Jmat.Real.erf

Percentage Accurate: 79.2% → 99.9%

Time: 1.2min

Alternatives: 13

Speedup: 1.6×

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: 79.2% 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: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\\ t_1 := \mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}\\ t_2 := \frac{0.254829592 + t_0}{t_1}\\ \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 - {\left(t_2 \cdot \frac{-0.254829592 - t_0}{t_1}\right)}^{2}}\right)}{1 + t_2 \cdot \frac{t_0 - -0.254829592}{t_1}}}{1 + t_2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0
         (/
          (+
           -0.284496736
           (/
            (+
             1.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x 1.0)))
              (fma 0.3275911 x 1.0)))
            (fma 0.3275911 x 1.0)))
          (fma 0.3275911 x 1.0)))
        (t_1 (* (fma 0.3275911 x 1.0) (pow (exp x) x)))
        (t_2 (/ (+ 0.254829592 t_0) t_1)))
   (if (<= (fabs x) 0.0001)
     (+
      1e-9
      (+
       (* -0.37545125292247583 (pow x 3.0))
       (+ (* -0.00011824294398844343 (pow x 2.0)) (* x 1.128386358070218))))
     (/
      (/
       (log (exp (- 1.0 (pow (* t_2 (/ (- -0.254829592 t_0) t_1)) 2.0))))
       (+ 1.0 (* t_2 (/ (- t_0 -0.254829592) t_1))))
      (+ 1.0 t_2)))))

C

x = abs(x);
double code(double x) {
	double t_0 = (-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0);
	double t_1 = fma(0.3275911, x, 1.0) * pow(exp(x), x);
	double t_2 = (0.254829592 + t_0) / t_1;
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x, 3.0)) + ((-0.00011824294398844343 * pow(x, 2.0)) + (x * 1.128386358070218)));
	} else {
		tmp = (log(exp((1.0 - pow((t_2 * ((-0.254829592 - t_0) / t_1)), 2.0)))) / (1.0 + (t_2 * ((t_0 - -0.254829592) / t_1)))) / (1.0 + t_2);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))
	t_1 = Float64(fma(0.3275911, x, 1.0) * (exp(x) ^ x))
	t_2 = Float64(Float64(0.254829592 + t_0) / t_1)
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(x * 1.128386358070218))));
	else
		tmp = Float64(Float64(log(exp(Float64(1.0 - (Float64(t_2 * Float64(Float64(-0.254829592 - t_0) / t_1)) ^ 2.0)))) / Float64(1.0 + Float64(t_2 * Float64(Float64(t_0 - -0.254829592) / t_1)))) / Float64(1.0 + t_2));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.3275911 * x + 1.0), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.254829592 + t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[Exp[N[(1.0 - N[Power[N[(t$95$2 * N[(N[(-0.254829592 - t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(t$95$2 * N[(N[(t$95$0 - -0.254829592), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   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%

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

   4. Taylor expanded in x around 0 55.7%

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

   6. Simplified 54.1%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 99.9%

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

   7. Simplified 99.1%

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

      1. flip-+ 99.1%

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

   9. Applied egg-rr 99.1%

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

       1. add-log-exp 99.1%

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

   11. Applied egg-rr 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{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 {\left(e^{x}\right)}^{x}} \cdot \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)}^{2}}\right)}{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 {\left(e^{x}\right)}^{x}} \cdot \frac{\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)} - -0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}}}{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 {\left(e^{x}\right)}^{x}}}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\\ t_1 := {\left(e^{x}\right)}^{x}\\ t_2 := t_1 \cdot \mathsf{fma}\left(x, 0.3275911, 1\right)\\ t_3 := \mathsf{fma}\left(0.3275911, x, 1\right) \cdot t_1\\ t_4 := \frac{0.254829592 + t_0}{t_3}\\ t_5 := \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\\ \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {\left(\frac{0.254829592 + t_5}{t_2} \cdot \frac{-0.254829592 - t_5}{t_2}\right)}^{2}}{1 + t_4 \cdot \frac{t_0 - -0.254829592}{t_3}}}{1 + t_4}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0
         (/
          (+
           -0.284496736
           (/
            (+
             1.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x 1.0)))
              (fma 0.3275911 x 1.0)))
            (fma 0.3275911 x 1.0)))
          (fma 0.3275911 x 1.0)))
        (t_1 (pow (exp x) x))
        (t_2 (* t_1 (fma x 0.3275911 1.0)))
        (t_3 (* (fma 0.3275911 x 1.0) t_1))
        (t_4 (/ (+ 0.254829592 t_0) t_3))
        (t_5
         (/
          (+
           -0.284496736
           (/
            (+
             1.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (fma x 0.3275911 1.0)))
              (fma x 0.3275911 1.0)))
            (fma x 0.3275911 1.0)))
          (fma x 0.3275911 1.0))))
   (if (<= (fabs x) 0.0001)
     (+
      1e-9
      (+
       (* -0.37545125292247583 (pow x 3.0))
       (+ (* -0.00011824294398844343 (pow x 2.0)) (* x 1.128386358070218))))
     (/
      (/
       (-
        1.0
        (pow (* (/ (+ 0.254829592 t_5) t_2) (/ (- -0.254829592 t_5) t_2)) 2.0))
       (+ 1.0 (* t_4 (/ (- t_0 -0.254829592) t_3))))
      (+ 1.0 t_4)))))

