Jmat.Real.erf

Percentage Accurate: 79.0% → 99.4%

Time: 36.4s

Alternatives: 8

Speedup: 8.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(-1.453152027 + N[(t$95$0 * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 79.0% 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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + x_m \cdot 0.3275911}\\ t_1 := 1 + \left|x_m\right| \cdot 0.3275911\\ \mathbf{if}\;\left|x_m\right| \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{10^{-27} + {x_m}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x_m \cdot 1.128386358070218\right)}^{2} - \left(x_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{t_1} \cdot \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(\left(-0.284496736 + t_0 \cdot \left(1.421413741 + t_0 \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 2 \cdot \log \left(\sqrt{{\left(e^{x_m}\right)}^{0.3275911}}\right)}\right)\right)\right) \cdot \frac{-1}{t_1} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* x_m 0.3275911))))
        (t_1 (+ 1.0 (* (fabs x_m) 0.3275911))))
   (if (<= (fabs x_m) 5e-13)
     (/
      (+ 1e-27 (* (pow x_m 3.0) 1.436724444676459))
      (+
       1e-18
       (-
        (pow (* x_m 1.128386358070218) 2.0)
        (* (* x_m 1.128386358070218) 1e-9))))
     (+
      1.0
      (*
       (/ 1.0 t_1)
       (*
        (exp (* x_m (- x_m)))
        (-
         (*
          (+
           -0.284496736
           (*
            t_0
            (+
             1.421413741
             (*
              t_0
              (+
               -1.453152027
               (/
                1.061405429
                (+ 1.0 (* 2.0 (log (sqrt (pow (exp x_m) 0.3275911)))))))))))
          (/ -1.0 t_1))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (x_m * 0.3275911));
	double t_1 = 1.0 + (fabs(x_m) * 0.3275911);
	double tmp;
	if (fabs(x_m) <= 5e-13) {
		tmp = (1e-27 + (pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + ((1.0 / t_1) * (exp((x_m * -x_m)) * (((-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (1.061405429 / (1.0 + (2.0 * log(sqrt(pow(exp(x_m), 0.3275911))))))))))) * (-1.0 / t_1)) - 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 = 1.0d0 / (1.0d0 + (x_m * 0.3275911d0))
    t_1 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    if (abs(x_m) <= 5d-13) then
        tmp = (1d-27 + ((x_m ** 3.0d0) * 1.436724444676459d0)) / (1d-18 + (((x_m * 1.128386358070218d0) ** 2.0d0) - ((x_m * 1.128386358070218d0) * 1d-9)))
    else
        tmp = 1.0d0 + ((1.0d0 / t_1) * (exp((x_m * -x_m)) * ((((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (1.061405429d0 / (1.0d0 + (2.0d0 * log(sqrt((exp(x_m) ** 0.3275911d0))))))))))) * ((-1.0d0) / t_1)) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (x_m * 0.3275911));
	double t_1 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double tmp;
	if (Math.abs(x_m) <= 5e-13) {
		tmp = (1e-27 + (Math.pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (Math.pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + ((1.0 / t_1) * (Math.exp((x_m * -x_m)) * (((-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (1.061405429 / (1.0 + (2.0 * Math.log(Math.sqrt(Math.pow(Math.exp(x_m), 0.3275911))))))))))) * (-1.0 / t_1)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 / (1.0 + (x_m * 0.3275911))
	t_1 = 1.0 + (math.fabs(x_m) * 0.3275911)
	tmp = 0
	if math.fabs(x_m) <= 5e-13:
		tmp = (1e-27 + (math.pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (math.pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)))
	else:
		tmp = 1.0 + ((1.0 / t_1) * (math.exp((x_m * -x_m)) * (((-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (1.061405429 / (1.0 + (2.0 * math.log(math.sqrt(math.pow(math.exp(x_m), 0.3275911))))))))))) * (-1.0 / t_1)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911)))
	t_1 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	tmp = 0.0
	if (abs(x_m) <= 5e-13)
		tmp = Float64(Float64(1e-27 + Float64((x_m ^ 3.0) * 1.436724444676459)) / Float64(1e-18 + Float64((Float64(x_m * 1.128386358070218) ^ 2.0) - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 / t_1) * Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * Float64(-1.453152027 + Float64(1.061405429 / Float64(1.0 + Float64(2.0 * log(sqrt((exp(x_m) ^ 0.3275911))))))))))) * Float64(-1.0 / t_1)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 / (1.0 + (x_m * 0.3275911));
	t_1 = 1.0 + (abs(x_m) * 0.3275911);
	tmp = 0.0;
	if (abs(x_m) <= 5e-13)
		tmp = (1e-27 + ((x_m ^ 3.0) * 1.436724444676459)) / (1e-18 + (((x_m * 1.128386358070218) ^ 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	else
		tmp = 1.0 + ((1.0 / t_1) * (exp((x_m * -x_m)) * (((-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (1.061405429 / (1.0 + (2.0 * log(sqrt((exp(x_m) ^ 0.3275911))))))))))) * (-1.0 / t_1)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 5e-13], N[(N[(1e-27 + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision]), $MachinePrecision] / N[(1e-18 + N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(-1.453152027 + N[(1.061405429 / N[(1.0 + N[(2.0 * N[Log[N[Sqrt[N[Power[N[Exp[x$95$m], $MachinePrecision], 0.3275911], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.9999999999999999e-13

