Jmat.Real.erf

Percentage Accurate: 79.3% → 99.2%

Time: 40.8s

Alternatives: 14

Speedup: 142.3×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right) \cdot e^{{x\_m}^{2}}}\\ t_1 := t\_0 + \left(1 + {t\_0}^{2}\right)\\ t_2 := \frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|}\\ \mathbf{if}\;\left(t\_2 \cdot \left(0.254829592 + t\_2 \cdot \left(-0.284496736 + t\_2 \cdot \left(1.421413741 + t\_2 \cdot \left(-1.453152027 + t\_2 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-x\_m \cdot x\_m} \leq 0.9995:\\ \;\;\;\;\frac{1 - \log \left(1 + \mathsf{expm1}\left({t\_0}^{3}\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + \left({x\_m}^{2} \cdot -3.820122044248399 + \left(0.3111712305105463 \cdot {x\_m}^{3} + x\_m \cdot 3.385159067440336\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (/
          (+
           0.254829592
           (/
            (+
             -0.284496736
             (/
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
                (fma 0.3275911 x_m 1.0)))
              (fma 0.3275911 x_m 1.0)))
            (fma 0.3275911 x_m 1.0)))
          (* (fma 0.3275911 x_m 1.0) (exp (pow x_m 2.0)))))
        (t_1 (+ t_0 (+ 1.0 (pow t_0 2.0))))
        (t_2 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m))))))
   (if (<=
        (*
         (*
          t_2
          (+
           0.254829592
           (*
            t_2
            (+
             -0.284496736
             (*
              t_2
              (+ 1.421413741 (* t_2 (+ -1.453152027 (* t_2 1.061405429)))))))))
         (exp (- (* x_m x_m))))
        0.9995)
     (/ (- 1.0 (log (+ 1.0 (expm1 (pow t_0 3.0))))) t_1)
     (/
      (+
       2.999999997e-9
       (+
        (* (pow x_m 2.0) -3.820122044248399)
        (+ (* 0.3111712305105463 (pow x_m 3.0)) (* x_m 3.385159067440336))))
      t_1))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / (fma(0.3275911, x_m, 1.0) * exp(pow(x_m, 2.0)));
	double t_1 = t_0 + (1.0 + pow(t_0, 2.0));
	double t_2 = 1.0 / (1.0 + (0.3275911 * fabs(x_m)));
	double tmp;
	if (((t_2 * (0.254829592 + (t_2 * (-0.284496736 + (t_2 * (1.421413741 + (t_2 * (-1.453152027 + (t_2 * 1.061405429))))))))) * exp(-(x_m * x_m))) <= 0.9995) {
		tmp = (1.0 - log((1.0 + expm1(pow(t_0, 3.0))))) / t_1;
	} else {
		tmp = (2.999999997e-9 + ((pow(x_m, 2.0) * -3.820122044248399) + ((0.3111712305105463 * pow(x_m, 3.0)) + (x_m * 3.385159067440336)))) / t_1;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / Float64(fma(0.3275911, x_m, 1.0) * exp((x_m ^ 2.0))))
	t_1 = Float64(t_0 + Float64(1.0 + (t_0 ^ 2.0)))
	t_2 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x_m))))
	tmp = 0.0
	if (Float64(Float64(t_2 * Float64(0.254829592 + Float64(t_2 * Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(t_2 * Float64(-1.453152027 + Float64(t_2 * 1.061405429))))))))) * exp(Float64(-Float64(x_m * x_m)))) <= 0.9995)
		tmp = Float64(Float64(1.0 - log(Float64(1.0 + expm1((t_0 ^ 3.0))))) / t_1);
	else
		tmp = Float64(Float64(2.999999997e-9 + Float64(Float64((x_m ^ 2.0) * -3.820122044248399) + Float64(Float64(0.3111712305105463 * (x_m ^ 3.0)) + Float64(x_m * 3.385159067440336)))) / t_1);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(1.0 + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[(0.254829592 + N[(t$95$2 * N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(t$95$2 * N[(-1.453152027 + N[(t$95$2 * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(x$95$m * x$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], 0.9995], N[(N[(1.0 - N[Log[N[(1.0 + N[(Exp[N[Power[t$95$0, 3.0], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.999999997e-9 + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -3.820122044248399), $MachinePrecision] + N[(N[(0.3111712305105463 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 3.385159067440336), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 0.99950000000000006

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. associate-/l/ 99.7%

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

      2. flip3-- 99.7%

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

   6. Applied egg-rr 99.7%

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

   8. Simplified 98.3%

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

      1. log1p-expm1-u 98.3%

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

      2. log1p-udef 98.3%

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

   10. Applied egg-rr 98.3%

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

   if 0.99950000000000006 < (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))