C

x = abs(x);
double code(double x) {
	double t_0 = (-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0);
	double t_1 = pow(exp(x), x);
	double t_2 = t_1 * fma(x, 0.3275911, 1.0);
	double t_3 = fma(0.3275911, x, 1.0) * t_1;
	double t_4 = (0.254829592 + t_0) / t_3;
	double t_5 = (-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0);
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x, 3.0)) + ((-0.00011824294398844343 * pow(x, 2.0)) + (x * 1.128386358070218)));
	} else {
		tmp = ((1.0 - pow((((0.254829592 + t_5) / t_2) * ((-0.254829592 - t_5) / t_2)), 2.0)) / (1.0 + (t_4 * ((t_0 - -0.254829592) / t_3)))) / (1.0 + t_4);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))
	t_1 = exp(x) ^ x
	t_2 = Float64(t_1 * fma(x, 0.3275911, 1.0))
	t_3 = Float64(fma(0.3275911, x, 1.0) * t_1)
	t_4 = Float64(Float64(0.254829592 + t_0) / t_3)
	t_5 = Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(x * 1.128386358070218))));
	else
		tmp = Float64(Float64(Float64(1.0 - (Float64(Float64(Float64(0.254829592 + t_5) / t_2) * Float64(Float64(-0.254829592 - t_5) / t_2)) ^ 2.0)) / Float64(1.0 + Float64(t_4 * Float64(Float64(t_0 - -0.254829592) / t_3)))) / Float64(1.0 + t_4));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.3275911 * x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.254829592 + t$95$0), $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Power[N[(N[(N[(0.254829592 + t$95$5), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(-0.254829592 - t$95$5), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$4 * N[(N[(t$95$0 - -0.254829592), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   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%

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

   4. Taylor expanded in x around 0 55.7%

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

   6. Simplified 54.1%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 99.9%

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

   7. Simplified 99.1%

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

      1. flip-+ 99.1%

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

   9. Applied egg-rr 99.1%

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

       1. sub-neg 99.1%

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

   11. Applied egg-rr 99.1%

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

       1. sub-neg 99.1%

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

   13. Simplified 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2}}{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 {\left(e^{x}\right)}^{x}} \cdot \frac{\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)} - -0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}}}{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 {\left(e^{x}\right)}^{x}}}\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\\ t_1 := \mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}\\ t_2 := \frac{0.254829592 + t_0}{t_1}\\ \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(1 + t_2 \cdot \frac{-0.254829592 - t_0}{t_1}\right)}}{1 + t_2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0
         (/
          (+
           -0.284496736
           (/
            (+
             1.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x 1.0)))
              (fma 0.3275911 x 1.0)))
            (fma 0.3275911 x 1.0)))
          (fma 0.3275911 x 1.0)))
        (t_1 (* (fma 0.3275911 x 1.0) (pow (exp x) x)))
        (t_2 (/ (+ 0.254829592 t_0) t_1)))
   (if (<= (fabs x) 0.0001)
     (+
      1e-9
      (+
       (* -0.37545125292247583 (pow x 3.0))
       (+ (* -0.00011824294398844343 (pow x 2.0)) (* x 1.128386358070218))))
     (/
      (exp (log (+ 1.0 (* t_2 (/ (- -0.254829592 t_0) t_1)))))
      (+ 1.0 t_2)))))

C

x = abs(x);
double code(double x) {
	double t_0 = (-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0);
	double t_1 = fma(0.3275911, x, 1.0) * pow(exp(x), x);
	double t_2 = (0.254829592 + t_0) / t_1;
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x, 3.0)) + ((-0.00011824294398844343 * pow(x, 2.0)) + (x * 1.128386358070218)));
	} else {
		tmp = exp(log((1.0 + (t_2 * ((-0.254829592 - t_0) / t_1))))) / (1.0 + t_2);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))
	t_1 = Float64(fma(0.3275911, x, 1.0) * (exp(x) ^ x))
	t_2 = Float64(Float64(0.254829592 + t_0) / t_1)
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(x * 1.128386358070218))));
	else
		tmp = Float64(exp(log(Float64(1.0 + Float64(t_2 * Float64(Float64(-0.254829592 - t_0) / t_1))))) / Float64(1.0 + t_2));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.3275911 * x + 1.0), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.254829592 + t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[Log[N[(1.0 + N[(t$95$2 * N[(N[(-0.254829592 - t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   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%