   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%

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

   5. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 95.5%

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

      1. *-commutative 95.5%

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

   8. Simplified 95.5%

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

      1. pow-pow 99.8%

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

      2. metadata-eval 99.8%

         \[\leadsto {\left(10^{-9} + x \cdot 1.128386358070218\right)}^{\color{blue}{1}} \]

      3. pow1 99.8%

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

      4. flip3-+ 99.8%

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

      5. metadata-eval 99.8%

         \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      6. unpow-prod-down 99.8%

         \[\leadsto \frac{10^{-27} + \color{blue}{{x}^{3} \cdot {1.128386358070218}^{3}}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      7. metadata-eval 99.8%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot \color{blue}{1.436724444676459}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{\color{blue}{10^{-18}} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      9. pow2 99.8%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left(\color{blue}{{\left(x \cdot 1.128386358070218\right)}^{2}} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

   10. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

   if 4.9999999999999999e-13 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. log1p-expm1-u 99.8%

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

      2. log1p-udef 99.9%

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

      3. add-sqr-sqrt 57.0%

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

      4. fabs-sqr 57.0%

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

      5. add-sqr-sqrt 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. add-sqr-sqrt 99.3%

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

      2. log-prod 99.3%

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

      3. add-exp-log 99.3%

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

      4. log1p-def 99.3%

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

      5. log1p-expm1-u 99.3%

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

      6. *-commutative 99.3%

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

      7. exp-prod 99.3%

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

      8. add-exp-log 99.3%

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

      9. log1p-def 99.3%

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

      10. log1p-expm1-u 99.3%

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

      11. *-commutative 99.3%

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

      12. exp-prod 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. count-2 99.3%

