Jmat.Real.erf

Percentage Accurate: 78.9% → 99.8%

Time: 19.4s

Alternatives: 15

Speedup: 142.3×

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: 78.9% 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: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := -0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}\\ t_1 := \mathsf{fma}\left(\frac{t\_0}{1 + {x\_m}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x\_m, -0.3275911, 1\right), 0.254829592\right)\\ t_2 := e^{-{x\_m}^{2}}\\ t_3 := \frac{t\_2}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}\\ t_4 := {\left(t\_1 \cdot t\_3\right)}^{2} + \mathsf{fma}\left(t\_2, \frac{t\_1}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}, 1\right)\\ \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + x\_m \cdot \left(3.385159067440336 + x\_m \cdot \left(x\_m \cdot 0.3111712305105463 - 3.820122044248399\right)\right)}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(t\_3 \cdot \mathsf{fma}\left(\frac{t\_0}{\mathsf{fma}\left({x\_m}^{2}, -0.10731592879921, 1\right)}, \mathsf{fma}\left(x\_m, -0.3275911, 1\right), 0.254829592\right)\right)}^{3}\right)\right)}{t\_4}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (+
          -0.284496736
          (/
           (+
            1.421413741
            (/
             (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
             (fma x_m 0.3275911 1.0)))
           (fma x_m 0.3275911 1.0))))
        (t_1
         (fma
          (/ t_0 (+ 1.0 (* (pow x_m 2.0) -0.10731592879921)))
          (fma x_m -0.3275911 1.0)
          0.254829592))
        (t_2 (exp (- (pow x_m 2.0))))
        (t_3 (/ t_2 (fma x_m 0.3275911 1.0)))
        (t_4
         (+
          (pow (* t_1 t_3) 2.0)
          (fma t_2 (/ t_1 (fma x_m 0.3275911 1.0)) 1.0))))
   (if (<= (fabs x_m) 2e-6)
     (/
      (+
       2.999999997e-9
       (*
        x_m
        (+
         3.385159067440336
         (* x_m (- (* x_m 0.3111712305105463) 3.820122044248399)))))
      t_4)
     (/
      (-
       1.0
       (expm1
        (log1p
         (pow
          (*
           t_3
           (fma
            (/ t_0 (fma (pow x_m 2.0) -0.10731592879921 1.0))
            (fma x_m -0.3275911 1.0)
            0.254829592))
          3.0))))
      t_4))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = -0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0));
	double t_1 = fma((t_0 / (1.0 + (pow(x_m, 2.0) * -0.10731592879921))), fma(x_m, -0.3275911, 1.0), 0.254829592);
	double t_2 = exp(-pow(x_m, 2.0));
	double t_3 = t_2 / fma(x_m, 0.3275911, 1.0);
	double t_4 = pow((t_1 * t_3), 2.0) + fma(t_2, (t_1 / fma(x_m, 0.3275911, 1.0)), 1.0);
	double tmp;
	if (fabs(x_m) <= 2e-6) {
		tmp = (2.999999997e-9 + (x_m * (3.385159067440336 + (x_m * ((x_m * 0.3111712305105463) - 3.820122044248399))))) / t_4;
	} else {
		tmp = (1.0 - expm1(log1p(pow((t_3 * fma((t_0 / fma(pow(x_m, 2.0), -0.10731592879921, 1.0)), fma(x_m, -0.3275911, 1.0), 0.254829592)), 3.0)))) / t_4;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0)))
	t_1 = fma(Float64(t_0 / Float64(1.0 + Float64((x_m ^ 2.0) * -0.10731592879921))), fma(x_m, -0.3275911, 1.0), 0.254829592)
	t_2 = exp(Float64(-(x_m ^ 2.0)))
	t_3 = Float64(t_2 / fma(x_m, 0.3275911, 1.0))
	t_4 = Float64((Float64(t_1 * t_3) ^ 2.0) + fma(t_2, Float64(t_1 / fma(x_m, 0.3275911, 1.0)), 1.0))
	tmp = 0.0
	if (abs(x_m) <= 2e-6)
		tmp = Float64(Float64(2.999999997e-9 + Float64(x_m * Float64(3.385159067440336 + Float64(x_m * Float64(Float64(x_m * 0.3111712305105463) - 3.820122044248399))))) / t_4);
	else
		tmp = Float64(Float64(1.0 - expm1(log1p((Float64(t_3 * fma(Float64(t_0 / fma((x_m ^ 2.0), -0.10731592879921, 1.0)), fma(x_m, -0.3275911, 1.0), 0.254829592)) ^ 3.0)))) / t_4);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.10731592879921), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * -0.3275911 + 1.0), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(t$95$1 * t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$2 * N[(t$95$1 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-6], N[(N[(2.999999997e-9 + N[(x$95$m * N[(3.385159067440336 + N[(x$95$m * N[(N[(x$95$m * 0.3111712305105463), $MachinePrecision] - 3.820122044248399), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(1.0 - N[(Exp[N[Log[1 + N[Power[N[(t$95$3 * N[(N[(t$95$0 / N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.10731592879921 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * -0.3275911 + 1.0), $MachinePrecision] + 0.254829592), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.99999999999999991e-6

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

      1. add-sqr-sqrt 28.1%

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

      2. fabs-sqr 28.1%

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

      3. add-sqr-sqrt 57.0%

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

      4. expm1-log1p-u 57.0%

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

      5. expm1-undefine 57.0%

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

   6. Applied egg-rr 57.0%

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

      1. flip-+ 57.1%

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

      2. metadata-eval 57.1%

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

      3. pow2 57.1%

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

   8. Applied egg-rr 57.1%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot \left|x\right|\right)}^{2}}{1 - 0.3275911 \cdot \left|x\right|}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. rem-square-sqrt 28.1%

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

      2. fabs-sqr 28.1%

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

      3. rem-square-sqrt 57.1%

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

      4. cancel-sign-sub-inv 57.1%

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

      5. metadata-eval 57.1%

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

      6. rem-square-sqrt 28.1%

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

      7. fabs-sqr 28.1%

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

      8. rem-square-sqrt 56.8%

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

      9. *-commutative 56.8%

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

   10. Simplified 56.8%

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

   12. Applied egg-rr 56.9%

       \[\leadsto \color{blue}{\frac{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{3}}{1 + \left({\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{2} + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}} \]

   14. Simplified 56.9%

       \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}} \]

   15. Taylor expanded in x around 0 97.8%

       \[\leadsto \frac{\color{blue}{2.999999997 \cdot 10^{-9} + x \cdot \left(3.385159067440336 + x \cdot \left(0.3111712305105463 \cdot x - 3.820122044248399\right)\right)}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]

   if 1.99999999999999991e-6 < (fabs.f64 x)

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 52.6%

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

      2. fabs-sqr 52.6%

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

      3. add-sqr-sqrt 99.2%

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

      4. expm1-log1p-u 52.6%

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

      5. expm1-undefine 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. flip-+ 52.6%

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

      2. metadata-eval 52.6%

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

      3. pow2 52.6%

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

   8. Applied egg-rr 52.6%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot \left|x\right|\right)}^{2}}{1 - 0.3275911 \cdot \left|x\right|}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. rem-square-sqrt 52.6%

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

      2. fabs-sqr 52.6%

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

      3. rem-square-sqrt 52.6%

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

      4. cancel-sign-sub-inv 52.6%

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

      5. metadata-eval 52.6%

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

      6. rem-square-sqrt 52.6%

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

      7. fabs-sqr 52.6%

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

      8. rem-square-sqrt 52.6%

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

      9. *-commutative 52.6%

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

   10. Simplified 52.6%

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

   12. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\frac{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{3}}{1 + \left({\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{2} + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}} \]

   14. Simplified 99.1%

       \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}} \]
   15. Step-by-step derivation

       1. expm1-log1p-u 99.0%

          \[\leadsto \frac{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}\right)\right)}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]

       2. expm1-undefine 99.0%

          \[\leadsto \frac{1 - \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}\right)} - 1\right)}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]

   16. Applied egg-rr 99.0%

       \[\leadsto \frac{1 - \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left({x}^{2}, -0.10731592879921, 1\right)}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}\right)} - 1\right)}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]
   17. Step-by-step derivation

       1. expm1-define 99.0%

          \[\leadsto \frac{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left({x}^{2}, -0.10731592879921, 1\right)}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}\right)\right)}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]