   1. Initial program 58.0%

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

   3. Simplified 58.0%

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

      1. associate-/l/ 58.0%

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

      2. flip3-- 58.1%

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

   6. Applied egg-rr 58.1%

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

   8. Simplified 56.5%

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

   9. Taylor expanded in x around 0 95.0%

      \[\leadsto \frac{\color{blue}{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + \left(0.3111712305105463 \cdot {x}^{3} + 3.385159067440336 \cdot x\right)\right)}}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}} + \left(1 + {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\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^{-x \cdot x} \leq 0.9995:\\ \;\;\;\;\frac{1 - \log \left(1 + \mathsf{expm1}\left({\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{3}\right)\right)}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}} + \left(1 + {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + \left({x}^{2} \cdot -3.820122044248399 + \left(0.3111712305105463 \cdot {x}^{3} + x \cdot 3.385159067440336\right)\right)}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}} + \left(1 + {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.3275911, x\_m, 1\right)\right)}^{3}\\ t_1 := 1 + 0.3275911 \cdot \left|x\_m\right|\\ t_2 := \frac{1}{t\_1}\\ t_3 := \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right) \cdot e^{{x\_m}^{2}}}\\ \mathbf{if}\;\left(t\_2 \cdot \left(0.254829592 + t\_2 \cdot \left(-0.284496736 + t\_2 \cdot \left(1.421413741 + t\_2 \cdot \left(-1.453152027 + t\_2 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-x\_m \cdot x\_m} \leq 0.9995:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \left(\frac{1.453152027}{t\_0} - \frac{\frac{1.061405429}{t\_0} + \left(-0.284496736 + \frac{1.421413741}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right), \frac{{\left(e^{x\_m}\right)}^{\left(-x\_m\right)}}{t\_1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + \left({x\_m}^{2} \cdot -3.820122044248399 + \left(0.3111712305105463 \cdot {x\_m}^{3} + x\_m \cdot 3.385159067440336\right)\right)}{t\_3 + \left(1 + {t\_3}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (pow (fma 0.3275911 x_m 1.0) 3.0))
        (t_1 (+ 1.0 (* 0.3275911 (fabs x_m))))
        (t_2 (/ 1.0 t_1))
        (t_3
         (/
          (+
           0.254829592
           (/
            (+
             -0.284496736
             (/
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
                (fma 0.3275911 x_m 1.0)))
              (fma 0.3275911 x_m 1.0)))
            (fma 0.3275911 x_m 1.0)))
          (* (fma 0.3275911 x_m 1.0) (exp (pow x_m 2.0))))))
   (if (<=
        (*
         (*
          t_2
          (+
           0.254829592
           (*
            t_2
            (+
             -0.284496736
             (*
              t_2
              (+ 1.421413741 (* t_2 (+ -1.453152027 (* t_2 1.061405429)))))))))
         (exp (- (* x_m x_m))))
        0.9995)
     (fma
      (+
       -0.254829592
       (-
        (/ 1.453152027 t_0)
        (/
         (+
          (/ 1.061405429 t_0)
          (+ -0.284496736 (/ 1.421413741 (fma 0.3275911 x_m 1.0))))
         (fma 0.3275911 x_m 1.0))))
      (/ (pow (exp x_m) (- x_m)) t_1)
      1.0)
     (/
      (+
       2.999999997e-9
       (+
        (* (pow x_m 2.0) -3.820122044248399)
        (+ (* 0.3111712305105463 (pow x_m 3.0)) (* x_m 3.385159067440336))))
      (+ t_3 (+ 1.0 (pow t_3 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = pow(fma(0.3275911, x_m, 1.0), 3.0);
	double t_1 = 1.0 + (0.3275911 * fabs(x_m));
	double t_2 = 1.0 / t_1;
	double t_3 = (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / (fma(0.3275911, x_m, 1.0) * exp(pow(x_m, 2.0)));
	double tmp;
	if (((t_2 * (0.254829592 + (t_2 * (-0.284496736 + (t_2 * (1.421413741 + (t_2 * (-1.453152027 + (t_2 * 1.061405429))))))))) * exp(-(x_m * x_m))) <= 0.9995) {
		tmp = fma((-0.254829592 + ((1.453152027 / t_0) - (((1.061405429 / t_0) + (-0.284496736 + (1.421413741 / fma(0.3275911, x_m, 1.0)))) / fma(0.3275911, x_m, 1.0)))), (pow(exp(x_m), -x_m) / t_1), 1.0);
	} else {
		tmp = (2.999999997e-9 + ((pow(x_m, 2.0) * -3.820122044248399) + ((0.3111712305105463 * pow(x_m, 3.0)) + (x_m * 3.385159067440336)))) / (t_3 + (1.0 + pow(t_3, 2.0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(0.3275911, x_m, 1.0) ^ 3.0
	t_1 = Float64(1.0 + Float64(0.3275911 * abs(x_m)))
	t_2 = Float64(1.0 / t_1)
	t_3 = Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / Float64(fma(0.3275911, x_m, 1.0) * exp((x_m ^ 2.0))))
	tmp = 0.0
	if (Float64(Float64(t_2 * Float64(0.254829592 + Float64(t_2 * Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(t_2 * Float64(-1.453152027 + Float64(t_2 * 1.061405429))))))))) * exp(Float64(-Float64(x_m * x_m)))) <= 0.9995)
		tmp = fma(Float64(-0.254829592 + Float64(Float64(1.453152027 / t_0) - Float64(Float64(Float64(1.061405429 / t_0) + Float64(-0.284496736 + Float64(1.421413741 / fma(0.3275911, x_m, 1.0)))) / fma(0.3275911, x_m, 1.0)))), Float64((exp(x_m) ^ Float64(-x_m)) / t_1), 1.0);
	else
		tmp = Float64(Float64(2.999999997e-9 + Float64(Float64((x_m ^ 2.0) * -3.820122044248399) + Float64(Float64(0.3111712305105463 * (x_m ^ 3.0)) + Float64(x_m * 3.385159067440336)))) / Float64(t_3 + Float64(1.0 + (t_3 ^ 2.0))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[(0.254829592 + N[(t$95$2 * N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(t$95$2 * N[(-1.453152027 + N[(t$95$2 * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(x$95$m * x$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], 0.9995], N[(N[(-0.254829592 + N[(N[(1.453152027 / t$95$0), $MachinePrecision] - N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] + N[(-0.284496736 + N[(1.421413741 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision] / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(2.999999997e-9 + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -3.820122044248399), $MachinePrecision] + N[(N[(0.3111712305105463 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 3.385159067440336), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 + N[(1.0 + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 0.99950000000000006

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate--r+ 99.7%

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

      2. div-sub 99.7%

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

   7. Simplified 98.3%

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

      1. fma-udef 98.3%

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

   9. Applied egg-rr 98.3%

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

   if 0.99950000000000006 < (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))

