Jmat.Real.erf

Percentage Accurate: 79.4% → 99.3%

Time: 21.2s

Alternatives: 12

Speedup: 167.0×

Specification

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (+
         -0.284496736
         (*
          t_0
          (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
     (exp (- (* (fabs x) (fabs x))))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (+
         -0.284496736
         (*
          t_0
          (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
     (exp (- (* (fabs x) (fabs x))))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ t_2 := e^{-x \cdot x}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 - t_1 \cdot \left(t_2 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{1 + {\left(\sqrt[3]{x \cdot 0.3275911}\right)}^{3}} \cdot \left(t_2 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_1 (/ 1.0 t_0))
        (t_2 (exp (- (* x x)))))
   (if (<= x -2.5e-17)
     (-
      1.0
      (*
       t_1
       (*
        t_2
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+ 1.421413741 (* t_1 (+ -1.453152027 (/ 1.061405429 t_0)))))))))))
     (if (<= x 1.65e-6)
       (+ 1e-9 (* x 1.128386358070218))
       (-
        1.0
        (*
         (/ 1.0 (+ 1.0 (pow (cbrt (* x 0.3275911)) 3.0)))
         (*
          t_2
          (+
           0.254829592
           (*
            t_1
            (+
             -0.284496736
             (*
              (/ 1.0 (+ 1.0 (* x 0.3275911)))
              (+
               (/ 1.061405429 (pow (fma 0.3275911 x 1.0) 2.0))
               (+
                1.421413741
                (/ -1.453152027 (fma 0.3275911 x 1.0)))))))))))))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = exp(-(x * x));
	double tmp;
	if (x <= -2.5e-17) {
		tmp = 1.0 - (t_1 * (t_2 * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))));
	} else if (x <= 1.65e-6) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 - ((1.0 / (1.0 + pow(cbrt((x * 0.3275911)), 3.0))) * (t_2 * (0.254829592 + (t_1 * (-0.284496736 + ((1.0 / (1.0 + (x * 0.3275911))) * ((1.061405429 / pow(fma(0.3275911, x, 1.0), 2.0)) + (1.421413741 + (-1.453152027 / fma(0.3275911, x, 1.0))))))))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	t_2 = exp(Float64(-Float64(x * x)))
	tmp = 0.0
	if (x <= -2.5e-17)
		tmp = Float64(1.0 - Float64(t_1 * Float64(t_2 * 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 <= 1.65e-6)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = Float64(1.0 - Float64(Float64(1.0 / Float64(1.0 + (cbrt(Float64(x * 0.3275911)) ^ 3.0))) * Float64(t_2 * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(Float64(1.0 / Float64(1.0 + Float64(x * 0.3275911))) * Float64(Float64(1.061405429 / (fma(0.3275911, x, 1.0) ^ 2.0)) + Float64(1.421413741 + Float64(-1.453152027 / fma(0.3275911, x, 1.0)))))))))));
	end
	return tmp
end

Wolfram

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]}, Block[{t$95$2 = N[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[x, -2.5e-17], N[(1.0 - N[(t$95$1 * N[(t$95$2 * 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, 1.65e-6], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(1.0 / N[(1.0 + N[Power[N[Power[N[(x * 0.3275911), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(N[(1.0 / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.061405429 / N[Power[N[(0.3275911 * x + 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(1.421413741 + N[(-1.453152027 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4999999999999999e-17

   1. Initial program 98.6%

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

      1. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

   if -2.4999999999999999e-17 < x < 1.65000000000000008e-6

   1. Initial program 57.7%

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

      1. associate-*l* 57.7%

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

   3. Simplified 57.7%

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

      1. add-exp-log 57.7%

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

      2. sub-neg 57.7%

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

   5. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

      2. associate-/r/ 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      3. distribute-lft-neg-in 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      4. exp-prod 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

   7. Simplified 57.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

   8. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 1.65000000000000008e-6 < x

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-lft-neg-in 99.8%

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

      6. associate-+l+ 99.8%

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

   6. Simplified 99.7%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right) \cdot e^{-x \cdot x}\right) \]
   7. Step-by-step derivation

      1. add-cube-cbrt 99.8%

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

      2. pow3 99.8%

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

      3. add-sqr-sqrt 99.8%

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

      4. fabs-sqr 99.8%

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

      5. add-sqr-sqrt 99.8%

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

   8. Applied egg-rr 99.8%

      \[\leadsto 1 - \frac{1}{1 + \color{blue}{{\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 99.8%

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

      4. add-exp-log 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   10. Applied egg-rr 99.8%

       \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   11. Step-by-step derivation

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. +-rgt-identity 99.8%

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

   12. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{1 + {\left(\sqrt[3]{x \cdot 0.3275911}\right)}^{3}} \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-11}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(\left(0.254829592 + \frac{1.061405429}{{t_0}^{4}}\right) + \left(\frac{1.421413741}{{t_0}^{2}} - \frac{0.284496736}{t_0}\right)\right) - \frac{1.453152027}{{t_0}^{3}}}{t_0} \cdot e^{-x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0)))
   (if (<= (fabs x) 2e-11)
     (+ 1e-9 (* x 1.128386358070218))
     (-
      1.0
      (*
       (/
        (-
         (+
          (+ 0.254829592 (/ 1.061405429 (pow t_0 4.0)))
          (- (/ 1.421413741 (pow t_0 2.0)) (/ 0.284496736 t_0)))
         (/ 1.453152027 (pow t_0 3.0)))
        t_0)
       (exp (- (* x x))))))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double tmp;
	if (fabs(x) <= 2e-11) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 - (((((0.254829592 + (1.061405429 / pow(t_0, 4.0))) + ((1.421413741 / pow(t_0, 2.0)) - (0.284496736 / t_0))) - (1.453152027 / pow(t_0, 3.0))) / t_0) * exp(-(x * x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	tmp = 0.0
	if (abs(x) <= 2e-11)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.254829592 + Float64(1.061405429 / (t_0 ^ 4.0))) + Float64(Float64(1.421413741 / (t_0 ^ 2.0)) - Float64(0.284496736 / t_0))) - Float64(1.453152027 / (t_0 ^ 3.0))) / t_0) * exp(Float64(-Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2e-11], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(N[(N[(0.254829592 + N[(1.061405429 / N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.421413741 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.284496736 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.453152027 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.99999999999999988e-11

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

   3. Simplified 57.6%

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

      1. add-exp-log 57.6%

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

      2. sub-neg 57.6%

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

   5. Applied egg-rr 57.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 57.5%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

      2. associate-/r/ 57.5%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      3. distribute-lft-neg-in 57.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      4. exp-prod 57.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

   7. Simplified 57.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

   8. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

   10. Simplified 99.5%

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

   if 1.99999999999999988e-11 < (fabs.f64 x)

