ \end{array} \]

Alternative 7
Accuracy	98.5%
Cost	54852

\[\begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 - t_1 \cdot \left(\left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(\left(1.421413741 + \frac{1.061405429}{{t_0}^{2}}\right) + 1.453152027 \cdot \frac{-1}{t_0}\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\right)\\ \end{array} \]

Alternative 8
Accuracy	98.4%
Cost	34948

\[\begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 + t_1 \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(t_1 \cdot \left(t_1 \cdot \left(0.031738286 - \frac{1.061405429}{t_0}\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \]

Alternative 9
Accuracy	98.4%
Cost	29188

\[\begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{1 + x \cdot 0.3275911}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + \frac{1}{t_0} \cdot \left(1.421413741 + \left(-1.453152027 + \frac{1.061405429}{t_0}\right) \cdot t_1\right)\right)\right)\right) \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \]

Alternative 10
Accuracy	98.4%
Cost	28676

\[\begin{array}{l} t_0 := \frac{1}{1 + \left|x\right| \cdot 0.3275911}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 + t_0 \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\left(-0.284496736 + t_0 \cdot \left(1.061405429 \cdot t_0 - 0.031738286\right)\right) \cdot \frac{-1}{1 + x \cdot 0.3275911} - 0.254829592\right)\right)\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \]

Alternative 11
Accuracy	98.3%
Cost	6792

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-10}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \]

Alternative 12
Accuracy	53.7%
Cost	64

\[10^{-9} \]

Reproduce

herbie shell --seed 2023178 
(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)))))))