   1. Initial program 58.0%

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

   3. Simplified 58.0%

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

      1. associate-/l/ 58.0%

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

      2. flip3-- 58.1%

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

   6. Applied egg-rr 58.1%

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

   8. Simplified 56.5%

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

   9. Taylor expanded in x around 0 95.0%

      \[\leadsto \frac{\color{blue}{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + \left(0.3111712305105463 \cdot {x}^{3} + 3.385159067440336 \cdot x\right)\right)}}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}} + \left(1 + {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \leq 0.9995:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \left(\frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \frac{\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} + \left(-0.284496736 + \frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{1 + 0.3275911 \cdot \left|x\right|}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + \left({x}^{2} \cdot -3.820122044248399 + \left(0.3111712305105463 \cdot {x}^{3} + x \cdot 3.385159067440336\right)\right)}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}} + \left(1 + {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.3275911, x\_m, 1\right)\right)}^{3}\\ t_1 := 1 + 0.3275911 \cdot \left|x\_m\right|\\ t_2 := \frac{1}{t\_1}\\ t_3 := \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right) \cdot e^{{x\_m}^{2}}}\\ \mathbf{if}\;\left(t\_2 \cdot \left(0.254829592 + t\_2 \cdot \left(-0.284496736 + t\_2 \cdot \left(1.421413741 + t\_2 \cdot \left(-1.453152027 + t\_2 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-x\_m \cdot x\_m} \leq 0.99998:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \left(\frac{1.453152027}{t\_0} - \frac{\frac{1.061405429}{t\_0} + \left(-0.284496736 + \frac{1.421413741}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right), \frac{{\left(e^{x\_m}\right)}^{\left(-x\_m\right)}}{t\_1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + \left({x\_m}^{2} \cdot -3.820122044248399 + x\_m \cdot 3.385159067440336\right)}{t\_3 + \left(1 + {t\_3}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (pow (fma 0.3275911 x_m 1.0) 3.0))
        (t_1 (+ 1.0 (* 0.3275911 (fabs x_m))))
        (t_2 (/ 1.0 t_1))
        (t_3
         (/
          (+
           0.254829592
           (/
            (+
             -0.284496736
             (/
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
                (fma 0.3275911 x_m 1.0)))
              (fma 0.3275911 x_m 1.0)))
            (fma 0.3275911 x_m 1.0)))
          (* (fma 0.3275911 x_m 1.0) (exp (pow x_m 2.0))))))
   (if (<=
        (*
         (*
          t_2
          (+
           0.254829592
           (*
            t_2
            (+
             -0.284496736
             (*
              t_2
              (+ 1.421413741 (* t_2 (+ -1.453152027 (* t_2 1.061405429)))))))))
         (exp (- (* x_m x_m))))
        0.99998)
     (fma
      (+
       -0.254829592
       (-
        (/ 1.453152027 t_0)
        (/
         (+
          (/ 1.061405429 t_0)
          (+ -0.284496736 (/ 1.421413741 (fma 0.3275911 x_m 1.0))))
         (fma 0.3275911 x_m 1.0))))
      (/ (pow (exp x_m) (- x_m)) t_1)
      1.0)
     (/
      (+
       2.999999997e-9
       (+ (* (pow x_m 2.0) -3.820122044248399) (* x_m 3.385159067440336)))
      (+ t_3 (+ 1.0 (pow t_3 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = pow(fma(0.3275911, x_m, 1.0), 3.0);
	double t_1 = 1.0 + (0.3275911 * fabs(x_m));
	double t_2 = 1.0 / t_1;
	double t_3 = (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / (fma(0.3275911, x_m, 1.0) * exp(pow(x_m, 2.0)));
	double tmp;
	if (((t_2 * (0.254829592 + (t_2 * (-0.284496736 + (t_2 * (1.421413741 + (t_2 * (-1.453152027 + (t_2 * 1.061405429))))))))) * exp(-(x_m * x_m))) <= 0.99998) {
		tmp = fma((-0.254829592 + ((1.453152027 / t_0) - (((1.061405429 / t_0) + (-0.284496736 + (1.421413741 / fma(0.3275911, x_m, 1.0)))) / fma(0.3275911, x_m, 1.0)))), (pow(exp(x_m), -x_m) / t_1), 1.0);
	} else {
		tmp = (2.999999997e-9 + ((pow(x_m, 2.0) * -3.820122044248399) + (x_m * 3.385159067440336))) / (t_3 + (1.0 + pow(t_3, 2.0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(0.3275911, x_m, 1.0) ^ 3.0
	t_1 = Float64(1.0 + Float64(0.3275911 * abs(x_m)))
	t_2 = Float64(1.0 / t_1)
	t_3 = Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / Float64(fma(0.3275911, x_m, 1.0) * exp((x_m ^ 2.0))))
	tmp = 0.0
	if (Float64(Float64(t_2 * Float64(0.254829592 + Float64(t_2 * Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(t_2 * Float64(-1.453152027 + Float64(t_2 * 1.061405429))))))))) * exp(Float64(-Float64(x_m * x_m)))) <= 0.99998)
		tmp = fma(Float64(-0.254829592 + Float64(Float64(1.453152027 / t_0) - Float64(Float64(Float64(1.061405429 / t_0) + Float64(-0.284496736 + Float64(1.421413741 / fma(0.3275911, x_m, 1.0)))) / fma(0.3275911, x_m, 1.0)))), Float64((exp(x_m) ^ Float64(-x_m)) / t_1), 1.0);
	else
		tmp = Float64(Float64(2.999999997e-9 + Float64(Float64((x_m ^ 2.0) * -3.820122044248399) + Float64(x_m * 3.385159067440336))) / Float64(t_3 + Float64(1.0 + (t_3 ^ 2.0))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[(0.254829592 + N[(t$95$2 * N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(t$95$2 * N[(-1.453152027 + N[(t$95$2 * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(x$95$m * x$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], 0.99998], N[(N[(-0.254829592 + N[(N[(1.453152027 / t$95$0), $MachinePrecision] - N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] + N[(-0.284496736 + N[(1.421413741 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision] / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(2.999999997e-9 + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -3.820122044248399), $MachinePrecision] + N[(x$95$m * 3.385159067440336), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 + N[(1.0 + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 0.99997999999999998