   1. Initial program 99.5%

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

      1. associate-*l* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 99.5%

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

   6. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-11}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(\left(0.254829592 + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}}\right) + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} - \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-11}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 - \left(1.453152027 \cdot t_1 + 1.061405429 \cdot \frac{-1}{{t_0}^{2}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x) 2e-11)
     (+ 1e-9 (* x 1.128386358070218))
     (+
      1.0
      (*
       (*
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (-
             1.421413741
             (+
              (* 1.453152027 t_1)
              (* 1.061405429 (/ -1.0 (pow t_0 2.0)))))))))
        (exp (- (* x x))))
       (/ -1.0 t_0))))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x) <= 2e-11) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 + (((0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 - ((1.453152027 * t_1) + (1.061405429 * (-1.0 / pow(t_0, 2.0))))))))) * exp(-(x * x))) * (-1.0 / t_0));
	}
	return tmp;
}

Fortran

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 (abs(x) <= 2d-11) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 1.0d0 + (((0.254829592d0 + (t_1 * ((-0.284496736d0) + (t_1 * (1.421413741d0 - ((1.453152027d0 * t_1) + (1.061405429d0 * ((-1.0d0) / (t_0 ** 2.0d0))))))))) * exp(-(x * x))) * ((-1.0d0) / t_0))
    end if
    code = tmp
end function

Java

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 (Math.abs(x) <= 2e-11) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 + (((0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 - ((1.453152027 * t_1) + (1.061405429 * (-1.0 / Math.pow(t_0, 2.0))))))))) * Math.exp(-(x * x))) * (-1.0 / t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 + (math.fabs(x) * 0.3275911)
	t_1 = 1.0 / t_0
	tmp = 0
	if math.fabs(x) <= 2e-11:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1.0 + (((0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 - ((1.453152027 * t_1) + (1.061405429 * (-1.0 / math.pow(t_0, 2.0))))))))) * math.exp(-(x * x))) * (-1.0 / t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x) <= 2e-11)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 - Float64(Float64(1.453152027 * t_1) + Float64(1.061405429 * Float64(-1.0 / (t_0 ^ 2.0))))))))) * exp(Float64(-Float64(x * x)))) * Float64(-1.0 / t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 + (abs(x) * 0.3275911);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (abs(x) <= 2e-11)
		tmp = 1e-9 + (x * 1.128386358070218);
	else
		tmp = 1.0 + (((0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 - ((1.453152027 * t_1) + (1.061405429 * (-1.0 / (t_0 ^ 2.0))))))))) * exp(-(x * x))) * (-1.0 / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

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[N[Abs[x], $MachinePrecision], 2e-11], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 - N[(N[(1.453152027 * t$95$1), $MachinePrecision] + N[(1.061405429 * N[(-1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.99999999999999988e-11

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

   3. Simplified 57.6%

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

      1. add-exp-log 57.6%

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

      2. sub-neg 57.6%

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

   5. Applied egg-rr 57.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 57.5%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

      2. associate-/r/ 57.5%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      3. distribute-lft-neg-in 57.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      4. exp-prod 57.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

   7. Simplified 57.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

   8. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

   10. Simplified 99.5%

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

   if 1.99999999999999988e-11 < (fabs.f64 x)

   1. Initial program 99.5%

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

      1. associate-*l* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 99.5%

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

      1. associate--l+ 99.6%

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

      2. associate-*r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. metadata-eval 99.6%

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

      5. +-commutative 99.6%

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

      6. fma-def 99.6%

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

      7. associate-*r/ 99.5%

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

      8. metadata-eval 99.5%

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

      9. fma-def 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-11}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 - \left(1.453152027 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911} + 1.061405429 \cdot \frac{-1}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{2}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ t_2 := \frac{1}{1 + x \cdot 0.3275911}\\ t_3 := e^{-x \cdot x}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 - t_1 \cdot \left(t_3 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - t_1 \cdot \left(t_3 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_2 \cdot \left(1.421413741 + t_2 \cdot \mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_1 (/ 1.0 t_0))
        (t_2 (/ 1.0 (+ 1.0 (* x 0.3275911))))
        (t_3 (exp (- (* x x)))))
   (if (<= x -2.5e-17)
     (-
      1.0
      (*
       t_1
       (*
        t_3
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+ 1.421413741 (* t_1 (+ -1.453152027 (/ 1.061405429 t_0)))))))))))
     (if (<= x 1.65e-6)
       (+ 1e-9 (* x 1.128386358070218))
       (-
        1.0
        (*
         t_1
         (*
          t_3
          (+
           0.254829592
           (*
            t_1
            (+
             -0.284496736
             (*
              t_2
              (+
               1.421413741
               (*
                t_2
                (fma
                 1.061405429
                 (/ 1.0 (fma 0.3275911 x 1.0))
                 -1.453152027))))))))))))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = 1.0 / (1.0 + (x * 0.3275911));
	double t_3 = exp(-(x * x));
	double tmp;
	if (x <= -2.5e-17) {
		tmp = 1.0 - (t_1 * (t_3 * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))));
	} else if (x <= 1.65e-6) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 - (t_1 * (t_3 * (0.254829592 + (t_1 * (-0.284496736 + (t_2 * (1.421413741 + (t_2 * fma(1.061405429, (1.0 / fma(0.3275911, x, 1.0)), -1.453152027)))))))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(1.0 / Float64(1.0 + Float64(x * 0.3275911)))
	t_3 = exp(Float64(-Float64(x * x)))
	tmp = 0.0
	if (x <= -2.5e-17)
		tmp = Float64(1.0 - Float64(t_1 * Float64(t_3 * 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 <= 1.65e-6)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = Float64(1.0 - Float64(t_1 * Float64(t_3 * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(t_2 * fma(1.061405429, Float64(1.0 / fma(0.3275911, x, 1.0)), -1.453152027))))))))));
	end
	return tmp
end

Wolfram

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]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[x, -2.5e-17], N[(1.0 - N[(t$95$1 * N[(t$95$3 * 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, 1.65e-6], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(t$95$1 * N[(t$95$3 * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(t$95$2 * N[(1.061405429 * N[(1.0 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4999999999999999e-17

   1. Initial program 98.6%

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

      1. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

   if -2.4999999999999999e-17 < x < 1.65000000000000008e-6

   1. Initial program 57.7%

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

      1. associate-*l* 57.7%

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

   3. Simplified 57.7%

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

      1. add-exp-log 57.7%

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

      2. sub-neg 57.7%

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

   5. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

      2. associate-/r/ 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      3. distribute-lft-neg-in 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      4. exp-prod 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

   7. Simplified 57.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

   8. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 1.65000000000000008e-6 < x

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 99.8%

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

      4. add-exp-log 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.7%

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

      1. fma-udef 99.8%

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

      2. associate--l+ 99.8%

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

      3. metadata-eval 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. +-rgt-identity 99.8%