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

   9. Simplified 99.3%

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

       1. expm1-log1p-u 99.3%

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

       2. expm1-udef 99.3%

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

       3. log1p-udef 99.3%

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

       4. +-commutative 99.3%

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

       5. fma-udef 99.3%

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

       6. add-exp-log 99.3%

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

       7. add-sqr-sqrt 57.1%

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

       8. fabs-sqr 57.1%

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

       9. add-sqr-sqrt 99.3%

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

   11. Applied egg-rr 99.3%

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

       1. fma-udef 99.3%

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

       2. associate--l+ 99.3%

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

       3. metadata-eval 99.3%

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

       4. +-rgt-identity 99.3%

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

   13. Simplified 99.3%

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

       1. expm1-log1p-u 99.3%

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

       2. expm1-udef 99.3%

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

       3. log1p-udef 99.3%

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

       4. +-commutative 99.3%

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

       5. fma-udef 99.3%

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

       6. add-exp-log 99.3%

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

       7. add-sqr-sqrt 57.1%

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

       8. fabs-sqr 57.1%

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

       9. add-sqr-sqrt 99.3%

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

   15. Applied egg-rr 99.3%

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

       1. fma-udef 99.3%

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

       2. associate--l+ 99.3%

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

       3. metadata-eval 99.3%

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

       4. +-rgt-identity 99.3%

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

   17. Simplified 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 2 \cdot \log \left(\sqrt{{\left(e^{x}\right)}^{0.3275911}}\right)}\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ t_2 := 1 + x_m \cdot 0.3275911\\ \mathbf{if}\;x_m \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{10^{-27} + {x_m}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x_m \cdot 1.128386358070218\right)}^{2} - \left(x_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + t_1 \cdot \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(\left(-0.284496736 + t_1 \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{t_2 \cdot \left(1 - x_m \cdot -0.3275911\right)}\right) + 1.453152027 \cdot \frac{-1}{t_2}\right)\right) \cdot \frac{-1}{t_0} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911)))
        (t_1 (/ 1.0 t_0))
        (t_2 (+ 1.0 (* x_m 0.3275911))))
   (if (<= x_m 6.8e-7)
     (/
      (+ 1e-27 (* (pow x_m 3.0) 1.436724444676459))
      (+
       1e-18
       (-
        (pow (* x_m 1.128386358070218) 2.0)
        (* (* x_m 1.128386358070218) 1e-9))))
     (+
      1.0
      (*
       t_1
       (*
        (exp (* x_m (- x_m)))
        (-
         (*
          (+
           -0.284496736
           (*
            t_1
            (+
             (+
              1.421413741
              (* 1.061405429 (/ 1.0 (* t_2 (- 1.0 (* x_m -0.3275911))))))
             (* 1.453152027 (/ -1.0 t_2)))))
          (/ -1.0 t_0))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = 1.0 + (x_m * 0.3275911);
	double tmp;
	if (x_m <= 6.8e-7) {
		tmp = (1e-27 + (pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * ((1.421413741 + (1.061405429 * (1.0 / (t_2 * (1.0 - (x_m * -0.3275911)))))) + (1.453152027 * (-1.0 / t_2))))) * (-1.0 / t_0)) - 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) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    t_2 = 1.0d0 + (x_m * 0.3275911d0)
    if (x_m <= 6.8d-7) then
        tmp = (1d-27 + ((x_m ** 3.0d0) * 1.436724444676459d0)) / (1d-18 + (((x_m * 1.128386358070218d0) ** 2.0d0) - ((x_m * 1.128386358070218d0) * 1d-9)))
    else
        tmp = 1.0d0 + (t_1 * (exp((x_m * -x_m)) * ((((-0.284496736d0) + (t_1 * ((1.421413741d0 + (1.061405429d0 * (1.0d0 / (t_2 * (1.0d0 - (x_m * (-0.3275911d0))))))) + (1.453152027d0 * ((-1.0d0) / t_2))))) * ((-1.0d0) / t_0)) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = 1.0 + (x_m * 0.3275911);
	double tmp;
	if (x_m <= 6.8e-7) {
		tmp = (1e-27 + (Math.pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (Math.pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (t_1 * (Math.exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * ((1.421413741 + (1.061405429 * (1.0 / (t_2 * (1.0 - (x_m * -0.3275911)))))) + (1.453152027 * (-1.0 / t_2))))) * (-1.0 / t_0)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (math.fabs(x_m) * 0.3275911)
	t_1 = 1.0 / t_0
	t_2 = 1.0 + (x_m * 0.3275911)
	tmp = 0
	if x_m <= 6.8e-7:
		tmp = (1e-27 + (math.pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (math.pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)))
	else:
		tmp = 1.0 + (t_1 * (math.exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * ((1.421413741 + (1.061405429 * (1.0 / (t_2 * (1.0 - (x_m * -0.3275911)))))) + (1.453152027 * (-1.0 / t_2))))) * (-1.0 / t_0)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(1.0 + Float64(x_m * 0.3275911))
	tmp = 0.0
	if (x_m <= 6.8e-7)
		tmp = Float64(Float64(1e-27 + Float64((x_m ^ 3.0) * 1.436724444676459)) / Float64(1e-18 + Float64((Float64(x_m * 1.128386358070218) ^ 2.0) - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = Float64(1.0 + Float64(t_1 * Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(Float64(-0.284496736 + Float64(t_1 * Float64(Float64(1.421413741 + Float64(1.061405429 * Float64(1.0 / Float64(t_2 * Float64(1.0 - Float64(x_m * -0.3275911)))))) + Float64(1.453152027 * Float64(-1.0 / t_2))))) * Float64(-1.0 / t_0)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (abs(x_m) * 0.3275911);
	t_1 = 1.0 / t_0;
	t_2 = 1.0 + (x_m * 0.3275911);
	tmp = 0.0;
	if (x_m <= 6.8e-7)
		tmp = (1e-27 + ((x_m ^ 3.0) * 1.436724444676459)) / (1e-18 + (((x_m * 1.128386358070218) ^ 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	else
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * ((1.421413741 + (1.061405429 * (1.0 / (t_2 * (1.0 - (x_m * -0.3275911)))))) + (1.453152027 * (-1.0 / t_2))))) * (-1.0 / t_0)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 6.8e-7], N[(N[(1e-27 + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision]), $MachinePrecision] / N[(1e-18 + N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$1 * N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.284496736 + N[(t$95$1 * N[(N[(1.421413741 + N[(1.061405429 * N[(1.0 / N[(t$95$2 * N[(1.0 - N[(x$95$m * -0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.453152027 * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.79999999999999948e-7

   1. Initial program 72.1%

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

   3. Simplified 72.1%

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

   5. Applied egg-rr 39.1%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 63.7%

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

      1. *-commutative 63.7%

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

   8. Simplified 63.7%

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

      1. pow-pow 66.2%

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

      2. metadata-eval 66.2%

         \[\leadsto {\left(10^{-9} + x \cdot 1.128386358070218\right)}^{\color{blue}{1}} \]