   18. Simplified 99.0%

       \[\leadsto \frac{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left({x}^{2}, -0.10731592879921, 1\right)}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}\right)\right)}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + x \cdot \left(3.385159067440336 + x \cdot \left(x \cdot 0.3111712305105463 - 3.820122044248399\right)\right)}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot \mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left({x}^{2}, -0.10731592879921, 1\right)}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)\right)}^{3}\right)\right)}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{-{x\_m}^{2}}\\ t_1 := 1 + {x\_m}^{2} \cdot -0.10731592879921\\ t_2 := \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}\\ t_3 := \mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + t\_2}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{t\_1}, \mathsf{fma}\left(x\_m, -0.3275911, 1\right), 0.254829592\right)\\ t_4 := \mathsf{fma}\left(t\_0, \frac{t\_3}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}, 1\right)\\ t_5 := \frac{t\_0}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}\\ t_6 := t\_3 \cdot t\_5\\ \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + x\_m \cdot \left(3.385159067440336 + x\_m \cdot \left(x\_m \cdot 0.3111712305105463 - 3.820122044248399\right)\right)}{{t\_6}^{2} + t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t\_6}^{3}}{t\_4 + {\left(t\_5 \cdot \mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + {\left(\sqrt[3]{t\_2}\right)}^{3}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{t\_1}, \mathsf{fma}\left(x\_m, -0.3275911, 1\right), 0.254829592\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (exp (- (pow x_m 2.0))))
        (t_1 (+ 1.0 (* (pow x_m 2.0) -0.10731592879921)))
        (t_2
         (/
          (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
          (fma x_m 0.3275911 1.0)))
        (t_3
         (fma
          (/
           (+ -0.284496736 (/ (+ 1.421413741 t_2) (fma x_m 0.3275911 1.0)))
           t_1)
          (fma x_m -0.3275911 1.0)
          0.254829592))
        (t_4 (fma t_0 (/ t_3 (fma x_m 0.3275911 1.0)) 1.0))
        (t_5 (/ t_0 (fma x_m 0.3275911 1.0)))
        (t_6 (* t_3 t_5)))
   (if (<= (fabs x_m) 2e-6)
     (/
      (+
       2.999999997e-9
       (*
        x_m
        (+
         3.385159067440336
         (* x_m (- (* x_m 0.3111712305105463) 3.820122044248399)))))
      (+ (pow t_6 2.0) t_4))
     (/
      (- 1.0 (pow t_6 3.0))
      (+
       t_4
       (pow
        (*
         t_5
         (fma
          (/
           (+
            -0.284496736
            (/ (+ 1.421413741 (pow (cbrt t_2) 3.0)) (fma x_m 0.3275911 1.0)))
           t_1)
          (fma x_m -0.3275911 1.0)
          0.254829592))
        2.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = exp(-pow(x_m, 2.0));
	double t_1 = 1.0 + (pow(x_m, 2.0) * -0.10731592879921);
	double t_2 = (-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0);
	double t_3 = fma(((-0.284496736 + ((1.421413741 + t_2) / fma(x_m, 0.3275911, 1.0))) / t_1), fma(x_m, -0.3275911, 1.0), 0.254829592);
	double t_4 = fma(t_0, (t_3 / fma(x_m, 0.3275911, 1.0)), 1.0);
	double t_5 = t_0 / fma(x_m, 0.3275911, 1.0);
	double t_6 = t_3 * t_5;
	double tmp;
	if (fabs(x_m) <= 2e-6) {
		tmp = (2.999999997e-9 + (x_m * (3.385159067440336 + (x_m * ((x_m * 0.3111712305105463) - 3.820122044248399))))) / (pow(t_6, 2.0) + t_4);
	} else {
		tmp = (1.0 - pow(t_6, 3.0)) / (t_4 + pow((t_5 * fma(((-0.284496736 + ((1.421413741 + pow(cbrt(t_2), 3.0)) / fma(x_m, 0.3275911, 1.0))) / t_1), fma(x_m, -0.3275911, 1.0), 0.254829592)), 2.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = exp(Float64(-(x_m ^ 2.0)))
	t_1 = Float64(1.0 + Float64((x_m ^ 2.0) * -0.10731592879921))
	t_2 = Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))
	t_3 = fma(Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + t_2) / fma(x_m, 0.3275911, 1.0))) / t_1), fma(x_m, -0.3275911, 1.0), 0.254829592)
	t_4 = fma(t_0, Float64(t_3 / fma(x_m, 0.3275911, 1.0)), 1.0)
	t_5 = Float64(t_0 / fma(x_m, 0.3275911, 1.0))
	t_6 = Float64(t_3 * t_5)
	tmp = 0.0
	if (abs(x_m) <= 2e-6)
		tmp = Float64(Float64(2.999999997e-9 + Float64(x_m * Float64(3.385159067440336 + Float64(x_m * Float64(Float64(x_m * 0.3111712305105463) - 3.820122044248399))))) / Float64((t_6 ^ 2.0) + t_4));
	else
		tmp = Float64(Float64(1.0 - (t_6 ^ 3.0)) / Float64(t_4 + (Float64(t_5 * fma(Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + (cbrt(t_2) ^ 3.0)) / fma(x_m, 0.3275911, 1.0))) / t_1), fma(x_m, -0.3275911, 1.0), 0.254829592)) ^ 2.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.10731592879921), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-0.284496736 + N[(N[(1.421413741 + t$95$2), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(x$95$m * -0.3275911 + 1.0), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * N[(t$95$3 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 * t$95$5), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-6], N[(N[(2.999999997e-9 + N[(x$95$m * N[(3.385159067440336 + N[(x$95$m * N[(N[(x$95$m * 0.3111712305105463), $MachinePrecision] - 3.820122044248399), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$6, 2.0], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$6, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$4 + N[Power[N[(t$95$5 * N[(N[(N[(-0.284496736 + N[(N[(1.421413741 + N[Power[N[Power[t$95$2, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(x$95$m * -0.3275911 + 1.0), $MachinePrecision] + 0.254829592), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.99999999999999991e-6

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

      1. add-sqr-sqrt 28.1%

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

      2. fabs-sqr 28.1%

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

      3. add-sqr-sqrt 57.0%

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

      4. expm1-log1p-u 57.0%

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

      5. expm1-undefine 57.0%

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

   6. Applied egg-rr 57.0%

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

      1. flip-+ 57.1%

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

      2. metadata-eval 57.1%

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

      3. pow2 57.1%

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

   8. Applied egg-rr 57.1%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot \left|x\right|\right)}^{2}}{1 - 0.3275911 \cdot \left|x\right|}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. rem-square-sqrt 28.1%

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

      2. fabs-sqr 28.1%

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

      3. rem-square-sqrt 57.1%

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

      4. cancel-sign-sub-inv 57.1%

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

      5. metadata-eval 57.1%

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

      6. rem-square-sqrt 28.1%

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

      7. fabs-sqr 28.1%

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

      8. rem-square-sqrt 56.8%

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

      9. *-commutative 56.8%

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

   10. Simplified 56.8%

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

   12. Applied egg-rr 56.9%

       \[\leadsto \color{blue}{\frac{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{3}}{1 + \left({\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{2} + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}} \]

   14. Simplified 56.9%

       \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}} \]

   15. Taylor expanded in x around 0 97.8%

       \[\leadsto \frac{\color{blue}{2.999999997 \cdot 10^{-9} + x \cdot \left(3.385159067440336 + x \cdot \left(0.3111712305105463 \cdot x - 3.820122044248399\right)\right)}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]

   if 1.99999999999999991e-6 < (fabs.f64 x)

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 52.6%

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

      2. fabs-sqr 52.6%

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

      3. add-sqr-sqrt 99.2%

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

      4. expm1-log1p-u 52.6%

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

      5. expm1-undefine 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. flip-+ 52.6%

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

      2. metadata-eval 52.6%

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

      3. pow2 52.6%

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

   8. Applied egg-rr 52.6%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot \left|x\right|\right)}^{2}}{1 - 0.3275911 \cdot \left|x\right|}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. rem-square-sqrt 52.6%

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

      2. fabs-sqr 52.6%

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

      3. rem-square-sqrt 52.6%

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

      4. cancel-sign-sub-inv 52.6%

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

      5. metadata-eval 52.6%

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

      6. rem-square-sqrt 52.6%

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

      7. fabs-sqr 52.6%

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

      8. rem-square-sqrt 52.6%

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

      9. *-commutative 52.6%

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

   10. Simplified 52.6%

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

   12. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\frac{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{3}}{1 + \left({\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{2} + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}} \]

   14. Simplified 99.1%

       \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}} \]
   15. Step-by-step derivation

       1. add-cube-cbrt 99.1%

          \[\leadsto \frac{1 - {\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \color{blue}{\left(\sqrt[3]{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}} \cdot \sqrt[3]{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}\right) \cdot \sqrt[3]{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]

       2. pow3 99.1%

          \[\leadsto \frac{1 - {\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \color{blue}{{\left(\sqrt[3]{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}\right)}^{3}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]