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

   7. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. div-inv 99.7%

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

      3. fma-def 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-def 99.8%

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

      6. add-sqr-sqrt 99.8%

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

      7. fabs-sqr 99.8%

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

      8. add-sqr-sqrt 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. expm1-log1p-u 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. expm1-udef 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. log1p-udef 99.8%

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

       4. add-exp-log 99.8%

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

       5. +-commutative 99.8%

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

       6. fma-def 99.8%

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

       7. add-sqr-sqrt 99.8%

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

       8. fabs-sqr 99.8%

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

       9. add-sqr-sqrt 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. +-rgt-identity 99.8%

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

   13. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x \cdot x}\\ t_1 := 1 + \left|x\right| \cdot 0.3275911\\ t_2 := \frac{1}{t_1}\\ t_3 := \frac{1}{1 + x \cdot 0.3275911}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-16}:\\ \;\;\;\;1 + t_2 \cdot \left(t_0 \cdot \left(\left(-0.284496736 + t_2 \cdot \left(1.421413741 + t_3 \cdot -0.391746598\right)\right) \cdot \frac{-1}{t_1} - 0.254829592\right)\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - t_2 \cdot \left(t_0 \cdot \left(0.254829592 + t_2 \cdot \left(-0.284496736 + t_3 \cdot \left(1.421413741 + t_3 \cdot \mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- (* x x))))
        (t_1 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_2 (/ 1.0 t_1))
        (t_3 (/ 1.0 (+ 1.0 (* x 0.3275911)))))
   (if (<= x -1.7e-16)
     (+
      1.0
      (*
       t_2
       (*
        t_0
        (-
         (*
          (+ -0.284496736 (* t_2 (+ 1.421413741 (* t_3 -0.391746598))))
          (/ -1.0 t_1))
         0.254829592))))
     (if (<= x 1.65e-6)
       (+ 1e-9 (* x 1.128386358070218))
       (-
        1.0
        (*
         t_2
         (*
          t_0
          (+
           0.254829592
           (*
            t_2
            (+
             -0.284496736
             (*
              t_3
              (+
               1.421413741
               (*
                t_3
                (fma
                 1.061405429
                 (/ 1.0 (fma 0.3275911 x 1.0))
                 -1.453152027))))))))))))))

C

double code(double x) {
	double t_0 = exp(-(x * x));
	double t_1 = 1.0 + (fabs(x) * 0.3275911);
	double t_2 = 1.0 / t_1;
	double t_3 = 1.0 / (1.0 + (x * 0.3275911));
	double tmp;
	if (x <= -1.7e-16) {
		tmp = 1.0 + (t_2 * (t_0 * (((-0.284496736 + (t_2 * (1.421413741 + (t_3 * -0.391746598)))) * (-1.0 / t_1)) - 0.254829592)));
	} else if (x <= 1.65e-6) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 - (t_2 * (t_0 * (0.254829592 + (t_2 * (-0.284496736 + (t_3 * (1.421413741 + (t_3 * fma(1.061405429, (1.0 / fma(0.3275911, x, 1.0)), -1.453152027)))))))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-Float64(x * x)))
	t_1 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_2 = Float64(1.0 / t_1)
	t_3 = Float64(1.0 / Float64(1.0 + Float64(x * 0.3275911)))
	tmp = 0.0
	if (x <= -1.7e-16)
		tmp = Float64(1.0 + Float64(t_2 * Float64(t_0 * Float64(Float64(Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(t_3 * -0.391746598)))) * Float64(-1.0 / t_1)) - 0.254829592))));
	elseif (x <= 1.65e-6)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = Float64(1.0 - Float64(t_2 * Float64(t_0 * Float64(0.254829592 + Float64(t_2 * Float64(-0.284496736 + Float64(t_3 * Float64(1.421413741 + Float64(t_3 * fma(1.061405429, Float64(1.0 / fma(0.3275911, x, 1.0)), -1.453152027))))))))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e-16], N[(1.0 + N[(t$95$2 * N[(t$95$0 * N[(N[(N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(t$95$3 * -0.391746598), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e-6], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(t$95$2 * N[(t$95$0 * N[(0.254829592 + N[(t$95$2 * N[(-0.284496736 + N[(t$95$3 * N[(1.421413741 + N[(t$95$3 * N[(1.061405429 * N[(1.0 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7e-16

   1. Initial program 98.6%

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

      1. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

      1. expm1-log1p-u 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 98.0%

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

      4. add-exp-log 98.0%

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

      5. +-commutative 98.0%

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

      6. fma-def 98.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 98.0%

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

   5. Applied egg-rr 98.1%

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

      1. fma-udef 98.0%

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

      2. associate--l+ 98.0%

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

      3. metadata-eval 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. +-rgt-identity 98.0%

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

   7. Simplified 98.1%

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

      1. +-commutative 98.1%

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

      2. div-inv 98.1%

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

      3. fma-def 98.1%

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

      4. +-commutative 98.1%

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

      5. fma-def 98.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 98.1%

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

   9. Applied egg-rr 98.1%

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

   10. Taylor expanded in x around 0 98.2%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \color{blue}{-0.391746598}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   if -1.7e-16 < x < 1.65000000000000008e-6

   1. Initial program 57.7%

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

      1. associate-*l* 57.7%

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

   3. Simplified 57.7%

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

      1. add-exp-log 57.7%

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

      2. sub-neg 57.7%

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

   5. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

      2. associate-/r/ 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      3. distribute-lft-neg-in 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      4. exp-prod 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

   7. Simplified 57.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

   8. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 1.65000000000000008e-6 < x

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 99.8%

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

      4. add-exp-log 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.7%

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

      1. fma-udef 99.8%

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

      2. associate--l+ 99.8%

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

      3. metadata-eval 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. +-rgt-identity 99.8%

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

   7. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. div-inv 99.7%

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

      3. fma-def 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-def 99.8%

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

      6. add-sqr-sqrt 99.8%

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

      7. fabs-sqr 99.8%

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

      8. add-sqr-sqrt 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. expm1-log1p-u 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. expm1-udef 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. log1p-udef 99.8%

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

       4. add-exp-log 99.8%

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

       5. +-commutative 99.8%

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

       6. fma-def 99.8%

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

       7. add-sqr-sqrt 99.8%

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

       8. fabs-sqr 99.8%

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

       9. add-sqr-sqrt 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. +-rgt-identity 99.8%