      3. pow1 66.2%

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

      4. flip3-+ 66.0%

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

      5. metadata-eval 66.0%

         \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      6. unpow-prod-down 66.0%

         \[\leadsto \frac{10^{-27} + \color{blue}{{x}^{3} \cdot {1.128386358070218}^{3}}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      7. metadata-eval 66.0%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot \color{blue}{1.436724444676459}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      8. metadata-eval 66.0%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{\color{blue}{10^{-18}} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      9. pow2 66.0%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left(\color{blue}{{\left(x \cdot 1.128386358070218\right)}^{2}} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

   10. Applied egg-rr 66.0%

       \[\leadsto \color{blue}{\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

   if 6.79999999999999948e-7 < x

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. log1p-expm1-u 99.6%

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

      2. log1p-udef 99.8%

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

      3. add-sqr-sqrt 99.8%

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

      4. fabs-sqr 99.8%

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

      5. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around -inf 99.8%

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

      1. expm1-log1p-u 99.8%

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

      2. expm1-udef 99.8%

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

      3. log1p-udef 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-udef 99.8%

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

      6. add-exp-log 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. associate--l+ 99.8%

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

      3. metadata-eval 99.8%

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

      4. +-rgt-identity 99.8%

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

   10. Simplified 99.8%

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

       1. expm1-log1p-u 99.8%

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

       2. expm1-udef 99.8%

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

       3. log1p-udef 99.8%

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

       4. +-commutative 99.8%

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

       5. fma-udef 99.8%

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

       6. add-exp-log 99.8%

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

       7. add-sqr-sqrt 99.8%

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

       8. fabs-sqr 99.8%

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

       9. add-sqr-sqrt 99.8%

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

   12. Applied egg-rr 99.8%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

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

       4. +-rgt-identity 99.8%

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

   14. Simplified 99.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{\left(1 + x \cdot 0.3275911\right) \cdot \left(1 - x \cdot -0.3275911\right)}\right) + 1.453152027 \cdot \frac{-1}{1 + x \cdot 0.3275911}\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x_m\right| \cdot 0.3275911\\ t_1 := 1 + x_m \cdot 0.3275911\\ t_2 := \frac{1}{t_0}\\ \mathbf{if}\;x_m \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{10^{-27} + {x_m}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x_m \cdot 1.128386358070218\right)}^{2} - \left(x_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(0.254829592 + t_2 \cdot \left(-0.284496736 + t_2 \cdot \left(1.421413741 + \frac{1}{t_1} \cdot \left(-1.453152027 + \frac{1.061405429}{t_1}\right)\right)\right)\right)\right) \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911)))
        (t_1 (+ 1.0 (* x_m 0.3275911)))
        (t_2 (/ 1.0 t_0)))
   (if (<= x_m 4.2e-7)
     (/
      (+ 1e-27 (* (pow x_m 3.0) 1.436724444676459))
      (+
       1e-18
       (-
        (pow (* x_m 1.128386358070218) 2.0)
        (* (* x_m 1.128386358070218) 1e-9))))
     (+
      1.0
      (*
       (*
        (exp (* x_m (- x_m)))
        (+
         0.254829592
         (*
          t_2
          (+
           -0.284496736
           (*
            t_2
            (+
             1.421413741
             (* (/ 1.0 t_1) (+ -1.453152027 (/ 1.061405429 t_1)))))))))
       (/ -1.0 t_0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_1 = 1.0 + (x_m * 0.3275911);
	double t_2 = 1.0 / t_0;
	double tmp;
	if (x_m <= 4.2e-7) {
		tmp = (1e-27 + (pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + ((exp((x_m * -x_m)) * (0.254829592 + (t_2 * (-0.284496736 + (t_2 * (1.421413741 + ((1.0 / t_1) * (-1.453152027 + (1.061405429 / t_1))))))))) * (-1.0 / t_0));
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    t_1 = 1.0d0 + (x_m * 0.3275911d0)
    t_2 = 1.0d0 / t_0
    if (x_m <= 4.2d-7) then
        tmp = (1d-27 + ((x_m ** 3.0d0) * 1.436724444676459d0)) / (1d-18 + (((x_m * 1.128386358070218d0) ** 2.0d0) - ((x_m * 1.128386358070218d0) * 1d-9)))
    else
        tmp = 1.0d0 + ((exp((x_m * -x_m)) * (0.254829592d0 + (t_2 * ((-0.284496736d0) + (t_2 * (1.421413741d0 + ((1.0d0 / t_1) * ((-1.453152027d0) + (1.061405429d0 / t_1))))))))) * ((-1.0d0) / t_0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double t_1 = 1.0 + (x_m * 0.3275911);
	double t_2 = 1.0 / t_0;
	double tmp;
	if (x_m <= 4.2e-7) {
		tmp = (1e-27 + (Math.pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (Math.pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + ((Math.exp((x_m * -x_m)) * (0.254829592 + (t_2 * (-0.284496736 + (t_2 * (1.421413741 + ((1.0 / t_1) * (-1.453152027 + (1.061405429 / t_1))))))))) * (-1.0 / t_0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (math.fabs(x_m) * 0.3275911)
	t_1 = 1.0 + (x_m * 0.3275911)
	t_2 = 1.0 / t_0
	tmp = 0
	if x_m <= 4.2e-7:
		tmp = (1e-27 + (math.pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (math.pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)))
	else:
		tmp = 1.0 + ((math.exp((x_m * -x_m)) * (0.254829592 + (t_2 * (-0.284496736 + (t_2 * (1.421413741 + ((1.0 / t_1) * (-1.453152027 + (1.061405429 / t_1))))))))) * (-1.0 / t_0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 + Float64(x_m * 0.3275911))
	t_2 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x_m <= 4.2e-7)
		tmp = Float64(Float64(1e-27 + Float64((x_m ^ 3.0) * 1.436724444676459)) / Float64(1e-18 + Float64((Float64(x_m * 1.128386358070218) ^ 2.0) - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(0.254829592 + Float64(t_2 * Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(Float64(1.0 / t_1) * Float64(-1.453152027 + Float64(1.061405429 / t_1))))))))) * Float64(-1.0 / t_0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (abs(x_m) * 0.3275911);
	t_1 = 1.0 + (x_m * 0.3275911);
	t_2 = 1.0 / t_0;
	tmp = 0.0;
	if (x_m <= 4.2e-7)
		tmp = (1e-27 + ((x_m ^ 3.0) * 1.436724444676459)) / (1e-18 + (((x_m * 1.128386358070218) ^ 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	else
		tmp = 1.0 + ((exp((x_m * -x_m)) * (0.254829592 + (t_2 * (-0.284496736 + (t_2 * (1.421413741 + ((1.0 / t_1) * (-1.453152027 + (1.061405429 / t_1))))))))) * (-1.0 / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 4.2e-7], N[(N[(1e-27 + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision]), $MachinePrecision] / N[(1e-18 + N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(0.254829592 + N[(t$95$2 * N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2e-7