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. associate--r+ 99.5%

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

      2. div-sub 99.5%

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

   7. Simplified 97.5%

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

      1. fma-udef 97.5%

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

   9. Applied egg-rr 97.5%

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

   if 0.99997999999999998 < (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))

   1. Initial program 57.9%

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

   3. Simplified 57.9%

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

      1. associate-/l/ 57.9%

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

      2. flip3-- 57.9%

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

   6. Applied egg-rr 57.9%

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

   8. Simplified 57.0%

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

   9. Taylor expanded in x around 0 95.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \leq 0.99998:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \left(\frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \frac{\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} + \left(-0.284496736 + \frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{1 + 0.3275911 \cdot \left|x\right|}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2.999999997 \cdot 10^{-9} + \left({x}^{2} \cdot -3.820122044248399 + x \cdot 3.385159067440336\right)}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}} + \left(1 + {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.3275911, x\_m, 1\right)\right)}^{-3}\\ \mathbf{if}\;\left|x\_m\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x\_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\mathsf{fma}\left(\left(-0.254829592 - \frac{\mathsf{fma}\left(1.061405429, t\_0, -0.284496736 + \frac{1.421413741}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right) + 1.453152027 \cdot t\_0, \frac{e^{-{x\_m}^{2}}}{\mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right)}, 1\right)}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (pow (fma 0.3275911 x_m 1.0) -3.0)))
   (if (<= (fabs x_m) 1e-17)
     (+ 1e-9 (* x_m 0.3275910996724089))
     (log
      (exp
       (fma
        (+
         (-
          -0.254829592
          (/
           (fma
            1.061405429
            t_0
            (+ -0.284496736 (/ 1.421413741 (fma 0.3275911 x_m 1.0))))
           (fma 0.3275911 x_m 1.0)))
         (* 1.453152027 t_0))
        (/ (exp (- (pow x_m 2.0))) (fma 0.3275911 (fabs x_m) 1.0))
        1.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = pow(fma(0.3275911, x_m, 1.0), -3.0);
	double tmp;
	if (fabs(x_m) <= 1e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = log(exp(fma(((-0.254829592 - (fma(1.061405429, t_0, (-0.284496736 + (1.421413741 / fma(0.3275911, x_m, 1.0)))) / fma(0.3275911, x_m, 1.0))) + (1.453152027 * t_0)), (exp(-pow(x_m, 2.0)) / fma(0.3275911, fabs(x_m), 1.0)), 1.0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(0.3275911, x_m, 1.0) ^ -3.0
	tmp = 0.0
	if (abs(x_m) <= 1e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = log(exp(fma(Float64(Float64(-0.254829592 - Float64(fma(1.061405429, t_0, Float64(-0.284496736 + Float64(1.421413741 / fma(0.3275911, x_m, 1.0)))) / fma(0.3275911, x_m, 1.0))) + Float64(1.453152027 * t_0)), Float64(exp(Float64(-(x_m ^ 2.0))) / fma(0.3275911, abs(x_m), 1.0)), 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], -3.0], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(N[(N[(-0.254829592 - N[(N[(1.061405429 * t$95$0 + N[(-0.284496736 + N[(1.421413741 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.453152027 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000007e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 57.8%

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

      1. distribute-lft-in 57.8%

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

      2. associate-*l/ 57.8%

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

      3. *-lft-identity 57.8%

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

   8. Simplified 57.8%

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

   9. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. pow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   11. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

       2. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

       3. sqr-pow 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

       4. fabs-sqr 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

       5. sqr-pow 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

       6. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   13. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 99.5%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   15. Step-by-step derivation

       1. *-commutative 99.5%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   16. Simplified 99.5%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 1.00000000000000007e-17 < (fabs.f64 x)

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 97.1%

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

      1. associate--r+ 97.1%

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

      2. div-sub 97.2%

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

   7. Simplified 94.2%

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

      1. add-log-exp 94.2%

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

   9. Applied egg-rr 94.3%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\mathsf{fma}\left(\left(-0.254829592 - \frac{\mathsf{fma}\left(1.061405429, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-3}, -0.284496736 + \frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + 1.453152027 \cdot {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-3}, \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.3275911, x\_m, 1\right)\right)}^{-3}\\ \mathbf{if}\;\left|x\_m\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x\_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.254829592 - \frac{\mathsf{fma}\left(1.061405429, t\_0, -0.284496736 + \frac{1.421413741}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right) + 1.453152027 \cdot t\_0, \frac{e^{-{x\_m}^{2}}}{\mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (pow (fma 0.3275911 x_m 1.0) -3.0)))
   (if (<= (fabs x_m) 1e-17)
     (+ 1e-9 (* x_m 0.3275910996724089))
     (fma
      (+
       (-
        -0.254829592
        (/
         (fma
          1.061405429
          t_0
          (+ -0.284496736 (/ 1.421413741 (fma 0.3275911 x_m 1.0))))
         (fma 0.3275911 x_m 1.0)))
       (* 1.453152027 t_0))
      (/ (exp (- (pow x_m 2.0))) (fma 0.3275911 (fabs x_m) 1.0))
      1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = pow(fma(0.3275911, x_m, 1.0), -3.0);
	double tmp;
	if (fabs(x_m) <= 1e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = fma(((-0.254829592 - (fma(1.061405429, t_0, (-0.284496736 + (1.421413741 / fma(0.3275911, x_m, 1.0)))) / fma(0.3275911, x_m, 1.0))) + (1.453152027 * t_0)), (exp(-pow(x_m, 2.0)) / fma(0.3275911, fabs(x_m), 1.0)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(0.3275911, x_m, 1.0) ^ -3.0
	tmp = 0.0
	if (abs(x_m) <= 1e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = fma(Float64(Float64(-0.254829592 - Float64(fma(1.061405429, t_0, Float64(-0.284496736 + Float64(1.421413741 / fma(0.3275911, x_m, 1.0)))) / fma(0.3275911, x_m, 1.0))) + Float64(1.453152027 * t_0)), Float64(exp(Float64(-(x_m ^ 2.0))) / fma(0.3275911, abs(x_m), 1.0)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], -3.0], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.254829592 - N[(N[(1.061405429 * t$95$0 + N[(-0.284496736 + N[(1.421413741 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.453152027 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000007e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 57.8%