   16. Applied egg-rr 99.1%

       \[\leadsto \frac{1 - {\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \color{blue}{{\left(\sqrt[3]{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}\right)}^{3}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + x \cdot \left(3.385159067440336 + x \cdot \left(x \cdot 0.3111712305105463 - 3.820122044248399\right)\right)}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}}{\mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right) + {\left(\frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot \mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + {\left(\sqrt[3]{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}\right)}^{3}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{1 + {x\_m}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x\_m, -0.3275911, 1\right), 0.254829592\right)\\ t_1 := e^{-{x\_m}^{2}}\\ t_2 := t\_0 \cdot \frac{t\_1}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}\\ t_3 := {t\_2}^{2} + \mathsf{fma}\left(t\_1, \frac{t\_0}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}, 1\right)\\ \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + x\_m \cdot \left(3.385159067440336 + x\_m \cdot \left(x\_m \cdot 0.3111712305105463 - 3.820122044248399\right)\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t\_2}^{3}}{t\_3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (fma
          (/
           (+
            -0.284496736
            (/
             (+
              1.421413741
              (/
               (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
               (fma x_m 0.3275911 1.0)))
             (fma x_m 0.3275911 1.0)))
           (+ 1.0 (* (pow x_m 2.0) -0.10731592879921)))
          (fma x_m -0.3275911 1.0)
          0.254829592))
        (t_1 (exp (- (pow x_m 2.0))))
        (t_2 (* t_0 (/ t_1 (fma x_m 0.3275911 1.0))))
        (t_3 (+ (pow t_2 2.0) (fma t_1 (/ t_0 (fma x_m 0.3275911 1.0)) 1.0))))
   (if (<= (fabs x_m) 2e-6)
     (/
      (+
       2.999999997e-9
       (*
        x_m
        (+
         3.385159067440336
         (* x_m (- (* x_m 0.3111712305105463) 3.820122044248399)))))
      t_3)
     (/ (- 1.0 (pow t_2 3.0)) t_3))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / (1.0 + (pow(x_m, 2.0) * -0.10731592879921))), fma(x_m, -0.3275911, 1.0), 0.254829592);
	double t_1 = exp(-pow(x_m, 2.0));
	double t_2 = t_0 * (t_1 / fma(x_m, 0.3275911, 1.0));
	double t_3 = pow(t_2, 2.0) + fma(t_1, (t_0 / fma(x_m, 0.3275911, 1.0)), 1.0);
	double tmp;
	if (fabs(x_m) <= 2e-6) {
		tmp = (2.999999997e-9 + (x_m * (3.385159067440336 + (x_m * ((x_m * 0.3111712305105463) - 3.820122044248399))))) / t_3;
	} else {
		tmp = (1.0 - pow(t_2, 3.0)) / t_3;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / Float64(1.0 + Float64((x_m ^ 2.0) * -0.10731592879921))), fma(x_m, -0.3275911, 1.0), 0.254829592)
	t_1 = exp(Float64(-(x_m ^ 2.0)))
	t_2 = Float64(t_0 * Float64(t_1 / fma(x_m, 0.3275911, 1.0)))
	t_3 = Float64((t_2 ^ 2.0) + fma(t_1, Float64(t_0 / fma(x_m, 0.3275911, 1.0)), 1.0))
	tmp = 0.0
	if (abs(x_m) <= 2e-6)
		tmp = Float64(Float64(2.999999997e-9 + Float64(x_m * Float64(3.385159067440336 + Float64(x_m * Float64(Float64(x_m * 0.3111712305105463) - 3.820122044248399))))) / t_3);
	else
		tmp = Float64(Float64(1.0 - (t_2 ^ 3.0)) / t_3);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.10731592879921), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * -0.3275911 + 1.0), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[t$95$2, 2.0], $MachinePrecision] + N[(t$95$1 * N[(t$95$0 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-6], N[(N[(2.999999997e-9 + N[(x$95$m * N[(3.385159067440336 + N[(x$95$m * N[(N[(x$95$m * 0.3111712305105463), $MachinePrecision] - 3.820122044248399), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.99999999999999991e-6

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

      1. add-sqr-sqrt 28.1%

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

      2. fabs-sqr 28.1%

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

      3. add-sqr-sqrt 57.0%

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

      4. expm1-log1p-u 57.0%

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

      5. expm1-undefine 57.0%

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

   6. Applied egg-rr 57.0%

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

      1. flip-+ 57.1%

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

      2. metadata-eval 57.1%

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

      3. pow2 57.1%

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

   8. Applied egg-rr 57.1%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot \left|x\right|\right)}^{2}}{1 - 0.3275911 \cdot \left|x\right|}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. rem-square-sqrt 28.1%

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

      2. fabs-sqr 28.1%

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

      3. rem-square-sqrt 57.1%

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

      4. cancel-sign-sub-inv 57.1%

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

      5. metadata-eval 57.1%

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

      6. rem-square-sqrt 28.1%

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

      7. fabs-sqr 28.1%

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

      8. rem-square-sqrt 56.8%

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

      9. *-commutative 56.8%

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

   10. Simplified 56.8%

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

   12. Applied egg-rr 56.9%

       \[\leadsto \color{blue}{\frac{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{3}}{1 + \left({\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{2} + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}} \]

   14. Simplified 56.9%

       \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}} \]

   15. Taylor expanded in x around 0 97.8%

       \[\leadsto \frac{\color{blue}{2.999999997 \cdot 10^{-9} + x \cdot \left(3.385159067440336 + x \cdot \left(0.3111712305105463 \cdot x - 3.820122044248399\right)\right)}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]

   if 1.99999999999999991e-6 < (fabs.f64 x)

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 52.6%

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

      2. fabs-sqr 52.6%

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

      3. add-sqr-sqrt 99.2%

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

      4. expm1-log1p-u 52.6%

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

      5. expm1-undefine 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. flip-+ 52.6%

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

      2. metadata-eval 52.6%

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

      3. pow2 52.6%

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

   8. Applied egg-rr 52.6%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot \left|x\right|\right)}^{2}}{1 - 0.3275911 \cdot \left|x\right|}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. rem-square-sqrt 52.6%

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

      2. fabs-sqr 52.6%

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

      3. rem-square-sqrt 52.6%

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

      4. cancel-sign-sub-inv 52.6%

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

      5. metadata-eval 52.6%

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

      6. rem-square-sqrt 52.6%

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

      7. fabs-sqr 52.6%

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

      8. rem-square-sqrt 52.6%

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

      9. *-commutative 52.6%

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

   10. Simplified 52.6%

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

   12. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\frac{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{3}}{1 + \left({\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{2} + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}} \]