   1. Initial program 72.1%

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

   3. Simplified 72.1%

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

   5. Applied egg-rr 39.1%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 63.7%

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

      1. *-commutative 63.7%

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

   8. Simplified 63.7%

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

      1. pow-pow 66.2%

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

      2. metadata-eval 66.2%

         \[\leadsto {\left(10^{-9} + x \cdot 1.128386358070218\right)}^{\color{blue}{1}} \]

      3. pow1 66.2%

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

      4. flip3-+ 66.0%

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

      5. metadata-eval 66.0%

         \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      6. unpow-prod-down 66.0%

         \[\leadsto \frac{10^{-27} + \color{blue}{{x}^{3} \cdot {1.128386358070218}^{3}}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      7. metadata-eval 66.0%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot \color{blue}{1.436724444676459}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      8. metadata-eval 66.0%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{\color{blue}{10^{-18}} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      9. pow2 66.0%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left(\color{blue}{{\left(x \cdot 1.128386358070218\right)}^{2}} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

   10. Applied egg-rr 66.0%

       \[\leadsto \color{blue}{\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

   if 4.2e-7 < x

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. log1p-expm1-u 99.6%

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

      2. log1p-udef 99.8%

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

      3. add-sqr-sqrt 99.8%

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

      4. fabs-sqr 99.8%

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

      5. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. log-prod 99.8%

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

      3. add-exp-log 99.8%

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

      4. log1p-def 99.8%

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

      5. log1p-expm1-u 99.8%

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

      6. *-commutative 99.8%

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

      7. exp-prod 99.8%

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

      8. add-exp-log 99.8%

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

      9. log1p-def 99.8%

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

      10. log1p-expm1-u 99.8%

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

      11. *-commutative 99.8%

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

      12. exp-prod 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. count-2 99.8%