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

      1. distribute-lft-in 57.8%

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

      2. associate-*l/ 57.8%

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

      3. *-lft-identity 57.8%

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

   8. Simplified 57.8%

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

   9. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. pow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   11. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

       2. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

       3. sqr-pow 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

       4. fabs-sqr 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

       5. sqr-pow 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

       6. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   13. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 99.5%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   15. Step-by-step derivation

       1. *-commutative 99.5%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   16. Simplified 99.5%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 1.00000000000000007e-17 < (fabs.f64 x)

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 97.1%

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

      1. associate--r+ 97.1%

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

      2. div-sub 97.2%

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

   7. Simplified 94.2%

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

      1. fma-udef 94.2%

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

   9. Applied egg-rr 94.3%

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

       1. fma-def 94.3%

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

       2. +-commutative 94.3%

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

       3. *-commutative 94.3%

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

   11. Simplified 94.3%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.254829592 - \frac{\mathsf{fma}\left(1.061405429, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-3}, -0.284496736 + \frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + 1.453152027 \cdot {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-3}, \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x\_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{e^{-{x\_m}^{2}}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} \cdot \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + \left(-0.254829592 + \left(\frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x\_m, 1\right)\right)}^{3}} - \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x\_m, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x\_m, 1\right)\right)}^{2}}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 1e-17)
   (+ 1e-9 (* x_m 0.3275910996724089))
   (+
    1.0
    (*
     (/ (exp (- (pow x_m 2.0))) (fma 0.3275911 x_m 1.0))
     (+
      (/ 0.284496736 (fma 0.3275911 x_m 1.0))
      (+
       -0.254829592
       (-
        (/ 1.453152027 (pow (fma 0.3275911 x_m 1.0) 3.0))
        (+
         (/ 1.061405429 (pow (fma 0.3275911 x_m 1.0) 4.0))
         (/ 1.421413741 (pow (fma 0.3275911 x_m 1.0) 2.0))))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 1e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0 + ((exp(-pow(x_m, 2.0)) / fma(0.3275911, x_m, 1.0)) * ((0.284496736 / fma(0.3275911, x_m, 1.0)) + (-0.254829592 + ((1.453152027 / pow(fma(0.3275911, x_m, 1.0), 3.0)) - ((1.061405429 / pow(fma(0.3275911, x_m, 1.0), 4.0)) + (1.421413741 / pow(fma(0.3275911, x_m, 1.0), 2.0)))))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 1e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(-(x_m ^ 2.0))) / fma(0.3275911, x_m, 1.0)) * Float64(Float64(0.284496736 / fma(0.3275911, x_m, 1.0)) + Float64(-0.254829592 + Float64(Float64(1.453152027 / (fma(0.3275911, x_m, 1.0) ^ 3.0)) - Float64(Float64(1.061405429 / (fma(0.3275911, x_m, 1.0) ^ 4.0)) + Float64(1.421413741 / (fma(0.3275911, x_m, 1.0) ^ 2.0))))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(0.284496736 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-0.254829592 + N[(N[(1.453152027 / N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - N[(N[(1.061405429 / N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.421413741 / N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000007e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 57.8%

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

      1. distribute-lft-in 57.8%

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

      2. associate-*l/ 57.8%

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

      3. *-lft-identity 57.8%

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

   8. Simplified 57.8%

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

   9. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. pow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   11. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

       2. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

       3. sqr-pow 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

       4. fabs-sqr 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

       5. sqr-pow 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

       6. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   13. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 99.5%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   15. Step-by-step derivation

       1. *-commutative 99.5%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   16. Simplified 99.5%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 1.00000000000000007e-17 < (fabs.f64 x)

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around inf 97.1%

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

   7. Simplified 94.3%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{e^{-{x}^{2}}}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \left(-0.254829592 + \left(\frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.3275911, x\_m, 1\right)\right)}^{3}\\ \mathbf{if}\;\left|x\_m\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x\_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \left(\frac{1.453152027}{t\_0} - \frac{\frac{1.061405429}{t\_0} + \left(-0.284496736 + \frac{1.421413741}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right), \frac{{\left(e^{x\_m}\right)}^{\left(-x\_m\right)}}{1 + 0.3275911 \cdot \left|x\_m\right|}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (pow (fma 0.3275911 x_m 1.0) 3.0)))
   (if (<= (fabs x_m) 1e-17)
     (+ 1e-9 (* x_m 0.3275910996724089))
     (fma
      (+
       -0.254829592
       (-
        (/ 1.453152027 t_0)
        (/
         (+
          (/ 1.061405429 t_0)
          (+ -0.284496736 (/ 1.421413741 (fma 0.3275911 x_m 1.0))))
         (fma 0.3275911 x_m 1.0))))
      (/ (pow (exp x_m) (- x_m)) (+ 1.0 (* 0.3275911 (fabs x_m))))
      1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = pow(fma(0.3275911, x_m, 1.0), 3.0);
	double tmp;
	if (fabs(x_m) <= 1e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = fma((-0.254829592 + ((1.453152027 / t_0) - (((1.061405429 / t_0) + (-0.284496736 + (1.421413741 / fma(0.3275911, x_m, 1.0)))) / fma(0.3275911, x_m, 1.0)))), (pow(exp(x_m), -x_m) / (1.0 + (0.3275911 * fabs(x_m)))), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(0.3275911, x_m, 1.0) ^ 3.0
	tmp = 0.0
	if (abs(x_m) <= 1e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = fma(Float64(-0.254829592 + Float64(Float64(1.453152027 / t_0) - Float64(Float64(Float64(1.061405429 / t_0) + Float64(-0.284496736 + Float64(1.421413741 / fma(0.3275911, x_m, 1.0)))) / fma(0.3275911, x_m, 1.0)))), Float64((exp(x_m) ^ Float64(-x_m)) / Float64(1.0 + Float64(0.3275911 * abs(x_m)))), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], 3.0], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[(N[(-0.254829592 + N[(N[(1.453152027 / t$95$0), $MachinePrecision] - N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] + N[(-0.284496736 + N[(1.421413741 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision] / N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000007e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 57.8%