   14. Simplified 99.1%

       \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + x \cdot \left(3.385159067440336 + x \cdot \left(x \cdot 0.3111712305105463 - 3.820122044248399\right)\right)}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{1 + {x\_m}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x\_m, -0.3275911, 1\right), 0.254829592\right)\\ t_1 := \left|x\_m\right| \cdot 0.3275911\\ t_2 := 1 + t\_1\\ t_3 := e^{-{x\_m}^{2}}\\ \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + x\_m \cdot \left(3.385159067440336 + x\_m \cdot \left(x\_m \cdot 0.3111712305105463 - 3.820122044248399\right)\right)}{{\left(t\_0 \cdot \frac{t\_3}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(t\_3, \frac{t\_0}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(e^{-\mathsf{log1p}\left(x\_m \cdot 0.3275911\right)} \cdot \left(\frac{1}{\frac{1 - {\left(x\_m \cdot 0.3275911\right)}^{2}}{1 + x\_m \cdot -0.3275911}} \cdot \left(\frac{1}{t\_2} \cdot \left(\left(-1.453152027 + \frac{1.061405429}{t\_2}\right) \cdot \frac{1}{-1 - t\_1} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (fma
          (/
           (+
            -0.284496736
            (/
             (+
              1.421413741
              (/
               (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
               (fma x_m 0.3275911 1.0)))
             (fma x_m 0.3275911 1.0)))
           (+ 1.0 (* (pow x_m 2.0) -0.10731592879921)))
          (fma x_m -0.3275911 1.0)
          0.254829592))
        (t_1 (* (fabs x_m) 0.3275911))
        (t_2 (+ 1.0 t_1))
        (t_3 (exp (- (pow x_m 2.0)))))
   (if (<= (fabs x_m) 2e-6)
     (/
      (+
       2.999999997e-9
       (*
        x_m
        (+
         3.385159067440336
         (* x_m (- (* x_m 0.3111712305105463) 3.820122044248399)))))
      (+
       (pow (* t_0 (/ t_3 (fma x_m 0.3275911 1.0))) 2.0)
       (fma t_3 (/ t_0 (fma x_m 0.3275911 1.0)) 1.0)))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (exp (- (log1p (* x_m 0.3275911))))
        (-
         (*
          (/
           1.0
           (/ (- 1.0 (pow (* x_m 0.3275911) 2.0)) (+ 1.0 (* x_m -0.3275911))))
          (-
           (*
            (/ 1.0 t_2)
            (-
             (* (+ -1.453152027 (/ 1.061405429 t_2)) (/ 1.0 (- -1.0 t_1)))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / (1.0 + (pow(x_m, 2.0) * -0.10731592879921))), fma(x_m, -0.3275911, 1.0), 0.254829592);
	double t_1 = fabs(x_m) * 0.3275911;
	double t_2 = 1.0 + t_1;
	double t_3 = exp(-pow(x_m, 2.0));
	double tmp;
	if (fabs(x_m) <= 2e-6) {
		tmp = (2.999999997e-9 + (x_m * (3.385159067440336 + (x_m * ((x_m * 0.3111712305105463) - 3.820122044248399))))) / (pow((t_0 * (t_3 / fma(x_m, 0.3275911, 1.0))), 2.0) + fma(t_3, (t_0 / fma(x_m, 0.3275911, 1.0)), 1.0));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * (exp(-log1p((x_m * 0.3275911))) * (((1.0 / ((1.0 - pow((x_m * 0.3275911), 2.0)) / (1.0 + (x_m * -0.3275911)))) * (((1.0 / t_2) * (((-1.453152027 + (1.061405429 / t_2)) * (1.0 / (-1.0 - t_1))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / Float64(1.0 + Float64((x_m ^ 2.0) * -0.10731592879921))), fma(x_m, -0.3275911, 1.0), 0.254829592)
	t_1 = Float64(abs(x_m) * 0.3275911)
	t_2 = Float64(1.0 + t_1)
	t_3 = exp(Float64(-(x_m ^ 2.0)))
	tmp = 0.0
	if (abs(x_m) <= 2e-6)
		tmp = Float64(Float64(2.999999997e-9 + Float64(x_m * Float64(3.385159067440336 + Float64(x_m * Float64(Float64(x_m * 0.3111712305105463) - 3.820122044248399))))) / Float64((Float64(t_0 * Float64(t_3 / fma(x_m, 0.3275911, 1.0))) ^ 2.0) + fma(t_3, Float64(t_0 / fma(x_m, 0.3275911, 1.0)), 1.0)));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(exp(Float64(-log1p(Float64(x_m * 0.3275911)))) * Float64(Float64(Float64(1.0 / Float64(Float64(1.0 - (Float64(x_m * 0.3275911) ^ 2.0)) / Float64(1.0 + Float64(x_m * -0.3275911)))) * Float64(Float64(Float64(1.0 / t_2) * Float64(Float64(Float64(-1.453152027 + Float64(1.061405429 / t_2)) * Float64(1.0 / Float64(-1.0 - t_1))) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.10731592879921), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * -0.3275911 + 1.0), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-6], N[(N[(2.999999997e-9 + N[(x$95$m * N[(3.385159067440336 + N[(x$95$m * N[(N[(x$95$m * 0.3111712305105463), $MachinePrecision] - 3.820122044248399), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(t$95$0 * N[(t$95$3 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$3 * N[(t$95$0 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[(-N[Log[1 + N[(x$95$m * 0.3275911), $MachinePrecision]], $MachinePrecision])], $MachinePrecision] * N[(N[(N[(1.0 / N[(N[(1.0 - N[Power[N[(x$95$m * 0.3275911), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x$95$m * -0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(N[(N[(-1.453152027 + N[(1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.99999999999999991e-6

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

      1. add-sqr-sqrt 28.1%

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

      2. fabs-sqr 28.1%

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

      3. add-sqr-sqrt 57.0%

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

      4. expm1-log1p-u 57.0%

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

      5. expm1-undefine 57.0%

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

   6. Applied egg-rr 57.0%

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

      1. flip-+ 57.1%

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

      2. metadata-eval 57.1%

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

      3. pow2 57.1%

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

   8. Applied egg-rr 57.1%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot \left|x\right|\right)}^{2}}{1 - 0.3275911 \cdot \left|x\right|}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. rem-square-sqrt 28.1%

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

      2. fabs-sqr 28.1%

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

      3. rem-square-sqrt 57.1%

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

      4. cancel-sign-sub-inv 57.1%

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

      5. metadata-eval 57.1%

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

      6. rem-square-sqrt 28.1%

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

      7. fabs-sqr 28.1%

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

      8. rem-square-sqrt 56.8%

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

      9. *-commutative 56.8%

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

   10. Simplified 56.8%

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

   12. Applied egg-rr 56.9%

       \[\leadsto \color{blue}{\frac{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{3}}{1 + \left({\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{2} + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}} \]

   14. Simplified 56.9%

       \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}} \]

   15. Taylor expanded in x around 0 97.8%

       \[\leadsto \frac{\color{blue}{2.999999997 \cdot 10^{-9} + x \cdot \left(3.385159067440336 + x \cdot \left(0.3111712305105463 \cdot x - 3.820122044248399\right)\right)}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]

   if 1.99999999999999991e-6 < (fabs.f64 x)

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 52.6%

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

      2. fabs-sqr 52.6%

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

      3. add-sqr-sqrt 99.2%

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

      4. expm1-log1p-u 52.6%

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

      5. expm1-undefine 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. flip-+ 52.6%

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

      2. metadata-eval 52.6%

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

      3. pow2 52.6%

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

   8. Applied egg-rr 52.6%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot \left|x\right|\right)}^{2}}{1 - 0.3275911 \cdot \left|x\right|}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. rem-square-sqrt 52.6%

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

      2. fabs-sqr 52.6%

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

      3. rem-square-sqrt 52.6%

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

      4. cancel-sign-sub-inv 52.6%

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

      5. metadata-eval 52.6%

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

      6. rem-square-sqrt 52.6%

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

      7. fabs-sqr 52.6%

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

      8. rem-square-sqrt 52.6%

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

      9. *-commutative 52.6%

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

   10. Simplified 52.6%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot x\right)}^{2}}{1 + x \cdot -0.3275911}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   11. Step-by-step derivation