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

   9. Simplified 99.8%

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

       1. expm1-log1p-u 99.8%

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

       2. expm1-udef 99.8%

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

       3. log1p-udef 99.8%

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

       4. +-commutative 99.8%

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

       5. fma-udef 99.8%

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

       6. add-exp-log 99.8%

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

       7. add-sqr-sqrt 99.8%

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

       8. fabs-sqr 99.8%

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

       9. add-sqr-sqrt 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

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

       4. +-rgt-identity 99.8%

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

   13. Simplified 99.8%

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

       1. expm1-log1p-u 3.6%

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

       2. expm1-udef 3.6%

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

       3. *-commutative 3.6%

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

       4. add-log-exp 3.6%

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

       5. exp-to-pow 3.6%

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

       6. pow2 3.6%

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

       7. add-sqr-sqrt 3.5%

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

       8. pow-exp 3.5%

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

       9. rem-log-exp 3.4%

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

   15. Applied egg-rr 3.4%

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

       1. expm1-def 3.4%

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

       2. expm1-log1p 99.6%

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

   17. Simplified 99.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x_m \leq 0.21:\\ \;\;\;\;\frac{10^{-27} + {x_m}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x_m \cdot 1.128386358070218\right)}^{2} - \left(x_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + t_1 \cdot \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(\left(-0.284496736 + t_1 \cdot \left(1.421413741 + \frac{1}{1 + x_m \cdot 0.3275911} \cdot \left(x_m \cdot -0.3477069720320819 - 0.391746598\right)\right)\right) \cdot \frac{-1}{t_0} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= x_m 0.21)
     (/
      (+ 1e-27 (* (pow x_m 3.0) 1.436724444676459))
      (+
       1e-18
       (-
        (pow (* x_m 1.128386358070218) 2.0)
        (* (* x_m 1.128386358070218) 1e-9))))
     (+
      1.0
      (*
       t_1
       (*
        (exp (* x_m (- x_m)))
        (-
         (*
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (*
              (/ 1.0 (+ 1.0 (* x_m 0.3275911)))
              (- (* x_m -0.3477069720320819) 0.391746598)))))
          (/ -1.0 t_0))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x_m <= 0.21) {
		tmp = (1e-27 + (pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + ((1.0 / (1.0 + (x_m * 0.3275911))) * ((x_m * -0.3477069720320819) - 0.391746598))))) * (-1.0 / t_0)) - 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 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    if (x_m <= 0.21d0) then
        tmp = (1d-27 + ((x_m ** 3.0d0) * 1.436724444676459d0)) / (1d-18 + (((x_m * 1.128386358070218d0) ** 2.0d0) - ((x_m * 1.128386358070218d0) * 1d-9)))
    else
        tmp = 1.0d0 + (t_1 * (exp((x_m * -x_m)) * ((((-0.284496736d0) + (t_1 * (1.421413741d0 + ((1.0d0 / (1.0d0 + (x_m * 0.3275911d0))) * ((x_m * (-0.3477069720320819d0)) - 0.391746598d0))))) * ((-1.0d0) / t_0)) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x_m <= 0.21) {
		tmp = (1e-27 + (Math.pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (Math.pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (t_1 * (Math.exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + ((1.0 / (1.0 + (x_m * 0.3275911))) * ((x_m * -0.3477069720320819) - 0.391746598))))) * (-1.0 / t_0)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (math.fabs(x_m) * 0.3275911)
	t_1 = 1.0 / t_0
	tmp = 0
	if x_m <= 0.21:
		tmp = (1e-27 + (math.pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (math.pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)))
	else:
		tmp = 1.0 + (t_1 * (math.exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + ((1.0 / (1.0 + (x_m * 0.3275911))) * ((x_m * -0.3477069720320819) - 0.391746598))))) * (-1.0 / t_0)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x_m <= 0.21)
		tmp = Float64(Float64(1e-27 + Float64((x_m ^ 3.0) * 1.436724444676459)) / Float64(1e-18 + Float64((Float64(x_m * 1.128386358070218) ^ 2.0) - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = Float64(1.0 + Float64(t_1 * Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911))) * Float64(Float64(x_m * -0.3477069720320819) - 0.391746598))))) * Float64(-1.0 / t_0)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (abs(x_m) * 0.3275911);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (x_m <= 0.21)
		tmp = (1e-27 + ((x_m ^ 3.0) * 1.436724444676459)) / (1e-18 + (((x_m * 1.128386358070218) ^ 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	else
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + ((1.0 / (1.0 + (x_m * 0.3275911))) * ((x_m * -0.3477069720320819) - 0.391746598))))) * (-1.0 / t_0)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.21], N[(N[(1e-27 + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision]), $MachinePrecision] / N[(1e-18 + N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$1 * N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * -0.3477069720320819), $MachinePrecision] - 0.391746598), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.209999999999999992

   1. Initial program 72.3%

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

   3. Simplified 72.3%

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

   5. Applied egg-rr 39.0%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 63.5%

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

      1. *-commutative 63.5%

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

   8. Simplified 63.5%

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

      1. pow-pow 65.9%

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

      2. metadata-eval 65.9%

         \[\leadsto {\left(10^{-9} + x \cdot 1.128386358070218\right)}^{\color{blue}{1}} \]