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

      1. distribute-lft-in 57.8%

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

      2. associate-*l/ 57.8%

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

      3. *-lft-identity 57.8%

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

   8. Simplified 57.8%

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

   9. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. pow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   11. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

       2. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

       3. sqr-pow 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

       4. fabs-sqr 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

       5. sqr-pow 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

       6. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   13. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 99.5%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   15. Step-by-step derivation

       1. *-commutative 99.5%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   16. Simplified 99.5%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 1.00000000000000007e-17 < (fabs.f64 x)

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 97.1%

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

      1. associate--r+ 97.1%

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

      2. div-sub 97.2%

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

   7. Simplified 94.2%

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

      1. fma-udef 94.2%

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

   9. Applied egg-rr 94.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \left(\frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \frac{\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} + \left(-0.284496736 + \frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{1 + 0.3275911 \cdot \left|x\right|}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x\_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{e^{{x\_m}^{2}}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 1e-17)
   (+ 1e-9 (* x_m 0.3275910996724089))
   (exp
    (log1p
     (/
      (/
       (-
        -0.254829592
        (/
         (+
          -0.284496736
          (/
           (+
            1.421413741
            (/
             (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
             (fma 0.3275911 x_m 1.0)))
           (fma 0.3275911 x_m 1.0)))
         (fma 0.3275911 x_m 1.0)))
       (fma 0.3275911 x_m 1.0))
      (exp (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 1e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = exp(log1p((((-0.254829592 - ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) / exp(pow(x_m, 2.0)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 1e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = exp(log1p(Float64(Float64(Float64(-0.254829592 - Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) / exp((x_m ^ 2.0)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[1 + N[(N[(N[(-0.254829592 - N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] / N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000007e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 57.8%

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

      1. distribute-lft-in 57.8%

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

      2. associate-*l/ 57.8%

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

      3. *-lft-identity 57.8%

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

   8. Simplified 57.8%

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

   9. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. pow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   11. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

       2. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

       3. sqr-pow 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

       4. fabs-sqr 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

       5. sqr-pow 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

       6. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   13. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 99.5%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   15. Step-by-step derivation

       1. *-commutative 99.5%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   16. Simplified 99.5%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 1.00000000000000007e-17 < (fabs.f64 x)

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

      1. associate-/l/ 97.2%

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

      2. add-exp-log 97.2%

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

      3. sub-neg 97.2%

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

      4. log1p-def 97.2%

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

   6. Applied egg-rr 97.2%

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

   8. Simplified 94.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x\_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right) \cdot e^{{x\_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 1e-17)
   (+ 1e-9 (* x_m 0.3275910996724089))
   (-
    1.0
    (/
     (+
      0.254829592
      (/
       (+
        -0.284496736
        (/
         (+
          1.421413741
          (/
           (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
           (fma 0.3275911 x_m 1.0)))
         (fma 0.3275911 x_m 1.0)))
       (fma 0.3275911 x_m 1.0)))
     (* (fma 0.3275911 x_m 1.0) (exp (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 1e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0 - ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / (fma(0.3275911, x_m, 1.0) * exp(pow(x_m, 2.0))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 1e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = Float64(1.0 - Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / Float64(fma(0.3275911, x_m, 1.0) * exp((x_m ^ 2.0)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000007e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 57.8%

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

      1. distribute-lft-in 57.8%

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

      2. associate-*l/ 57.8%

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

      3. *-lft-identity 57.8%

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

   8. Simplified 57.8%

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

   9. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. pow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   11. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

       2. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

       3. sqr-pow 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

       4. fabs-sqr 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

       5. sqr-pow 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

       6. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   13. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 99.5%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   15. Step-by-step derivation

       1. *-commutative 99.5%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   16. Simplified 99.5%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 1.00000000000000007e-17 < (fabs.f64 x)