       1. add-exp-log 52.6%

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

       2. +-commutative 52.6%

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

       3. expm1-define 52.6%

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

       4. expm1-log1p-u 52.6%

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

       5. fma-undefine 52.6%

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

       6. log-rec 52.6%

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

       7. fma-undefine 52.6%

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

       8. expm1-log1p-u 52.6%

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

       9. expm1-define 52.6%

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

       10. +-commutative 52.6%

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

       11. expm1-define 52.6%

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

       12. expm1-log1p-u 52.6%

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

       13. log1p-define 52.6%

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

   12. Applied egg-rr 52.6%

       \[\leadsto 1 - \left(\color{blue}{e^{-\mathsf{log1p}\left(0.3275911 \cdot x\right)}} \cdot \left(0.254829592 + \frac{1}{\frac{1 - {\left(0.3275911 \cdot x\right)}^{2}}{1 + x \cdot -0.3275911}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + x \cdot \left(3.385159067440336 + x \cdot \left(x \cdot 0.3111712305105463 - 3.820122044248399\right)\right)}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(e^{-\mathsf{log1p}\left(x \cdot 0.3275911\right)} \cdot \left(\frac{1}{\frac{1 - {\left(x \cdot 0.3275911\right)}^{2}}{1 + x \cdot -0.3275911}} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}\right) \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{1 + {x\_m}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x\_m, -0.3275911, 1\right), 0.254829592\right)\\ t_1 := \left|x\_m\right| \cdot 0.3275911\\ t_2 := 1 + t\_1\\ t_3 := e^{-{x\_m}^{2}}\\ \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + x\_m \cdot \left(3.385159067440336 + x\_m \cdot -3.820122044248399\right)}{{\left(t\_0 \cdot \frac{t\_3}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(t\_3, \frac{t\_0}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(e^{-\mathsf{log1p}\left(x\_m \cdot 0.3275911\right)} \cdot \left(\frac{1}{\frac{1 - {\left(x\_m \cdot 0.3275911\right)}^{2}}{1 + x\_m \cdot -0.3275911}} \cdot \left(\frac{1}{t\_2} \cdot \left(\left(-1.453152027 + \frac{1.061405429}{t\_2}\right) \cdot \frac{1}{-1 - t\_1} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (fma
          (/
           (+
            -0.284496736
            (/
             (+
              1.421413741
              (/
               (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
               (fma x_m 0.3275911 1.0)))
             (fma x_m 0.3275911 1.0)))
           (+ 1.0 (* (pow x_m 2.0) -0.10731592879921)))
          (fma x_m -0.3275911 1.0)
          0.254829592))
        (t_1 (* (fabs x_m) 0.3275911))
        (t_2 (+ 1.0 t_1))
        (t_3 (exp (- (pow x_m 2.0)))))
   (if (<= (fabs x_m) 2e-6)
     (/
      (+
       2.999999997e-9
       (* x_m (+ 3.385159067440336 (* x_m -3.820122044248399))))
      (+
       (pow (* t_0 (/ t_3 (fma x_m 0.3275911 1.0))) 2.0)
       (fma t_3 (/ t_0 (fma x_m 0.3275911 1.0)) 1.0)))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (exp (- (log1p (* x_m 0.3275911))))
        (-
         (*
          (/
           1.0
           (/ (- 1.0 (pow (* x_m 0.3275911) 2.0)) (+ 1.0 (* x_m -0.3275911))))
          (-
           (*
            (/ 1.0 t_2)
            (-
             (* (+ -1.453152027 (/ 1.061405429 t_2)) (/ 1.0 (- -1.0 t_1)))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / (1.0 + (pow(x_m, 2.0) * -0.10731592879921))), fma(x_m, -0.3275911, 1.0), 0.254829592);
	double t_1 = fabs(x_m) * 0.3275911;
	double t_2 = 1.0 + t_1;
	double t_3 = exp(-pow(x_m, 2.0));
	double tmp;
	if (fabs(x_m) <= 2e-6) {
		tmp = (2.999999997e-9 + (x_m * (3.385159067440336 + (x_m * -3.820122044248399)))) / (pow((t_0 * (t_3 / fma(x_m, 0.3275911, 1.0))), 2.0) + fma(t_3, (t_0 / fma(x_m, 0.3275911, 1.0)), 1.0));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * (exp(-log1p((x_m * 0.3275911))) * (((1.0 / ((1.0 - pow((x_m * 0.3275911), 2.0)) / (1.0 + (x_m * -0.3275911)))) * (((1.0 / t_2) * (((-1.453152027 + (1.061405429 / t_2)) * (1.0 / (-1.0 - t_1))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / Float64(1.0 + Float64((x_m ^ 2.0) * -0.10731592879921))), fma(x_m, -0.3275911, 1.0), 0.254829592)
	t_1 = Float64(abs(x_m) * 0.3275911)
	t_2 = Float64(1.0 + t_1)
	t_3 = exp(Float64(-(x_m ^ 2.0)))
	tmp = 0.0
	if (abs(x_m) <= 2e-6)
		tmp = Float64(Float64(2.999999997e-9 + Float64(x_m * Float64(3.385159067440336 + Float64(x_m * -3.820122044248399)))) / Float64((Float64(t_0 * Float64(t_3 / fma(x_m, 0.3275911, 1.0))) ^ 2.0) + fma(t_3, Float64(t_0 / fma(x_m, 0.3275911, 1.0)), 1.0)));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(exp(Float64(-log1p(Float64(x_m * 0.3275911)))) * Float64(Float64(Float64(1.0 / Float64(Float64(1.0 - (Float64(x_m * 0.3275911) ^ 2.0)) / Float64(1.0 + Float64(x_m * -0.3275911)))) * Float64(Float64(Float64(1.0 / t_2) * Float64(Float64(Float64(-1.453152027 + Float64(1.061405429 / t_2)) * Float64(1.0 / Float64(-1.0 - t_1))) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.10731592879921), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * -0.3275911 + 1.0), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-6], N[(N[(2.999999997e-9 + N[(x$95$m * N[(3.385159067440336 + N[(x$95$m * -3.820122044248399), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(t$95$0 * N[(t$95$3 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$3 * N[(t$95$0 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[(-N[Log[1 + N[(x$95$m * 0.3275911), $MachinePrecision]], $MachinePrecision])], $MachinePrecision] * N[(N[(N[(1.0 / N[(N[(1.0 - N[Power[N[(x$95$m * 0.3275911), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x$95$m * -0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(N[(N[(-1.453152027 + N[(1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.99999999999999991e-6

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

      1. add-sqr-sqrt 28.1%

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

      2. fabs-sqr 28.1%

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

      3. add-sqr-sqrt 57.0%

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

      4. expm1-log1p-u 57.0%

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

      5. expm1-undefine 57.0%

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

   6. Applied egg-rr 57.0%

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

      1. flip-+ 57.1%

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

      2. metadata-eval 57.1%

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

      3. pow2 57.1%

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

   8. Applied egg-rr 57.1%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot \left|x\right|\right)}^{2}}{1 - 0.3275911 \cdot \left|x\right|}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. rem-square-sqrt 28.1%

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

      2. fabs-sqr 28.1%

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

      3. rem-square-sqrt 57.1%

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

      4. cancel-sign-sub-inv 57.1%

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

      5. metadata-eval 57.1%

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

      6. rem-square-sqrt 28.1%

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

      7. fabs-sqr 28.1%

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

      8. rem-square-sqrt 56.8%

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

      9. *-commutative 56.8%

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

   10. Simplified 56.8%

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

   12. Applied egg-rr 56.9%

       \[\leadsto \color{blue}{\frac{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{3}}{1 + \left({\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}^{2} + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1 - {x}^{2} \cdot 0.10731592879921}{\mathsf{fma}\left(x, -0.3275911, 1\right)}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{-{x}^{2}}\right)}} \]

   14. Simplified 56.9%

       \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{3}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}} \]

   15. Taylor expanded in x around 0 97.8%

       \[\leadsto \frac{\color{blue}{2.999999997 \cdot 10^{-9} + x \cdot \left(3.385159067440336 + -3.820122044248399 \cdot x\right)}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]
   16. Step-by-step derivation

       1. *-commutative 97.8%

          \[\leadsto \frac{2.999999997 \cdot 10^{-9} + x \cdot \left(3.385159067440336 + \color{blue}{x \cdot -3.820122044248399}\right)}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]

   17. Simplified 97.8%

       \[\leadsto \frac{\color{blue}{2.999999997 \cdot 10^{-9} + x \cdot \left(3.385159067440336 + x \cdot -3.820122044248399\right)}}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)} \]

   if 1.99999999999999991e-6 < (fabs.f64 x)

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 52.6%

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

      2. fabs-sqr 52.6%

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

      3. add-sqr-sqrt 99.2%

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

      4. expm1-log1p-u 52.6%

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

      5. expm1-undefine 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. flip-+ 52.6%

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

      2. metadata-eval 52.6%

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

      3. pow2 52.6%

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

   8. Applied egg-rr 52.6%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot \left|x\right|\right)}^{2}}{1 - 0.3275911 \cdot \left|x\right|}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. rem-square-sqrt 52.6%

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

      2. fabs-sqr 52.6%

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

      3. rem-square-sqrt 52.6%

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

      4. cancel-sign-sub-inv 52.6%

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

      5. metadata-eval 52.6%

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

      6. rem-square-sqrt 52.6%

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

      7. fabs-sqr 52.6%

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

      8. rem-square-sqrt 52.6%

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

      9. *-commutative 52.6%

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

   10. Simplified 52.6%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot x\right)}^{2}}{1 + x \cdot -0.3275911}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   11. Step-by-step derivation