      3. pow1 65.9%

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

      4. flip3-+ 65.8%

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

      5. metadata-eval 65.8%

         \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      6. unpow-prod-down 65.8%

         \[\leadsto \frac{10^{-27} + \color{blue}{{x}^{3} \cdot {1.128386358070218}^{3}}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      7. metadata-eval 65.8%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot \color{blue}{1.436724444676459}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      8. metadata-eval 65.8%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{\color{blue}{10^{-18}} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      9. pow2 65.8%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left(\color{blue}{{\left(x \cdot 1.128386358070218\right)}^{2}} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

   10. Applied egg-rr 65.8%

       \[\leadsto \color{blue}{\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

   if 0.209999999999999992 < x

   1. Initial program 99.9%

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

   3. Simplified 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. log1p-udef 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. fabs-sqr 100.0%

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

      5. add-sqr-sqrt 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. add-exp-log 100.0%

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

      4. log1p-def 100.0%

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

      5. log1p-expm1-u 100.0%

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

      6. *-commutative 100.0%

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

      7. exp-prod 100.0%

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

      8. add-exp-log 100.0%

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

      9. log1p-def 100.0%

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

      10. log1p-expm1-u 100.0%

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

      11. *-commutative 100.0%

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

      12. exp-prod 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   9. Simplified 100.0%

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

       1. expm1-log1p-u 100.0%

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

       2. expm1-udef 100.0%

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

       3. log1p-udef 100.0%

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

       4. +-commutative 100.0%

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

       5. fma-udef 100.0%

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

       6. add-exp-log 100.0%

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

       7. add-sqr-sqrt 100.0%

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

       8. fabs-sqr 100.0%

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

       9. add-sqr-sqrt 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. fma-udef 100.0%

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

       2. associate--l+ 100.0%

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

       3. metadata-eval 100.0%

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

       4. +-rgt-identity 100.0%

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

   13. Simplified 100.0%

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

   14. Taylor expanded in x around 0 99.1%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \color{blue}{\left(-0.3477069720320819 \cdot x - 0.391746598\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.21:\\ \;\;\;\;\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(x \cdot -0.3477069720320819 - 0.391746598\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]

Alternative 5: 99.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.88:\\ \;\;\;\;\frac{10^{-27} + {x_m}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x_m \cdot 1.128386358070218\right)}^{2} - \left(x_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88)
   (/
    (+ 1e-27 (* (pow x_m 3.0) 1.436724444676459))
    (+
     1e-18
     (-
      (pow (* x_m 1.128386358070218) 2.0)
      (* (* x_m 1.128386358070218) 1e-9))))
   (pow 1.0 0.3333333333333333)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = (1e-27 + (pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = pow(1.0, 0.3333333333333333);
	}
	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-27 + ((x_m ** 3.0d0) * 1.436724444676459d0)) / (1d-18 + (((x_m * 1.128386358070218d0) ** 2.0d0) - ((x_m * 1.128386358070218d0) * 1d-9)))
    else
        tmp = 1.0d0 ** 0.3333333333333333d0
    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-27 + (Math.pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (Math.pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = Math.pow(1.0, 0.3333333333333333);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.88:
		tmp = (1e-27 + (math.pow(x_m, 3.0) * 1.436724444676459)) / (1e-18 + (math.pow((x_m * 1.128386358070218), 2.0) - ((x_m * 1.128386358070218) * 1e-9)))
	else:
		tmp = math.pow(1.0, 0.3333333333333333)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.88)
		tmp = Float64(Float64(1e-27 + Float64((x_m ^ 3.0) * 1.436724444676459)) / Float64(1e-18 + Float64((Float64(x_m * 1.128386358070218) ^ 2.0) - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = 1.0 ^ 0.3333333333333333;
	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-27 + ((x_m ^ 3.0) * 1.436724444676459)) / (1e-18 + (((x_m * 1.128386358070218) ^ 2.0) - ((x_m * 1.128386358070218) * 1e-9)));
	else
		tmp = 1.0 ^ 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.88], N[(N[(1e-27 + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision]), $MachinePrecision] / N[(1e-18 + N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[1.0, 0.3333333333333333], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 72.4%

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

   3. Simplified 72.4%

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

   5. Applied egg-rr 38.9%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 63.3%

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

      1. *-commutative 63.3%

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

   8. Simplified 63.3%

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

      1. pow-pow 65.7%

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

      2. metadata-eval 65.7%

         \[\leadsto {\left(10^{-9} + x \cdot 1.128386358070218\right)}^{\color{blue}{1}} \]