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   6. Applied egg-rr 97.1%

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

   8. Simplified 94.3%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\_m\right|\\ \mathbf{if}\;\left|x\_m\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x\_m \cdot 0.3275910996724089\\ \mathbf{elif}\;\left|x\_m\right| \leq 0.2:\\ \;\;\;\;\left(1 + \left(0.36953108532122814 \cdot \frac{{x\_m}^{2}}{t\_0} + 0.8007952583978091 \cdot \frac{x\_m}{t\_0}\right)\right) + 0.999999999 \cdot \frac{-1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.3275911 (fabs x_m)))))
   (if (<= (fabs x_m) 1e-17)
     (+ 1e-9 (* x_m 0.3275910996724089))
     (if (<= (fabs x_m) 0.2)
       (+
        (+
         1.0
         (+
          (* 0.36953108532122814 (/ (pow x_m 2.0) t_0))
          (* 0.8007952583978091 (/ x_m t_0))))
        (* 0.999999999 (/ -1.0 t_0)))
       1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (0.3275911 * fabs(x_m));
	double tmp;
	if (fabs(x_m) <= 1e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else if (fabs(x_m) <= 0.2) {
		tmp = (1.0 + ((0.36953108532122814 * (pow(x_m, 2.0) / t_0)) + (0.8007952583978091 * (x_m / t_0)))) + (0.999999999 * (-1.0 / t_0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (0.3275911d0 * abs(x_m))
    if (abs(x_m) <= 1d-17) then
        tmp = 1d-9 + (x_m * 0.3275910996724089d0)
    else if (abs(x_m) <= 0.2d0) then
        tmp = (1.0d0 + ((0.36953108532122814d0 * ((x_m ** 2.0d0) / t_0)) + (0.8007952583978091d0 * (x_m / t_0)))) + (0.999999999d0 * ((-1.0d0) / t_0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (0.3275911 * Math.abs(x_m));
	double tmp;
	if (Math.abs(x_m) <= 1e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else if (Math.abs(x_m) <= 0.2) {
		tmp = (1.0 + ((0.36953108532122814 * (Math.pow(x_m, 2.0) / t_0)) + (0.8007952583978091 * (x_m / t_0)))) + (0.999999999 * (-1.0 / t_0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (0.3275911 * math.fabs(x_m))
	tmp = 0
	if math.fabs(x_m) <= 1e-17:
		tmp = 1e-9 + (x_m * 0.3275910996724089)
	elif math.fabs(x_m) <= 0.2:
		tmp = (1.0 + ((0.36953108532122814 * (math.pow(x_m, 2.0) / t_0)) + (0.8007952583978091 * (x_m / t_0)))) + (0.999999999 * (-1.0 / t_0))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x_m)))
	tmp = 0.0
	if (abs(x_m) <= 1e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	elseif (abs(x_m) <= 0.2)
		tmp = Float64(Float64(1.0 + Float64(Float64(0.36953108532122814 * Float64((x_m ^ 2.0) / t_0)) + Float64(0.8007952583978091 * Float64(x_m / t_0)))) + Float64(0.999999999 * Float64(-1.0 / t_0)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (0.3275911 * abs(x_m));
	tmp = 0.0;
	if (abs(x_m) <= 1e-17)
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	elseif (abs(x_m) <= 0.2)
		tmp = (1.0 + ((0.36953108532122814 * ((x_m ^ 2.0) / t_0)) + (0.8007952583978091 * (x_m / t_0)))) + (0.999999999 * (-1.0 / t_0));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.2], N[(N[(1.0 + N[(N[(0.36953108532122814 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.8007952583978091 * N[(x$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.999999999 * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (fabs.f64 x) < 1.00000000000000007e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 57.8%

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

      1. distribute-lft-in 57.8%

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

      2. associate-*l/ 57.8%

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

      3. *-lft-identity 57.8%

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

   8. Simplified 57.8%

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

   9. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. pow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   11. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

       2. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

       3. sqr-pow 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

       4. fabs-sqr 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

       5. sqr-pow 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

       6. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   13. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 99.5%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   15. Step-by-step derivation

       1. *-commutative 99.5%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   16. Simplified 99.5%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 1.00000000000000007e-17 < (fabs.f64 x) < 0.20000000000000001

   1. Initial program 67.6%

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

   3. Simplified 68.0%

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

   6. Applied egg-rr 68.3%

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

      1. distribute-lft-in 67.8%

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

      2. associate-*l/ 68.0%

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

      3. *-lft-identity 68.0%

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

   8. Simplified 36.3%

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

   9. Taylor expanded in x around 0 35.5%

      \[\leadsto \color{blue}{\left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]

   if 0.20000000000000001 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 99.1%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. pow1 99.1%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   11. Applied egg-rr 99.1%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 99.1%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

       2. unpow1 99.1%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

       3. sqr-pow 52.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

       4. fabs-sqr 52.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

       5. sqr-pow 99.2%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

       6. unpow1 99.2%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   13. Simplified 99.2%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around inf 100.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{elif}\;\left|x\right| \leq 0.2:\\ \;\;\;\;\left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) + 0.999999999 \cdot \frac{-1}{1 + 0.3275911 \cdot \left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + 0.3275911 \cdot x\_m\\ t_1 := \frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|}\\ \mathbf{if}\;\left|x\_m\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x\_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 - \left(t\_1 \cdot \left(0.254829592 + t\_1 \cdot \left(-0.284496736 + t\_1 \cdot \left(1.421413741 + \frac{1}{t\_0} \cdot \left(-1.453152027 + \frac{1.061405429}{t\_0}\right)\right)\right)\right)\right) \cdot e^{-x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.3275911 x_m)))
        (t_1 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m))))))
   (if (<= (fabs x_m) 1e-17)
     (+ 1e-9 (* x_m 0.3275910996724089))
     (-
      1.0
      (*
       (*
        t_1
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (* (/ 1.0 t_0) (+ -1.453152027 (/ 1.061405429 t_0)))))))))
       (exp (- (* x_m x_m))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (0.3275911 * x_m);
	double t_1 = 1.0 / (1.0 + (0.3275911 * fabs(x_m)));
	double tmp;
	if (fabs(x_m) <= 1e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0 - ((t_1 * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + ((1.0 / t_0) * (-1.453152027 + (1.061405429 / t_0))))))))) * exp(-(x_m * x_m)));
	}
	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 + (0.3275911d0 * x_m)
    t_1 = 1.0d0 / (1.0d0 + (0.3275911d0 * abs(x_m)))
    if (abs(x_m) <= 1d-17) then
        tmp = 1d-9 + (x_m * 0.3275910996724089d0)
    else
        tmp = 1.0d0 - ((t_1 * (0.254829592d0 + (t_1 * ((-0.284496736d0) + (t_1 * (1.421413741d0 + ((1.0d0 / t_0) * ((-1.453152027d0) + (1.061405429d0 / t_0))))))))) * exp(-(x_m * x_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (0.3275911 * x_m);
	double t_1 = 1.0 / (1.0 + (0.3275911 * Math.abs(x_m)));
	double tmp;
	if (Math.abs(x_m) <= 1e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0 - ((t_1 * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + ((1.0 / t_0) * (-1.453152027 + (1.061405429 / t_0))))))))) * Math.exp(-(x_m * x_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (0.3275911 * x_m)
	t_1 = 1.0 / (1.0 + (0.3275911 * math.fabs(x_m)))
	tmp = 0
	if math.fabs(x_m) <= 1e-17:
		tmp = 1e-9 + (x_m * 0.3275910996724089)
	else:
		tmp = 1.0 - ((t_1 * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + ((1.0 / t_0) * (-1.453152027 + (1.061405429 / t_0))))))))) * math.exp(-(x_m * x_m)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(0.3275911 * x_m))
	t_1 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x_m))))
	tmp = 0.0
	if (abs(x_m) <= 1e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = Float64(1.0 - Float64(Float64(t_1 * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(Float64(1.0 / t_0) * Float64(-1.453152027 + Float64(1.061405429 / t_0))))))))) * exp(Float64(-Float64(x_m * x_m)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (0.3275911 * x_m);
	t_1 = 1.0 / (1.0 + (0.3275911 * abs(x_m)));
	tmp = 0.0;
	if (abs(x_m) <= 1e-17)
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	else
		tmp = 1.0 - ((t_1 * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + ((1.0 / t_0) * (-1.453152027 + (1.061405429 / t_0))))))))) * exp(-(x_m * x_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(0.3275911 * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(t$95$1 * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(x$95$m * x$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000007e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 57.8%