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

   13. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-16}:\\ \;\;\;\;1 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{-x \cdot x} \cdot \left(\left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot -0.391746598\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 6: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := e^{-x \cdot x}\\ t_2 := 1 + \left|x\right| \cdot 0.3275911\\ t_3 := \frac{1}{t_2}\\ t_4 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-16}:\\ \;\;\;\;1 + t_3 \cdot \left(t_1 \cdot \left(\left(-0.284496736 + t_3 \cdot \left(1.421413741 + t_4 \cdot -0.391746598\right)\right) \cdot \frac{-1}{t_2} - 0.254829592\right)\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - t_3 \cdot \left(t_1 \cdot \left(0.254829592 + t_4 \cdot \left(-0.284496736 + t_3 \cdot \left(1.421413741 + t_4 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x 0.3275911)))
        (t_1 (exp (- (* x x))))
        (t_2 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_3 (/ 1.0 t_2))
        (t_4 (/ 1.0 t_0)))
   (if (<= x -1.7e-16)
     (+
      1.0
      (*
       t_3
       (*
        t_1
        (-
         (*
          (+ -0.284496736 (* t_3 (+ 1.421413741 (* t_4 -0.391746598))))
          (/ -1.0 t_2))
         0.254829592))))
     (if (<= x 1.65e-6)
       (+ 1e-9 (* x 1.128386358070218))
       (-
        1.0
        (*
         t_3
         (*
          t_1
          (+
           0.254829592
           (*
            t_4
            (+
             -0.284496736
             (*
              t_3
              (+
               1.421413741
               (* t_4 (+ -1.453152027 (/ 1.061405429 t_0)))))))))))))))

C

double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = exp(-(x * x));
	double t_2 = 1.0 + (fabs(x) * 0.3275911);
	double t_3 = 1.0 / t_2;
	double t_4 = 1.0 / t_0;
	double tmp;
	if (x <= -1.7e-16) {
		tmp = 1.0 + (t_3 * (t_1 * (((-0.284496736 + (t_3 * (1.421413741 + (t_4 * -0.391746598)))) * (-1.0 / t_2)) - 0.254829592)));
	} else if (x <= 1.65e-6) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 - (t_3 * (t_1 * (0.254829592 + (t_4 * (-0.284496736 + (t_3 * (1.421413741 + (t_4 * (-1.453152027 + (1.061405429 / t_0))))))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = 1.0d0 + (x * 0.3275911d0)
    t_1 = exp(-(x * x))
    t_2 = 1.0d0 + (abs(x) * 0.3275911d0)
    t_3 = 1.0d0 / t_2
    t_4 = 1.0d0 / t_0
    if (x <= (-1.7d-16)) then
        tmp = 1.0d0 + (t_3 * (t_1 * ((((-0.284496736d0) + (t_3 * (1.421413741d0 + (t_4 * (-0.391746598d0))))) * ((-1.0d0) / t_2)) - 0.254829592d0)))
    else if (x <= 1.65d-6) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 1.0d0 - (t_3 * (t_1 * (0.254829592d0 + (t_4 * ((-0.284496736d0) + (t_3 * (1.421413741d0 + (t_4 * ((-1.453152027d0) + (1.061405429d0 / t_0))))))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = Math.exp(-(x * x));
	double t_2 = 1.0 + (Math.abs(x) * 0.3275911);
	double t_3 = 1.0 / t_2;
	double t_4 = 1.0 / t_0;
	double tmp;
	if (x <= -1.7e-16) {
		tmp = 1.0 + (t_3 * (t_1 * (((-0.284496736 + (t_3 * (1.421413741 + (t_4 * -0.391746598)))) * (-1.0 / t_2)) - 0.254829592)));
	} else if (x <= 1.65e-6) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 - (t_3 * (t_1 * (0.254829592 + (t_4 * (-0.284496736 + (t_3 * (1.421413741 + (t_4 * (-1.453152027 + (1.061405429 / t_0))))))))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 + (x * 0.3275911)
	t_1 = math.exp(-(x * x))
	t_2 = 1.0 + (math.fabs(x) * 0.3275911)
	t_3 = 1.0 / t_2
	t_4 = 1.0 / t_0
	tmp = 0
	if x <= -1.7e-16:
		tmp = 1.0 + (t_3 * (t_1 * (((-0.284496736 + (t_3 * (1.421413741 + (t_4 * -0.391746598)))) * (-1.0 / t_2)) - 0.254829592)))
	elif x <= 1.65e-6:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1.0 - (t_3 * (t_1 * (0.254829592 + (t_4 * (-0.284496736 + (t_3 * (1.421413741 + (t_4 * (-1.453152027 + (1.061405429 / t_0))))))))))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * 0.3275911))
	t_1 = exp(Float64(-Float64(x * x)))
	t_2 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_3 = Float64(1.0 / t_2)
	t_4 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x <= -1.7e-16)
		tmp = Float64(1.0 + Float64(t_3 * Float64(t_1 * Float64(Float64(Float64(-0.284496736 + Float64(t_3 * Float64(1.421413741 + Float64(t_4 * -0.391746598)))) * Float64(-1.0 / t_2)) - 0.254829592))));
	elseif (x <= 1.65e-6)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = Float64(1.0 - Float64(t_3 * Float64(t_1 * Float64(0.254829592 + Float64(t_4 * Float64(-0.284496736 + Float64(t_3 * Float64(1.421413741 + Float64(t_4 * Float64(-1.453152027 + Float64(1.061405429 / t_0)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 + (x * 0.3275911);
	t_1 = exp(-(x * x));
	t_2 = 1.0 + (abs(x) * 0.3275911);
	t_3 = 1.0 / t_2;
	t_4 = 1.0 / t_0;
	tmp = 0.0;
	if (x <= -1.7e-16)
		tmp = 1.0 + (t_3 * (t_1 * (((-0.284496736 + (t_3 * (1.421413741 + (t_4 * -0.391746598)))) * (-1.0 / t_2)) - 0.254829592)));
	elseif (x <= 1.65e-6)
		tmp = 1e-9 + (x * 1.128386358070218);
	else
		tmp = 1.0 - (t_3 * (t_1 * (0.254829592 + (t_4 * (-0.284496736 + (t_3 * (1.421413741 + (t_4 * (-1.453152027 + (1.061405429 / t_0))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x, -1.7e-16], N[(1.0 + N[(t$95$3 * N[(t$95$1 * N[(N[(N[(-0.284496736 + N[(t$95$3 * N[(1.421413741 + N[(t$95$4 * -0.391746598), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e-6], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(t$95$3 * N[(t$95$1 * N[(0.254829592 + N[(t$95$4 * N[(-0.284496736 + N[(t$95$3 * N[(1.421413741 + N[(t$95$4 * N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7e-16

   1. Initial program 98.6%

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

      1. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

      1. expm1-log1p-u 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 98.0%

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

      4. add-exp-log 98.0%

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

      5. +-commutative 98.0%

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

      6. fma-def 98.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 98.0%

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

   5. Applied egg-rr 98.1%

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

      1. fma-udef 98.0%

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

      2. associate--l+ 98.0%

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

      3. metadata-eval 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. +-rgt-identity 98.0%

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

   7. Simplified 98.1%

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

      1. +-commutative 98.1%

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

      2. div-inv 98.1%

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

      3. fma-def 98.1%

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

      4. +-commutative 98.1%

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

      5. fma-def 98.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 98.1%

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

   9. Applied egg-rr 98.1%

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

   10. Taylor expanded in x around 0 98.2%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \color{blue}{-0.391746598}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   if -1.7e-16 < x < 1.65000000000000008e-6