       1. add-exp-log 52.6%

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

       2. +-commutative 52.6%

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

       3. expm1-define 52.6%

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

       4. expm1-log1p-u 52.6%

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

       5. fma-undefine 52.6%

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

       6. log-rec 52.6%

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

       7. fma-undefine 52.6%

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

       8. expm1-log1p-u 52.6%

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

       9. expm1-define 52.6%

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

       10. +-commutative 52.6%

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

       11. expm1-define 52.6%

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

       12. expm1-log1p-u 52.6%

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

       13. log1p-define 52.6%

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

   12. Applied egg-rr 52.6%

       \[\leadsto 1 - \left(\color{blue}{e^{-\mathsf{log1p}\left(0.3275911 \cdot x\right)}} \cdot \left(0.254829592 + \frac{1}{\frac{1 - {\left(0.3275911 \cdot x\right)}^{2}}{1 + x \cdot -0.3275911}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + x \cdot \left(3.385159067440336 + x \cdot -3.820122044248399\right)}{{\left(\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right) \cdot \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} + \mathsf{fma}\left(e^{-{x}^{2}}, \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{1 + {x}^{2} \cdot -0.10731592879921}, \mathsf{fma}\left(x, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(e^{-\mathsf{log1p}\left(x \cdot 0.3275911\right)} \cdot \left(\frac{1}{\frac{1 - {\left(x \cdot 0.3275911\right)}^{2}}{1 + x \cdot -0.3275911}} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}\right) \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\left|x\_m\right| \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(e^{-\mathsf{log1p}\left(x\_m \cdot 0.3275911\right)} \cdot \left(\frac{1}{\frac{1 - {\left(x\_m \cdot 0.3275911\right)}^{2}}{1 + x\_m \cdot -0.3275911}} \cdot \left(\frac{1}{t\_1} \cdot \left(\left(-1.453152027 + \frac{1.061405429}{t\_1}\right) \cdot \frac{1}{-1 - t\_0} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)) (t_1 (+ 1.0 t_0)))
   (if (<= (fabs x_m) 1.5e-6)
     (+ 1e-9 (* x_m 1.128386358070218))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (exp (- (log1p (* x_m 0.3275911))))
        (-
         (*
          (/
           1.0
           (/ (- 1.0 (pow (* x_m 0.3275911) 2.0)) (+ 1.0 (* x_m -0.3275911))))
          (-
           (*
            (/ 1.0 t_1)
            (-
             (* (+ -1.453152027 (/ 1.061405429 t_1)) (/ 1.0 (- -1.0 t_0)))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double t_1 = 1.0 + t_0;
	double tmp;
	if (fabs(x_m) <= 1.5e-6) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * (exp(-log1p((x_m * 0.3275911))) * (((1.0 / ((1.0 - pow((x_m * 0.3275911), 2.0)) / (1.0 + (x_m * -0.3275911)))) * (((1.0 / t_1) * (((-1.453152027 + (1.061405429 / t_1)) * (1.0 / (-1.0 - t_0))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.abs(x_m) * 0.3275911;
	double t_1 = 1.0 + t_0;
	double tmp;
	if (Math.abs(x_m) <= 1.5e-6) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + (Math.exp((x_m * -x_m)) * (Math.exp(-Math.log1p((x_m * 0.3275911))) * (((1.0 / ((1.0 - Math.pow((x_m * 0.3275911), 2.0)) / (1.0 + (x_m * -0.3275911)))) * (((1.0 / t_1) * (((-1.453152027 + (1.061405429 / t_1)) * (1.0 / (-1.0 - t_0))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.fabs(x_m) * 0.3275911
	t_1 = 1.0 + t_0
	tmp = 0
	if math.fabs(x_m) <= 1.5e-6:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 1.0 + (math.exp((x_m * -x_m)) * (math.exp(-math.log1p((x_m * 0.3275911))) * (((1.0 / ((1.0 - math.pow((x_m * 0.3275911), 2.0)) / (1.0 + (x_m * -0.3275911)))) * (((1.0 / t_1) * (((-1.453152027 + (1.061405429 / t_1)) * (1.0 / (-1.0 - t_0))) - 1.421413741)) - -0.284496736)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	t_1 = Float64(1.0 + t_0)
	tmp = 0.0
	if (abs(x_m) <= 1.5e-6)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(exp(Float64(-log1p(Float64(x_m * 0.3275911)))) * Float64(Float64(Float64(1.0 / Float64(Float64(1.0 - (Float64(x_m * 0.3275911) ^ 2.0)) / Float64(1.0 + Float64(x_m * -0.3275911)))) * Float64(Float64(Float64(1.0 / t_1) * Float64(Float64(Float64(-1.453152027 + Float64(1.061405429 / t_1)) * Float64(1.0 / Float64(-1.0 - t_0))) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1.5e-6], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[(-N[Log[1 + N[(x$95$m * 0.3275911), $MachinePrecision]], $MachinePrecision])], $MachinePrecision] * N[(N[(N[(1.0 / N[(N[(1.0 - N[Power[N[(x$95$m * 0.3275911), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x$95$m * -0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(N[(N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.5e-6

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

   6. Applied egg-rr 52.5%

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

   7. Taylor expanded in x around 0 98.6%

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

      1. *-commutative 98.6%

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

   9. Simplified 98.6%

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

   if 1.5e-6 < (fabs.f64 x)

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

      1. add-sqr-sqrt 52.2%

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

      2. fabs-sqr 52.2%

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

      3. add-sqr-sqrt 98.6%

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

      4. expm1-log1p-u 52.3%

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

      5. expm1-undefine 52.3%

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

   6. Applied egg-rr 52.3%

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

      1. flip-+ 52.3%

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

      2. metadata-eval 52.3%

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

      3. pow2 52.3%

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

   8. Applied egg-rr 52.3%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot \left|x\right|\right)}^{2}}{1 - 0.3275911 \cdot \left|x\right|}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. rem-square-sqrt 52.2%

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

      2. fabs-sqr 52.2%

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

      3. rem-square-sqrt 52.3%

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

      4. cancel-sign-sub-inv 52.3%

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

      5. metadata-eval 52.3%

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

      6. rem-square-sqrt 52.2%

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

      7. fabs-sqr 52.2%

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

      8. rem-square-sqrt 52.2%

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

      9. *-commutative 52.2%

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

   10. Simplified 52.2%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot x\right)}^{2}}{1 + x \cdot -0.3275911}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   11. Step-by-step derivation

       1. add-exp-log 52.2%

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

       2. +-commutative 52.2%

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

       3. expm1-define 52.2%

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

       4. expm1-log1p-u 52.2%

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

       5. fma-undefine 52.2%

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

       6. log-rec 52.2%

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

       7. fma-undefine 52.2%

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

       8. expm1-log1p-u 52.2%

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

       9. expm1-define 52.2%

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

       10. +-commutative 52.2%

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

       11. expm1-define 52.2%

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

       12. expm1-log1p-u 52.2%

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

       13. log1p-define 52.2%

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

   12. Applied egg-rr 52.2%

       \[\leadsto 1 - \left(\color{blue}{e^{-\mathsf{log1p}\left(0.3275911 \cdot x\right)}} \cdot \left(0.254829592 + \frac{1}{\frac{1 - {\left(0.3275911 \cdot x\right)}^{2}}{1 + x \cdot -0.3275911}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(e^{-\mathsf{log1p}\left(x \cdot 0.3275911\right)} \cdot \left(\frac{1}{\frac{1 - {\left(x \cdot 0.3275911\right)}^{2}}{1 + x \cdot -0.3275911}} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}\right) \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ \mathbf{if}\;\left|x\_m\right| \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left(\left(x\_m + 1\right) + -1\right)} \cdot \left(\left(-0.284496736 + \frac{1}{1 + t\_0} \cdot {\left(\sqrt[3]{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}\right)}^{3}\right) \cdot \frac{1}{-1 - t\_0} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)))
   (if (<= (fabs x_m) 1.5e-6)
     (+ 1e-9 (* x_m 1.128386358070218))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (/ 1.0 (+ 1.0 (* 0.3275911 (+ (+ x_m 1.0) -1.0))))
        (-
         (*
          (+
           -0.284496736
           (*
            (/ 1.0 (+ 1.0 t_0))
            (pow
             (cbrt
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
                (fma 0.3275911 x_m 1.0))))
             3.0)))
          (/ 1.0 (- -1.0 t_0)))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double tmp;
	if (fabs(x_m) <= 1.5e-6) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((1.0 / (1.0 + (0.3275911 * ((x_m + 1.0) + -1.0)))) * (((-0.284496736 + ((1.0 / (1.0 + t_0)) * pow(cbrt((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)))), 3.0))) * (1.0 / (-1.0 - t_0))) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	tmp = 0.0
	if (abs(x_m) <= 1.5e-6)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * Float64(Float64(x_m + 1.0) + -1.0)))) * Float64(Float64(Float64(-0.284496736 + Float64(Float64(1.0 / Float64(1.0 + t_0)) * (cbrt(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)))) ^ 3.0))) * Float64(1.0 / Float64(-1.0 - t_0))) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1.5e-6], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[(N[(x$95$m + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.284496736 + N[(N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.5e-6

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

   6. Applied egg-rr 52.5%

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

   7. Taylor expanded in x around 0 98.6%

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

      1. *-commutative 98.6%

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

   9. Simplified 98.6%

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

   if 1.5e-6 < (fabs.f64 x)