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

      1. distribute-lft-in 57.8%

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

      2. associate-*l/ 57.8%

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

      3. *-lft-identity 57.8%

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

   8. Simplified 57.8%

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

   9. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. pow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   11. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

       2. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

       3. sqr-pow 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

       4. fabs-sqr 26.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

       5. sqr-pow 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

       6. unpow1 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   13. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 99.5%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   15. Step-by-step derivation

       1. *-commutative 99.5%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   16. Simplified 99.5%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 1.00000000000000007e-17 < (fabs.f64 x)

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

      1. pow1 92.2%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   6. Applied egg-rr 97.1%

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

      1. unpow1 92.2%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

      2. unpow1 92.2%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

      3. sqr-pow 48.7%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

      4. fabs-sqr 48.7%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

      5. sqr-pow 91.7%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

      6. unpow1 91.7%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   8. Simplified 94.8%

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

      1. pow1 92.2%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   10. Applied egg-rr 94.8%

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

       1. unpow1 92.2%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

       2. unpow1 92.2%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

       3. sqr-pow 48.7%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

       4. fabs-sqr 48.7%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

       5. sqr-pow 91.7%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

       6. unpow1 91.7%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   12. Simplified 94.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot x}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.4% accurate, 7.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.2:\\ \;\;\;\;10^{-9} + x\_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.2) (+ 1e-9 (* x_m 0.3275910996724089)) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.2) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (abs(x_m) <= 0.2d0) then
        tmp = 1d-9 + (x_m * 0.3275910996724089d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.2) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.2:
		tmp = 1e-9 + (x_m * 0.3275910996724089)
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.2)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.2)
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.2], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.20000000000000001

   1. Initial program 58.6%

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

   3. Simplified 58.7%

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

   6. Applied egg-rr 58.7%

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

      1. distribute-lft-in 58.7%

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

      2. associate-*l/ 58.7%

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

      3. *-lft-identity 58.7%

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

   8. Simplified 55.9%

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

   9. Taylor expanded in x around 0 54.6%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. pow1 54.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   11. Applied egg-rr 54.6%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 54.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

       2. unpow1 54.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

       3. sqr-pow 25.1%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

       4. fabs-sqr 25.1%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

       5. sqr-pow 54.1%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

       6. unpow1 54.1%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   13. Simplified 54.1%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 92.1%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   15. Step-by-step derivation

       1. *-commutative 92.1%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   16. Simplified 92.1%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 0.20000000000000001 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 99.1%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. pow1 99.1%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   11. Applied egg-rr 99.1%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 99.1%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

       2. unpow1 99.1%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

       3. sqr-pow 52.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

       4. fabs-sqr 52.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

       5. sqr-pow 99.2%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

       6. unpow1 99.2%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   13. Simplified 99.2%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around inf 100.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.4% accurate, 142.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

   6. Applied egg-rr 70.9%

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

      1. distribute-lft-in 70.9%

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

      2. associate-*l/ 70.9%

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

      3. *-lft-identity 70.9%

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

   8. Simplified 68.9%

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

   9. Taylor expanded in x around 0 67.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. pow1 67.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   11. Applied egg-rr 67.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 67.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

       2. unpow1 67.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

       3. sqr-pow 17.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

       4. fabs-sqr 17.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

       5. sqr-pow 67.5%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

       6. unpow1 67.5%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   13. Simplified 67.5%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 68.5%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. associate-*l/ 99.7%

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

      3. *-lft-identity 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around 0 98.6%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. pow1 98.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

   11. Applied egg-rr 98.6%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 98.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

       2. unpow1 98.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

       3. sqr-pow 98.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

       4. fabs-sqr 98.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

       5. sqr-pow 98.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

       6. unpow1 98.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

   13. Simplified 98.6%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around inf 98.7%

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

4. Final simplification 76.2%

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

Alternative 14: 52.6% accurate, 856.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

3. Simplified 78.2%

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

6. Applied egg-rr 78.2%

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

   1. distribute-lft-in 78.2%

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

   2. associate-*l/ 78.2%

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

   3. *-lft-identity 78.2%

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

8. Simplified 76.7%

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

9. Taylor expanded in x around 0 75.6%

   \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
10. Step-by-step derivation

    1. pow1 75.6%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]

11. Applied egg-rr 75.6%

    \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \]
12. Step-by-step derivation

    1. unpow1 75.6%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}} \]

    2. unpow1 75.6%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{1}}\right|} \]

    3. sqr-pow 38.2%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

    4. fabs-sqr 38.2%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \]

    5. sqr-pow 75.4%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{{x}^{1}}} \]

    6. unpow1 75.4%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \color{blue}{x}} \]

13. Simplified 75.4%

    \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

14. Taylor expanded in x around 0 53.9%

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

15. Final simplification 53.9%

    \[\leadsto 10^{-9} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024029 
(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)))))))