   1. Initial program 57.7%

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

      1. associate-*l* 57.7%

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

   3. Simplified 57.7%

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

      1. add-exp-log 57.7%

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

      2. sub-neg 57.7%

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

   5. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

      2. associate-/r/ 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      3. distribute-lft-neg-in 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      4. exp-prod 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

   7. Simplified 57.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

   8. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 1.65000000000000008e-6 < x

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 99.8%

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

      4. add-exp-log 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.7%

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

      1. fma-udef 99.8%

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

      2. associate--l+ 99.8%

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

      3. metadata-eval 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. +-rgt-identity 99.8%

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

   7. Simplified 99.7%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 99.8%

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

      4. add-exp-log 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   9. Applied egg-rr 99.7%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. +-rgt-identity 99.8%

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

   11. Simplified 99.7%

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

       1. expm1-log1p-u 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. expm1-udef 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. log1p-udef 99.8%

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

       4. add-exp-log 99.8%

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

       5. +-commutative 99.8%

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

       6. fma-def 99.8%

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

       7. add-sqr-sqrt 99.8%

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

       8. fabs-sqr 99.8%

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

       9. add-sqr-sqrt 99.8%

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

   13. Applied egg-rr 99.7%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. +-rgt-identity 99.8%

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

   15. Simplified 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-16}:\\ \;\;\;\;1 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{-x \cdot x} \cdot \left(\left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot -0.391746598\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ t_2 := 1 + \left|x\right| \cdot 0.3275911\\ t_3 := \frac{1}{t_2}\\ t_4 := e^{-x \cdot x}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 - t_3 \cdot \left(t_4 \cdot \left(0.254829592 + \frac{t_3 \cdot 1.029667143 - 0.284496736}{t_2}\right)\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - t_3 \cdot \left(t_4 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_3 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x 0.3275911)))
        (t_1 (/ 1.0 t_0))
        (t_2 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_3 (/ 1.0 t_2))
        (t_4 (exp (- (* x x)))))
   (if (<= x -2.5e-17)
     (-
      1.0
      (*
       t_3
       (* t_4 (+ 0.254829592 (/ (- (* t_3 1.029667143) 0.284496736) t_2)))))
     (if (<= x 1.65e-6)
       (+ 1e-9 (* x 1.128386358070218))
       (-
        1.0
        (*
         t_3
         (*
          t_4
          (+
           0.254829592
           (*
            t_1
            (+
             -0.284496736
             (*
              t_3
              (+
               1.421413741
               (* t_1 (+ -1.453152027 (/ 1.061405429 t_0)))))))))))))))

C

double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = 1.0 + (fabs(x) * 0.3275911);
	double t_3 = 1.0 / t_2;
	double t_4 = exp(-(x * x));
	double tmp;
	if (x <= -2.5e-17) {
		tmp = 1.0 - (t_3 * (t_4 * (0.254829592 + (((t_3 * 1.029667143) - 0.284496736) / t_2))));
	} else if (x <= 1.65e-6) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 - (t_3 * (t_4 * (0.254829592 + (t_1 * (-0.284496736 + (t_3 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = 1.0d0 + (x * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    t_2 = 1.0d0 + (abs(x) * 0.3275911d0)
    t_3 = 1.0d0 / t_2
    t_4 = exp(-(x * x))
    if (x <= (-2.5d-17)) then
        tmp = 1.0d0 - (t_3 * (t_4 * (0.254829592d0 + (((t_3 * 1.029667143d0) - 0.284496736d0) / t_2))))
    else if (x <= 1.65d-6) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 1.0d0 - (t_3 * (t_4 * (0.254829592d0 + (t_1 * ((-0.284496736d0) + (t_3 * (1.421413741d0 + (t_1 * ((-1.453152027d0) + (1.061405429d0 / t_0))))))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = 1.0 + (Math.abs(x) * 0.3275911);
	double t_3 = 1.0 / t_2;
	double t_4 = Math.exp(-(x * x));
	double tmp;
	if (x <= -2.5e-17) {
		tmp = 1.0 - (t_3 * (t_4 * (0.254829592 + (((t_3 * 1.029667143) - 0.284496736) / t_2))));
	} else if (x <= 1.65e-6) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 - (t_3 * (t_4 * (0.254829592 + (t_1 * (-0.284496736 + (t_3 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 + (x * 0.3275911)
	t_1 = 1.0 / t_0
	t_2 = 1.0 + (math.fabs(x) * 0.3275911)
	t_3 = 1.0 / t_2
	t_4 = math.exp(-(x * x))
	tmp = 0
	if x <= -2.5e-17:
		tmp = 1.0 - (t_3 * (t_4 * (0.254829592 + (((t_3 * 1.029667143) - 0.284496736) / t_2))))
	elif x <= 1.65e-6:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1.0 - (t_3 * (t_4 * (0.254829592 + (t_1 * (-0.284496736 + (t_3 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_3 = Float64(1.0 / t_2)
	t_4 = exp(Float64(-Float64(x * x)))
	tmp = 0.0
	if (x <= -2.5e-17)
		tmp = Float64(1.0 - Float64(t_3 * Float64(t_4 * Float64(0.254829592 + Float64(Float64(Float64(t_3 * 1.029667143) - 0.284496736) / t_2)))));
	elseif (x <= 1.65e-6)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = Float64(1.0 - Float64(t_3 * Float64(t_4 * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_3 * Float64(1.421413741 + Float64(t_1 * Float64(-1.453152027 + Float64(1.061405429 / t_0)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 + (x * 0.3275911);
	t_1 = 1.0 / t_0;
	t_2 = 1.0 + (abs(x) * 0.3275911);
	t_3 = 1.0 / t_2;
	t_4 = exp(-(x * x));
	tmp = 0.0;
	if (x <= -2.5e-17)
		tmp = 1.0 - (t_3 * (t_4 * (0.254829592 + (((t_3 * 1.029667143) - 0.284496736) / t_2))));
	elseif (x <= 1.65e-6)
		tmp = 1e-9 + (x * 1.128386358070218);
	else
		tmp = 1.0 - (t_3 * (t_4 * (0.254829592 + (t_1 * (-0.284496736 + (t_3 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[x, -2.5e-17], N[(1.0 - N[(t$95$3 * N[(t$95$4 * N[(0.254829592 + N[(N[(N[(t$95$3 * 1.029667143), $MachinePrecision] - 0.284496736), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e-6], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(t$95$3 * N[(t$95$4 * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$3 * 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]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4999999999999999e-17

   1. Initial program 98.6%

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

      1. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

      1. expm1-log1p-u 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 98.0%

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

      4. add-exp-log 98.0%

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

      5. +-commutative 98.0%

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

      6. fma-def 98.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 98.0%

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

   5. Applied egg-rr 98.1%

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

      1. fma-udef 98.0%

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

      2. associate--l+ 98.0%

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

      3. metadata-eval 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. +-rgt-identity 98.0%