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

      1. add-sqr-sqrt 52.2%

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

      2. fabs-sqr 52.2%

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

      3. add-sqr-sqrt 98.6%

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

      4. expm1-log1p-u 52.3%

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

      5. expm1-undefine 52.3%

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

   6. Applied egg-rr 52.3%

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

   7. Taylor expanded in x around 0 98.6%

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

      1. associate-*l/ 98.6%

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

      2. *-un-lft-identity 98.6%

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

      3. +-commutative 98.6%

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

      4. fma-undefine 98.6%

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

      5. +-commutative 98.6%

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

      6. fma-undefine 98.6%

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

      7. add-cube-cbrt 98.6%

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

      8. pow3 98.6%

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

   9. Applied egg-rr 98.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left(\left(x + 1\right) + -1\right)} \cdot \left(\left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot {\left(\sqrt[3]{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}^{3}\right) \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 5 \cdot 10^{-11}:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\frac{\mathsf{fma}\left(0.3275911, x\_m, 1\right) \cdot e^{{x\_m}^{2}}}{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 5e-11)
   (+ 1e-9 (* x_m 1.128386358070218))
   (+
    1.0
    (/
     -1.0
     (/
      (* (fma 0.3275911 x_m 1.0) (exp (pow x_m 2.0)))
      (+
       0.254829592
       (/
        (+
         -0.284496736
         (/
          (+
           1.421413741
           (/
            (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
            (fma 0.3275911 x_m 1.0)))
          (fma 0.3275911 x_m 1.0)))
        (fma 0.3275911 x_m 1.0))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 5e-11) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + (-1.0 / ((fma(0.3275911, x_m, 1.0) * exp(pow(x_m, 2.0))) / (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 5e-11)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(Float64(fma(0.3275911, x_m, 1.0) * exp((x_m ^ 2.0))) / Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 5e-11], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.00000000000000018e-11

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   6. Applied egg-rr 52.8%

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

   7. Taylor expanded in x around 0 99.3%

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

      1. *-commutative 99.3%

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

   9. Simplified 99.3%

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

   if 5.00000000000000018e-11 < (fabs.f64 x)

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

      1. add-cbrt-cube 99.3%

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

      2. pow3 99.3%

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

      3. pow-exp 99.3%

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

      4. pow2 99.3%

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

      5. add-sqr-sqrt 51.8%

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

      6. fabs-sqr 51.8%

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

      7. add-sqr-sqrt 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. rem-cbrt-cube 97.9%

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

      2. clear-num 97.9%

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

      3. inv-pow 97.9%

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

      4. *-commutative 97.9%

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

   8. Applied egg-rr 97.9%

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

      1. unpow-1 97.9%

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

   10. Simplified 97.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-11}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ \mathbf{if}\;\left|x\_m\right| \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left(\left(x\_m + 1\right) + -1\right)} \cdot \left(\frac{1}{\frac{1 - {\left(x\_m \cdot 0.3275911\right)}^{2}}{1 + x\_m \cdot -0.3275911}} \cdot \left(\frac{1}{1 + x\_m \cdot 0.3275911} \cdot \left(\left(-1.453152027 + \frac{1.061405429}{1 + t\_0}\right) \cdot \frac{1}{-1 - t\_0} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)))
   (if (<= (fabs x_m) 1.5e-6)
     (+ 1e-9 (* x_m 1.128386358070218))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (/ 1.0 (+ 1.0 (* 0.3275911 (+ (+ x_m 1.0) -1.0))))
        (-
         (*
          (/
           1.0
           (/ (- 1.0 (pow (* x_m 0.3275911) 2.0)) (+ 1.0 (* x_m -0.3275911))))
          (-
           (*
            (/ 1.0 (+ 1.0 (* x_m 0.3275911)))
            (-
             (*
              (+ -1.453152027 (/ 1.061405429 (+ 1.0 t_0)))
              (/ 1.0 (- -1.0 t_0)))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double tmp;
	if (fabs(x_m) <= 1.5e-6) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((1.0 / (1.0 + (0.3275911 * ((x_m + 1.0) + -1.0)))) * (((1.0 / ((1.0 - pow((x_m * 0.3275911), 2.0)) / (1.0 + (x_m * -0.3275911)))) * (((1.0 / (1.0 + (x_m * 0.3275911))) * (((-1.453152027 + (1.061405429 / (1.0 + t_0))) * (1.0 / (-1.0 - t_0))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(x_m) * 0.3275911d0
    if (abs(x_m) <= 1.5d-6) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 1.0d0 + (exp((x_m * -x_m)) * ((1.0d0 / (1.0d0 + (0.3275911d0 * ((x_m + 1.0d0) + (-1.0d0))))) * (((1.0d0 / ((1.0d0 - ((x_m * 0.3275911d0) ** 2.0d0)) / (1.0d0 + (x_m * (-0.3275911d0))))) * (((1.0d0 / (1.0d0 + (x_m * 0.3275911d0))) * ((((-1.453152027d0) + (1.061405429d0 / (1.0d0 + t_0))) * (1.0d0 / ((-1.0d0) - t_0))) - 1.421413741d0)) - (-0.284496736d0))) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.abs(x_m) * 0.3275911;
	double tmp;
	if (Math.abs(x_m) <= 1.5e-6) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + (Math.exp((x_m * -x_m)) * ((1.0 / (1.0 + (0.3275911 * ((x_m + 1.0) + -1.0)))) * (((1.0 / ((1.0 - Math.pow((x_m * 0.3275911), 2.0)) / (1.0 + (x_m * -0.3275911)))) * (((1.0 / (1.0 + (x_m * 0.3275911))) * (((-1.453152027 + (1.061405429 / (1.0 + t_0))) * (1.0 / (-1.0 - t_0))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.fabs(x_m) * 0.3275911
	tmp = 0
	if math.fabs(x_m) <= 1.5e-6:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 1.0 + (math.exp((x_m * -x_m)) * ((1.0 / (1.0 + (0.3275911 * ((x_m + 1.0) + -1.0)))) * (((1.0 / ((1.0 - math.pow((x_m * 0.3275911), 2.0)) / (1.0 + (x_m * -0.3275911)))) * (((1.0 / (1.0 + (x_m * 0.3275911))) * (((-1.453152027 + (1.061405429 / (1.0 + t_0))) * (1.0 / (-1.0 - t_0))) - 1.421413741)) - -0.284496736)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	tmp = 0.0
	if (abs(x_m) <= 1.5e-6)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * Float64(Float64(x_m + 1.0) + -1.0)))) * Float64(Float64(Float64(1.0 / Float64(Float64(1.0 - (Float64(x_m * 0.3275911) ^ 2.0)) / Float64(1.0 + Float64(x_m * -0.3275911)))) * Float64(Float64(Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911))) * Float64(Float64(Float64(-1.453152027 + Float64(1.061405429 / Float64(1.0 + t_0))) * Float64(1.0 / Float64(-1.0 - t_0))) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = abs(x_m) * 0.3275911;
	tmp = 0.0;
	if (abs(x_m) <= 1.5e-6)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 1.0 + (exp((x_m * -x_m)) * ((1.0 / (1.0 + (0.3275911 * ((x_m + 1.0) + -1.0)))) * (((1.0 / ((1.0 - ((x_m * 0.3275911) ^ 2.0)) / (1.0 + (x_m * -0.3275911)))) * (((1.0 / (1.0 + (x_m * 0.3275911))) * (((-1.453152027 + (1.061405429 / (1.0 + t_0))) * (1.0 / (-1.0 - t_0))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1.5e-6], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[(N[(x$95$m + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[(N[(1.0 - N[Power[N[(x$95$m * 0.3275911), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x$95$m * -0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-1.453152027 + N[(1.061405429 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.5e-6

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

   6. Applied egg-rr 52.5%

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

   7. Taylor expanded in x around 0 98.6%

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

      1. *-commutative 98.6%

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

   9. Simplified 98.6%

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

   if 1.5e-6 < (fabs.f64 x)

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

      1. add-sqr-sqrt 52.2%

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

      2. fabs-sqr 52.2%

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

      3. add-sqr-sqrt 98.6%

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

      4. expm1-log1p-u 52.3%

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

      5. expm1-undefine 52.3%

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

   6. Applied egg-rr 52.3%

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

      1. flip-+ 52.3%

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

      2. metadata-eval 52.3%

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

      3. pow2 52.3%

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

   8. Applied egg-rr 52.3%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot \left|x\right|\right)}^{2}}{1 - 0.3275911 \cdot \left|x\right|}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. rem-square-sqrt 52.2%