      3. pow1 65.7%

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

      4. flip3-+ 65.5%

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

      5. metadata-eval 65.5%

         \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(x \cdot 1.128386358070218\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      6. unpow-prod-down 65.5%

         \[\leadsto \frac{10^{-27} + \color{blue}{{x}^{3} \cdot {1.128386358070218}^{3}}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      7. metadata-eval 65.5%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot \color{blue}{1.436724444676459}}{10^{-9} \cdot 10^{-9} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      8. metadata-eval 65.5%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{\color{blue}{10^{-18}} + \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

      9. pow2 65.5%

         \[\leadsto \frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left(\color{blue}{{\left(x \cdot 1.128386358070218\right)}^{2}} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)} \]

   10. Applied egg-rr 65.5%

       \[\leadsto \color{blue}{\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - 10^{-9} \cdot \left(x \cdot 1.128386358070218\right)\right)}} \]

   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%

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

   5. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto {\left({\color{blue}{1}}^{3}\right)}^{0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{10^{-27} + {x}^{3} \cdot 1.436724444676459}{10^{-18} + \left({\left(x \cdot 1.128386358070218\right)}^{2} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \]

Alternative 6: 99.3% accurate, 8.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.88:\\ \;\;\;\;x_m \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88)
   (+ (* x_m 1.128386358070218) 1e-9)
   (pow 1.0 0.3333333333333333)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = pow(1.0, 0.3333333333333333);
	}
	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 = (x_m * 1.128386358070218d0) + 1d-9
    else
        tmp = 1.0d0 ** 0.3333333333333333d0
    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 = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = Math.pow(1.0, 0.3333333333333333);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.88:
		tmp = (x_m * 1.128386358070218) + 1e-9
	else:
		tmp = math.pow(1.0, 0.3333333333333333)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.88)
		tmp = Float64(Float64(x_m * 1.128386358070218) + 1e-9);
	else
		tmp = 1.0 ^ 0.3333333333333333;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.88)
		tmp = (x_m * 1.128386358070218) + 1e-9;
	else
		tmp = 1.0 ^ 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.88], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], N[Power[1.0, 0.3333333333333333], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 72.4%

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

   3. Simplified 72.4%

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

   5. Applied egg-rr 38.9%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around 0 63.3%

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

      1. *-commutative 63.3%

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

   8. Simplified 63.3%

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

      1. pow-pow 65.7%

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

      2. metadata-eval 65.7%

         \[\leadsto {\left(10^{-9} + x \cdot 1.128386358070218\right)}^{\color{blue}{1}} \]

      3. pow1 65.7%

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

      4. +-commutative 65.7%

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

   10. Applied egg-rr 65.7%

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

   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%

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

   5. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto {\left({\color{blue}{1}}^{3}\right)}^{0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \]

Alternative 7: 52.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \sqrt[3]{10^{-27}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (cbrt 1e-27))

C

x_m = fabs(x);
double code(double x_m) {
	return cbrt(1e-27);
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.cbrt(1e-27);
}

Julia

x_m = abs(x)
function code(x_m)
	return cbrt(1e-27)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[1e-27, 1/3], $MachinePrecision]

Derivation

1. Initial program 80.7%

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

3. Simplified 80.7%

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

5. Applied egg-rr 28.1%

   \[\leadsto \color{blue}{{\left({\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Taylor expanded in x around 0 50.4%

   \[\leadsto \color{blue}{\sqrt[3]{10^{-27}}} \]

7. Final simplification 50.4%

   \[\leadsto \sqrt[3]{10^{-27}} \]

Alternative 8: 52.2% accurate, 171.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_m \cdot 1.128386358070218 + 10^{-9} \end{array} \]

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * 1.128386358070218) + 1e-9;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (x_m * 1.128386358070218d0) + 1d-9
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * 1.128386358070218) + 1e-9;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * 1.128386358070218) + 1e-9

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * 1.128386358070218) + 1e-9)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * 1.128386358070218) + 1e-9;
end

Wolfram

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

Derivation

1. Initial program 80.7%

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

3. Simplified 80.7%

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

5. Applied egg-rr 28.1%

   \[\leadsto \color{blue}{{\left({\left(1 - \frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) \cdot e^{{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Taylor expanded in x around 0 45.8%

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

   1. *-commutative 45.8%

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

8. Simplified 45.8%

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

   1. pow-pow 47.7%

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

   2. metadata-eval 47.7%

      \[\leadsto {\left(10^{-9} + x \cdot 1.128386358070218\right)}^{\color{blue}{1}} \]

   3. pow1 47.7%

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

   4. +-commutative 47.7%

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

10. Applied egg-rr 47.7%

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

11. Final simplification 47.7%

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

Reproduce

herbie shell --seed 2023319 
(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)))))))