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

   7. Simplified 98.1%

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

      1. +-commutative 98.1%

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

      2. div-inv 98.1%

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

      3. fma-def 98.1%

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

      4. +-commutative 98.1%

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

      5. fma-def 98.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 98.1%

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

   9. Applied egg-rr 98.1%

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

   10. Taylor expanded in x around 0 98.1%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{1.029667143}\right)\right) \cdot e^{-x \cdot x}\right) \]

   11. Taylor expanded in x around inf 98.1%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \color{blue}{\frac{1.029667143 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1} - 0.284496736}{0.3275911 \cdot \left|x\right| + 1}}\right) \cdot e^{-x \cdot x}\right) \]

   if -2.4999999999999999e-17 < x < 1.65000000000000008e-6

   1. Initial program 57.7%

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

      1. associate-*l* 57.7%

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

   3. Simplified 57.7%

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

      1. add-exp-log 57.7%

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

      2. sub-neg 57.7%

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

   5. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

      2. associate-/r/ 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      3. distribute-lft-neg-in 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      4. exp-prod 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

   7. Simplified 57.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

   8. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 1.65000000000000008e-6 < x

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 99.8%

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

      4. add-exp-log 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.7%

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

      1. fma-udef 99.8%

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

      2. associate--l+ 99.8%

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

      3. metadata-eval 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. +-rgt-identity 99.8%

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

   7. Simplified 99.7%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 99.8%

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

      4. add-exp-log 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   9. Applied egg-rr 99.7%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. +-rgt-identity 99.8%

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

   11. Simplified 99.7%

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

       1. expm1-log1p-u 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. expm1-udef 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. log1p-udef 99.8%

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

       4. add-exp-log 99.8%

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

       5. +-commutative 99.8%

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

       6. fma-def 99.8%

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

       7. add-sqr-sqrt 99.8%

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

       8. fabs-sqr 99.8%

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

       9. add-sqr-sqrt 99.8%

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

   13. Applied egg-rr 99.7%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. +-rgt-identity 99.8%

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

   15. Simplified 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 1.029667143 - 0.284496736}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 99.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ t_2 := \frac{1}{1 + \left|x\right| \cdot 0.3275911}\\ t_3 := e^{-x \cdot x}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 - t_2 \cdot \left(t_3 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_2 \cdot 1.029667143\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - t_2 \cdot \left(t_3 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_2 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x 0.3275911)))
        (t_1 (/ 1.0 t_0))
        (t_2 (/ 1.0 (+ 1.0 (* (fabs x) 0.3275911))))
        (t_3 (exp (- (* x x)))))
   (if (<= x -2.5e-17)
     (-
      1.0
      (*
       t_2
       (* t_3 (+ 0.254829592 (* t_1 (+ -0.284496736 (* t_2 1.029667143)))))))
     (if (<= x 1.65e-6)
       (+ 1e-9 (* x 1.128386358070218))
       (-
        1.0
        (*
         t_2
         (*
          t_3
          (+
           0.254829592
           (*
            t_1
            (+
             -0.284496736
             (*
              t_2
              (+
               1.421413741
               (* t_1 (+ -1.453152027 (/ 1.061405429 t_0)))))))))))))))

C

double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = 1.0 / (1.0 + (fabs(x) * 0.3275911));
	double t_3 = exp(-(x * x));
	double tmp;
	if (x <= -2.5e-17) {
		tmp = 1.0 - (t_2 * (t_3 * (0.254829592 + (t_1 * (-0.284496736 + (t_2 * 1.029667143))))));
	} else if (x <= 1.65e-6) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 - (t_2 * (t_3 * (0.254829592 + (t_1 * (-0.284496736 + (t_2 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 1.0d0 + (x * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    t_2 = 1.0d0 / (1.0d0 + (abs(x) * 0.3275911d0))
    t_3 = exp(-(x * x))
    if (x <= (-2.5d-17)) then
        tmp = 1.0d0 - (t_2 * (t_3 * (0.254829592d0 + (t_1 * ((-0.284496736d0) + (t_2 * 1.029667143d0))))))
    else if (x <= 1.65d-6) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 1.0d0 - (t_2 * (t_3 * (0.254829592d0 + (t_1 * ((-0.284496736d0) + (t_2 * (1.421413741d0 + (t_1 * ((-1.453152027d0) + (1.061405429d0 / t_0))))))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = 1.0 / (1.0 + (Math.abs(x) * 0.3275911));
	double t_3 = Math.exp(-(x * x));
	double tmp;
	if (x <= -2.5e-17) {
		tmp = 1.0 - (t_2 * (t_3 * (0.254829592 + (t_1 * (-0.284496736 + (t_2 * 1.029667143))))));
	} else if (x <= 1.65e-6) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0 - (t_2 * (t_3 * (0.254829592 + (t_1 * (-0.284496736 + (t_2 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 + (x * 0.3275911)
	t_1 = 1.0 / t_0
	t_2 = 1.0 / (1.0 + (math.fabs(x) * 0.3275911))
	t_3 = math.exp(-(x * x))
	tmp = 0
	if x <= -2.5e-17:
		tmp = 1.0 - (t_2 * (t_3 * (0.254829592 + (t_1 * (-0.284496736 + (t_2 * 1.029667143))))))
	elif x <= 1.65e-6:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1.0 - (t_2 * (t_3 * (0.254829592 + (t_1 * (-0.284496736 + (t_2 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(1.0 / Float64(1.0 + Float64(abs(x) * 0.3275911)))
	t_3 = exp(Float64(-Float64(x * x)))
	tmp = 0.0
	if (x <= -2.5e-17)
		tmp = Float64(1.0 - Float64(t_2 * Float64(t_3 * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_2 * 1.029667143)))))));
	elseif (x <= 1.65e-6)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = Float64(1.0 - Float64(t_2 * Float64(t_3 * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(t_1 * Float64(-1.453152027 + Float64(1.061405429 / t_0)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 + (x * 0.3275911);
	t_1 = 1.0 / t_0;
	t_2 = 1.0 / (1.0 + (abs(x) * 0.3275911));
	t_3 = exp(-(x * x));
	tmp = 0.0;
	if (x <= -2.5e-17)
		tmp = 1.0 - (t_2 * (t_3 * (0.254829592 + (t_1 * (-0.284496736 + (t_2 * 1.029667143))))));
	elseif (x <= 1.65e-6)
		tmp = 1e-9 + (x * 1.128386358070218);
	else
		tmp = 1.0 - (t_2 * (t_3 * (0.254829592 + (t_1 * (-0.284496736 + (t_2 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[x, -2.5e-17], N[(1.0 - N[(t$95$2 * N[(t$95$3 * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$2 * 1.029667143), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e-6], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(t$95$2 * N[(t$95$3 * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$2 * 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]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4999999999999999e-17