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

      2. fabs-sqr 52.2%

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

      3. rem-square-sqrt 52.3%

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

      4. cancel-sign-sub-inv 52.3%

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

      5. metadata-eval 52.3%

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

      6. rem-square-sqrt 52.2%

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

      7. fabs-sqr 52.2%

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

      8. rem-square-sqrt 52.2%

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

      9. *-commutative 52.2%

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

   10. Simplified 52.2%

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

   11. Taylor expanded in x around 0 98.4%

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

       1. expm1-log1p-u 98.4%

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

       2. log1p-define 98.4%

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

       3. +-commutative 98.4%

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

       4. fma-undefine 98.4%

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

       5. expm1-undefine 98.4%

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

       6. add-exp-log 98.4%

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

       7. add-sqr-sqrt 52.2%

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

       8. fabs-sqr 52.2%

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

       9. add-sqr-sqrt 98.3%

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

   13. Applied egg-rr 98.3%

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

       1. fma-undefine 98.3%

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

       2. associate--l+ 98.3%

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

       3. metadata-eval 98.3%

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

       4. metadata-eval 98.3%

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

       5. distribute-lft-in 98.3%

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

       6. +-rgt-identity 98.3%

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

       7. *-commutative 98.3%

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

   15. Simplified 98.3%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left(\left(x + 1\right) + -1\right)} \cdot \left(\frac{1}{\frac{1 - {\left(x \cdot 0.3275911\right)}^{2}}{1 + x \cdot -0.3275911}} \cdot \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(\left(-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}\right) \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ t_1 := 1 + t\_0\\ \mathbf{if}\;x\_m \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left(\left(x\_m + 1\right) + -1\right)} \cdot \left(\frac{1}{1 + x\_m \cdot 0.3275911} \cdot \left(\frac{1}{t\_1} \cdot \left(\left(-1.453152027 + \frac{1.061405429}{t\_1}\right) \cdot \frac{1}{-1 - t\_0} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)) (t_1 (+ 1.0 t_0)))
   (if (<= x_m 1.7e-6)
     (+ 1e-9 (* x_m 1.128386358070218))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (/ 1.0 (+ 1.0 (* 0.3275911 (+ (+ x_m 1.0) -1.0))))
        (-
         (*
          (/ 1.0 (+ 1.0 (* x_m 0.3275911)))
          (-
           (*
            (/ 1.0 t_1)
            (-
             (* (+ -1.453152027 (/ 1.061405429 t_1)) (/ 1.0 (- -1.0 t_0)))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.70000000000000003e-6

   1. Initial program 70.6%

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

   3. Simplified 70.6%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in x around 0 68.4%

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

      1. *-commutative 68.4%

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

   9. Simplified 68.4%

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

   if 1.70000000000000003e-6 < x

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

      1. add-sqr-sqrt 99.5%

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

      2. fabs-sqr 99.5%

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

      3. add-sqr-sqrt 99.5%

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

      4. expm1-log1p-u 99.5%

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

      5. expm1-undefine 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 99.5%

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

      1. expm1-log1p-u 99.6%

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

      2. log1p-define 99.6%

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

      3. +-commutative 99.6%

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

      4. fma-undefine 99.6%

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

      5. expm1-undefine 99.6%

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

      6. add-exp-log 99.6%

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

      7. add-sqr-sqrt 99.6%

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

      8. fabs-sqr 99.6%

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

      9. add-sqr-sqrt 99.6%

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

   9. Applied egg-rr 99.5%

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

       1. fma-undefine 99.6%

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

       2. associate--l+ 99.6%

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

       3. metadata-eval 99.6%

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

       4. metadata-eval 99.6%

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

       5. distribute-lft-in 99.6%

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

       6. +-rgt-identity 99.6%

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

       7. *-commutative 99.6%

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

   11. Simplified 99.5%

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

4. Final simplification 76.3%

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

Alternative 11: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.8:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{e^{-{x\_m}^{2}}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} \cdot -0.254829592\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.8)
   (+ 1e-9 (* x_m 1.128386358070218))
   (+
    1.0
    (* (/ (exp (- (pow x_m 2.0))) (fma 0.3275911 x_m 1.0)) -0.254829592))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.8) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + ((exp(-pow(x_m, 2.0)) / fma(0.3275911, x_m, 1.0)) * -0.254829592);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.8)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(-(x_m ^ 2.0))) / fma(0.3275911, x_m, 1.0)) * -0.254829592));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.8], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] * -0.254829592), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in x around 0 68.2%

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

      1. *-commutative 68.2%

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

   9. Simplified 68.2%

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

   if 0.80000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 + -0.254829592 \cdot \frac{e^{-1 \cdot {x}^{2}}}{1 + 0.3275911 \cdot \left|x\right|}} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto 1 + \color{blue}{\frac{e^{-1 \cdot {x}^{2}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot -0.254829592} \]

      2. neg-mul-1 100.0%

         \[\leadsto 1 + \frac{e^{\color{blue}{-{x}^{2}}}}{1 + 0.3275911 \cdot \left|x\right|} \cdot -0.254829592 \]

      3. +-commutative 100.0%

         \[\leadsto 1 + \frac{e^{-{x}^{2}}}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \cdot -0.254829592 \]

      4. fma-undefine 100.0%

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

      5. rem-square-sqrt 100.0%

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

      6. fabs-sqr 100.0%

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

      7. rem-square-sqrt 100.0%

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

   10. Simplified 100.0%

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

Alternative 12: 99.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.88:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-0.7778892405807117}{e^{{x\_m}^{2}}}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88)
   (+ 1e-9 (* x_m 1.128386358070218))
   (+ 1.0 (/ (/ -0.7778892405807117 (exp (pow x_m 2.0))) x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + ((-0.7778892405807117 / exp(pow(x_m, 2.0))) / x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.88d0) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 1.0d0 + (((-0.7778892405807117d0) / exp((x_m ** 2.0d0))) / x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + ((-0.7778892405807117 / Math.exp(Math.pow(x_m, 2.0))) / x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.88:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 1.0 + ((-0.7778892405807117 / math.exp(math.pow(x_m, 2.0))) / x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.88)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = Float64(1.0 + Float64(Float64(-0.7778892405807117 / exp((x_m ^ 2.0))) / x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.88)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 1.0 + ((-0.7778892405807117 / exp((x_m ^ 2.0))) / x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.88], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-0.7778892405807117 / N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in x around 0 68.2%

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

      1. *-commutative 68.2%

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

   9. Simplified 68.2%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. expm1-log1p-u 100.0%

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

      5. expm1-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. flip-+ 100.0%

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

      2. metadata-eval 100.0%

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

      3. pow2 100.0%

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

   8. Applied egg-rr 100.0%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot \left(0.254829592 + \frac{1}{\color{blue}{\frac{1 - {\left(0.3275911 \cdot \left|x\right|\right)}^{2}}{1 - 0.3275911 \cdot \left|x\right|}}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. metadata-eval 100.0%

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

      6. rem-square-sqrt 100.0%

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

      7. fabs-sqr 100.0%

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

      8. rem-square-sqrt 100.0%

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

      9. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around inf 100.0%

       \[\leadsto \color{blue}{1 + -0.7778892405807117 \cdot \frac{e^{-{x}^{2}}}{x}} \]
   12. Step-by-step derivation

       1. associate-*r/ 100.0%

          \[\leadsto 1 + \color{blue}{\frac{-0.7778892405807117 \cdot e^{-{x}^{2}}}{x}} \]

       2. exp-neg 100.0%

          \[\leadsto 1 + \frac{-0.7778892405807117 \cdot \color{blue}{\frac{1}{e^{{x}^{2}}}}}{x} \]

       3. associate-*r/ 100.0%

          \[\leadsto 1 + \frac{\color{blue}{\frac{-0.7778892405807117 \cdot 1}{e^{{x}^{2}}}}}{x} \]

       4. metadata-eval 100.0%

          \[\leadsto 1 + \frac{\frac{\color{blue}{-0.7778892405807117}}{e^{{x}^{2}}}}{x} \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{1 + \frac{\frac{-0.7778892405807117}{e^{{x}^{2}}}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 99.3% accurate, 85.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.9:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.9) (+ 1e-9 (* x_m 1.128386358070218)) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.9d0) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.9:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.9)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.9)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.9], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in x around 0 68.2%

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

      1. *-commutative 68.2%

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

   9. Simplified 68.2%

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

   if 0.900000000000000022 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow3 100.0%

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

      3. pow-exp 100.0%

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

      4. pow2 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. fabs-sqr 100.0%

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

      7. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 97.6% accurate, 142.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.8e-5) 1e-9 1.0))

C

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

Fortran

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

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.8e-5:
		tmp = 1e-9
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.8e-5], 1e-9, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 70.6%

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

   3. Simplified 70.6%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in x around 0 70.6%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

      1. add-cbrt-cube 99.6%

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

      2. pow3 99.6%

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

      3. pow-exp 99.6%

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

      4. pow2 99.6%

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

      5. add-sqr-sqrt 99.6%

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

      6. fabs-sqr 99.6%

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

      7. add-sqr-sqrt 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around inf 97.3%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 53.6% accurate, 856.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 10^{-9} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1e-9)

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1e-9;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1e-9

Julia

x_m = abs(x)
function code(x_m)
	return 1e-9
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1e-9

Derivation

1. Initial program 78.0%

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

3. Simplified 78.0%

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

6. Applied egg-rr 28.2%

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

7. Taylor expanded in x around 0 55.5%

   \[\leadsto \color{blue}{10^{-9}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024182 
(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)))))))