   1. Initial program 98.6%

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

      1. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

      1. expm1-log1p-u 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 98.0%

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

      4. add-exp-log 98.0%

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

      5. +-commutative 98.0%

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

      6. fma-def 98.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 98.0%

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

   5. Applied egg-rr 98.1%

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

      1. fma-udef 98.0%

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

      2. associate--l+ 98.0%

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

      3. metadata-eval 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. +-rgt-identity 98.0%

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

   7. Simplified 98.1%

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

      1. +-commutative 98.1%

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

      2. div-inv 98.1%

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

      3. fma-def 98.1%

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

      4. +-commutative 98.1%

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

      5. fma-def 98.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 98.1%

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

   9. Applied egg-rr 98.1%

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

   10. Taylor expanded in x around 0 98.1%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{1.029667143}\right)\right) \cdot e^{-x \cdot x}\right) \]
   11. Step-by-step derivation

       1. expm1-log1p-u 98.0%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. expm1-udef 98.0%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. log1p-udef 98.0%

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

       4. add-exp-log 98.0%

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

       5. +-commutative 98.0%

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

       6. fma-def 98.0%

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

       7. add-sqr-sqrt 0.0%

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

       8. fabs-sqr 0.0%

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

       9. add-sqr-sqrt 98.0%

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

   12. Applied egg-rr 98.1%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.029667143\right)\right) \cdot e^{-x \cdot x}\right) \]
   13. Step-by-step derivation

       1. fma-udef 98.0%

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

       2. associate--l+ 98.0%

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

       3. metadata-eval 98.0%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. +-rgt-identity 98.0%

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

   14. Simplified 98.1%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.029667143\right)\right) \cdot e^{-x \cdot x}\right) \]

   if -2.4999999999999999e-17 < x < 1.65000000000000008e-6

   1. Initial program 57.7%

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

      1. associate-*l* 57.7%

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

   3. Simplified 57.7%

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

      1. add-exp-log 57.7%

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

      2. sub-neg 57.7%

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

   5. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

      2. associate-/r/ 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      3. distribute-lft-neg-in 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      4. exp-prod 57.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

   7. Simplified 57.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

   8. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 1.65000000000000008e-6 < x

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 99.8%

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

      4. add-exp-log 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.7%

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

      1. fma-udef 99.8%

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

      2. associate--l+ 99.8%

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

      3. metadata-eval 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. +-rgt-identity 99.8%

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

   7. Simplified 99.7%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 99.8%

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

      4. add-exp-log 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   9. Applied egg-rr 99.7%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. +-rgt-identity 99.8%

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

   11. Simplified 99.7%

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

       1. expm1-log1p-u 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. expm1-udef 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. log1p-udef 99.8%

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

       4. add-exp-log 99.8%

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

       5. +-commutative 99.8%

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

       6. fma-def 99.8%

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

       7. add-sqr-sqrt 99.8%

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

       8. fabs-sqr 99.8%

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

       9. add-sqr-sqrt 99.8%

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

   13. Applied egg-rr 99.7%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. +-rgt-identity 99.8%

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

   15. Simplified 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 1.029667143\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 9: 98.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x\right| \cdot 0.3275911}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 - t_0 \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-0.284496736 + t_0 \cdot 1.029667143\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x) 0.3275911)))))
   (if (<= x -2.5e-17)
     (-
      1.0
      (*
       t_0
       (*
        (exp (- (* x x)))
        (+
         0.254829592
         (*
          (/ 1.0 (+ 1.0 (* x 0.3275911)))
          (+ -0.284496736 (* t_0 1.029667143)))))))
     (if (<= x 0.88) (+ 1e-9 (* x 1.128386358070218)) 1.0))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (fabs(x) * 0.3275911));
	double tmp;
	if (x <= -2.5e-17) {
		tmp = 1.0 - (t_0 * (exp(-(x * x)) * (0.254829592 + ((1.0 / (1.0 + (x * 0.3275911))) * (-0.284496736 + (t_0 * 1.029667143))))));
	} else if (x <= 0.88) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (1.0d0 + (abs(x) * 0.3275911d0))
    if (x <= (-2.5d-17)) then
        tmp = 1.0d0 - (t_0 * (exp(-(x * x)) * (0.254829592d0 + ((1.0d0 / (1.0d0 + (x * 0.3275911d0))) * ((-0.284496736d0) + (t_0 * 1.029667143d0))))))
    else if (x <= 0.88d0) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (Math.abs(x) * 0.3275911));
	double tmp;
	if (x <= -2.5e-17) {
		tmp = 1.0 - (t_0 * (Math.exp(-(x * x)) * (0.254829592 + ((1.0 / (1.0 + (x * 0.3275911))) * (-0.284496736 + (t_0 * 1.029667143))))));
	} else if (x <= 0.88) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + (math.fabs(x) * 0.3275911))
	tmp = 0
	if x <= -2.5e-17:
		tmp = 1.0 - (t_0 * (math.exp(-(x * x)) * (0.254829592 + ((1.0 / (1.0 + (x * 0.3275911))) * (-0.284496736 + (t_0 * 1.029667143))))))
	elif x <= 0.88:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x) * 0.3275911)))
	tmp = 0.0
	if (x <= -2.5e-17)
		tmp = Float64(1.0 - Float64(t_0 * Float64(exp(Float64(-Float64(x * x))) * Float64(0.254829592 + Float64(Float64(1.0 / Float64(1.0 + Float64(x * 0.3275911))) * Float64(-0.284496736 + Float64(t_0 * 1.029667143)))))));
	elseif (x <= 0.88)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / (1.0 + (abs(x) * 0.3275911));
	tmp = 0.0;
	if (x <= -2.5e-17)
		tmp = 1.0 - (t_0 * (exp(-(x * x)) * (0.254829592 + ((1.0 / (1.0 + (x * 0.3275911))) * (-0.284496736 + (t_0 * 1.029667143))))));
	elseif (x <= 0.88)
		tmp = 1e-9 + (x * 1.128386358070218);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-17], N[(1.0 - N[(t$95$0 * N[(N[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision] * N[(0.254829592 + N[(N[(1.0 / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.284496736 + N[(t$95$0 * 1.029667143), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.88], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4999999999999999e-17

   1. Initial program 98.6%

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

      1. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

      1. expm1-log1p-u 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. expm1-udef 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. log1p-udef 98.0%

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

      4. add-exp-log 98.0%

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

      5. +-commutative 98.0%

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

      6. fma-def 98.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 98.0%

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

   5. Applied egg-rr 98.1%

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

      1. fma-udef 98.0%

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

      2. associate--l+ 98.0%

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

      3. metadata-eval 98.0%

         \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. +-rgt-identity 98.0%

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

   7. Simplified 98.1%

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

      1. +-commutative 98.1%

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

      2. div-inv 98.1%

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

      3. fma-def 98.1%

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

      4. +-commutative 98.1%

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

      5. fma-def 98.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 98.1%

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

   9. Applied egg-rr 98.1%

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

   10. Taylor expanded in x around 0 98.1%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{1.029667143}\right)\right) \cdot e^{-x \cdot x}\right) \]
   11. Step-by-step derivation

       1. expm1-log1p-u 98.0%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. expm1-udef 98.0%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. log1p-udef 98.0%

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

       4. add-exp-log 98.0%

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

       5. +-commutative 98.0%

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

       6. fma-def 98.0%

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

       7. add-sqr-sqrt 0.0%

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

       8. fabs-sqr 0.0%

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

       9. add-sqr-sqrt 98.0%

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

   12. Applied egg-rr 98.1%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.029667143\right)\right) \cdot e^{-x \cdot x}\right) \]
   13. Step-by-step derivation

       1. fma-udef 98.0%

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

       2. associate--l+ 98.0%

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

       3. metadata-eval 98.0%

          \[\leadsto 1 - \frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. +-rgt-identity 98.0%

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

   14. Simplified 98.1%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.029667143\right)\right) \cdot e^{-x \cdot x}\right) \]

   if -2.4999999999999999e-17 < x < 0.880000000000000004

   1. Initial program 58.2%

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

      1. associate-*l* 58.2%

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

   3. Simplified 58.2%

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

      1. add-exp-log 58.2%

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

      2. sub-neg 58.2%

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

   5. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 57.1%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

      2. associate-/r/ 57.1%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      3. distribute-lft-neg-in 57.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      4. exp-prod 57.1%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

   7. Simplified 57.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

   8. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

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

   10. Simplified 99.0%

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

   if 0.880000000000000004 < 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. Step-by-step derivation

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. sub-neg 100.0%

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

   5. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

      2. associate-/r/ 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      3. distribute-lft-neg-in 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      4. exp-prod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

   7. Simplified 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;1 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 1.029667143\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 98.7% accurate, 93.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -8.8e-10)
   1.0
   (if (<= x 0.88) (+ 1e-9 (* x 1.128386358070218)) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -8.8e-10) {
		tmp = 1.0;
	} else if (x <= 0.88) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-8.8d-10)) then
        tmp = 1.0d0
    else if (x <= 0.88d0) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -8.8e-10) {
		tmp = 1.0;
	} else if (x <= 0.88) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -8.8e-10:
		tmp = 1.0
	elif x <= 0.88:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -8.8e-10)
		tmp = 1.0;
	elseif (x <= 0.88)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -8.8e-10)
		tmp = 1.0;
	elseif (x <= 0.88)
		tmp = 1e-9 + (x * 1.128386358070218);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -8.8e-10], 1.0, If[LessEqual[x, 0.88], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -8.7999999999999996e-10 or 0.880000000000000004 < 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. Step-by-step derivation

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. sub-neg 100.0%

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

   5. Applied egg-rr 1.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 1.5%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

      2. associate-/r/ 1.5%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      3. distribute-lft-neg-in 1.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      4. exp-prod 1.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

   7. Simplified 1.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1} \]

   if -8.7999999999999996e-10 < x < 0.880000000000000004

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

   3. Simplified 58.1%

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

      1. add-exp-log 58.1%

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

      2. sub-neg 58.1%

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

   5. Applied egg-rr 56.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 56.7%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

      2. associate-/r/ 56.7%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      3. distribute-lft-neg-in 56.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      4. exp-prod 56.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

   7. Simplified 56.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

   8. Taylor expanded in x around 0 97.9%

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

      1. *-commutative 97.9%

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

   10. Simplified 97.9%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 97.8% accurate, 167.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.85e-5) 1.0 (if (<= x 2.8e-5) 1e-9 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -2.85e-5) {
		tmp = 1.0;
	} else if (x <= 2.8e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.85d-5)) then
        tmp = 1.0d0
    else if (x <= 2.8d-5) then
        tmp = 1d-9
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.85e-5) {
		tmp = 1.0;
	} else if (x <= 2.8e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.85e-5:
		tmp = 1.0
	elif x <= 2.8e-5:
		tmp = 1e-9
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.85e-5)
		tmp = 1.0;
	elseif (x <= 2.8e-5)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.85e-5)
		tmp = 1.0;
	elseif (x <= 2.8e-5)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.85e-5], 1.0, If[LessEqual[x, 2.8e-5], 1e-9, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8500000000000002e-5 or 2.79999999999999996e-5 < x

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

      1. add-exp-log 99.9%

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

      2. sub-neg 99.9%

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

   5. Applied egg-rr 1.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 1.9%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

      2. associate-/r/ 1.9%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      3. distribute-lft-neg-in 1.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      4. exp-prod 1.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

   7. Simplified 1.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

   8. Taylor expanded in x around inf 98.8%

      \[\leadsto \color{blue}{1} \]

   if -2.8500000000000002e-5 < x < 2.79999999999999996e-5

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

   3. Simplified 57.6%

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

      1. add-exp-log 57.6%

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

      2. sub-neg 57.6%

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

   5. Applied egg-rr 57.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 57.2%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

      2. associate-/r/ 57.2%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      3. distribute-lft-neg-in 57.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

      4. exp-prod 57.2%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

   7. Simplified 57.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

   8. Taylor expanded in x around 0 97.3%

      \[\leadsto \color{blue}{10^{-9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

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

Alternative 12: 52.7% accurate, 856.0× speedup

Math

\[\begin{array}{l} \\ 10^{-9} \end{array} \]

FPCore

(FPCore (x) :precision binary64 1e-9)

C

double code(double x) {
	return 1e-9;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1d-9
end function

Java

public static double code(double x) {
	return 1e-9;
}

Python

def code(x):
	return 1e-9

Julia

function code(x)
	return 1e-9
end

MATLAB

function tmp = code(x)
	tmp = 1e-9;
end

Wolfram

code[x_] := 1e-9

Derivation

1. Initial program 80.1%

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

   1. associate-*l* 80.1%

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

3. Simplified 80.1%

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

   1. add-exp-log 80.1%

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

   2. sub-neg 80.1%

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

5. Applied egg-rr 27.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}} \]
6. Step-by-step derivation

   1. associate-/l* 27.8%

      \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}}\right)} \]

   2. associate-/r/ 27.8%

      \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot {\left(e^{x}\right)}^{x}}\right)} \]

   3. distribute-lft-neg-in 27.8%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]

   4. exp-prod 27.8%

      \[\leadsto e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot \color{blue}{e^{x \cdot x}}\right)} \]

7. Simplified 27.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{x \cdot x}\right)}} \]

8. Taylor expanded in x around 0 51.5%

   \[\leadsto \color{blue}{10^{-9}} \]

9. Final simplification 51.5%

   \[\leadsto 10^{-9} \]

Reproduce

herbie shell --seed 2023172 
(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)))))))