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

   4. Taylor expanded in x around 0 55.7%

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

   6. Simplified 54.1%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 99.9%

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

   7. Simplified 99.1%

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

      1. add-exp-log 99.1%

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

   9. Applied egg-rr 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}} \cdot \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}}\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\\ t_1 := \mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}\\ t_2 := \frac{0.254829592 + t_0}{t_1}\\ \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_2 \cdot \frac{-0.254829592 - t_0}{t_1}}{1 + t_2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0
         (/
          (+
           -0.284496736
           (/
            (+
             1.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x 1.0)))
              (fma 0.3275911 x 1.0)))
            (fma 0.3275911 x 1.0)))
          (fma 0.3275911 x 1.0)))
        (t_1 (* (fma 0.3275911 x 1.0) (pow (exp x) x)))
        (t_2 (/ (+ 0.254829592 t_0) t_1)))
   (if (<= (fabs x) 0.0001)
     (+
      1e-9
      (+
       (* -0.37545125292247583 (pow x 3.0))
       (+ (* -0.00011824294398844343 (pow x 2.0)) (* x 1.128386358070218))))
     (/ (+ 1.0 (* t_2 (/ (- -0.254829592 t_0) t_1))) (+ 1.0 t_2)))))

C

x = abs(x);
double code(double x) {
	double t_0 = (-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0);
	double t_1 = fma(0.3275911, x, 1.0) * pow(exp(x), x);
	double t_2 = (0.254829592 + t_0) / t_1;
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x, 3.0)) + ((-0.00011824294398844343 * pow(x, 2.0)) + (x * 1.128386358070218)));
	} else {
		tmp = (1.0 + (t_2 * ((-0.254829592 - t_0) / t_1))) / (1.0 + t_2);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))
	t_1 = Float64(fma(0.3275911, x, 1.0) * (exp(x) ^ x))
	t_2 = Float64(Float64(0.254829592 + t_0) / t_1)
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(x * 1.128386358070218))));
	else
		tmp = Float64(Float64(1.0 + Float64(t_2 * Float64(Float64(-0.254829592 - t_0) / t_1))) / Float64(1.0 + t_2));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.3275911 * x + 1.0), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.254829592 + t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t$95$2 * N[(N[(-0.254829592 - t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   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%

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

   4. Taylor expanded in x around 0 55.7%

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

   6. Simplified 54.1%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 99.9%

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

   7. Simplified 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}} \cdot \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}}\\ \end{array} \]

Alternative 5: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t_0} \cdot \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{t_0}}{t_0}}{t_0}\right), {\left(e^{x}\right)}^{\left(-x\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0)))
   (if (<= (fabs x) 0.0001)
     (+
      1e-9
      (+
       (* -0.37545125292247583 (pow x 3.0))
       (+ (* -0.00011824294398844343 (pow x 2.0)) (* x 1.128386358070218))))
     (fma
      (*
       (/ -1.0 t_0)
       (+
        0.254829592
        (/
         (+
          -0.284496736
          (/
           (+
            1.421413741
            (/ (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x 1.0))) t_0))
           t_0))
         t_0)))
      (pow (exp x) (- x))
      1.0))))

C

x = abs(x);
double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x, 3.0)) + ((-0.00011824294398844343 * pow(x, 2.0)) + (x * 1.128386358070218)));
	} else {
		tmp = fma(((-1.0 / t_0) * (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x, 1.0))) / t_0)) / t_0)) / t_0))), pow(exp(x), -x), 1.0);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(x * 1.128386358070218))));
	else
		tmp = fma(Float64(Float64(-1.0 / t_0) * Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x, 1.0))) / t_0)) / t_0)) / t_0))), (exp(x) ^ Float64(-x)), 1.0);
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   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%

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

   4. Taylor expanded in x around 0 55.7%

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

   6. Simplified 54.1%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. pow1 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. unpow1 99.9%

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

      2. unpow1 99.9%

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

      3. sqr-pow 49.5%

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

      4. fabs-sqr 49.5%

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

      5. sqr-pow 99.3%

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

      6. unpow1 99.3%

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

   7. Simplified 99.3%

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

   9. Applied egg-rr 99.3%

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

       1. +-commutative 99.3%

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

       2. associate-*r* 99.3%

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

       3. fma-def 99.3%

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

       4. distribute-neg-frac 99.3%

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

       5. metadata-eval 99.3%

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

   11. Simplified 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right), {\left(e^{x}\right)}^{\left(-x\right)}, 1\right)\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + t_1 \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(t_1 \cdot \left(\left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right) \cdot \frac{-1}{t_0} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x) 0.0001)
     (+
      1e-9
      (+
       (* -0.37545125292247583 (pow x 3.0))
       (+ (* -0.00011824294398844343 (pow x 2.0)) (* x 1.128386358070218))))
     (+
      1.0
      (*
       t_1
       (*
        (exp (* x (- x)))
        (-
         (*
          t_1
          (-
           (*
            (+
             1.421413741
             (* t_1 (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* x 0.3275911))))))
            (/ -1.0 t_0))
           -0.284496736))
         0.254829592)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x, 3.0)) + ((-0.00011824294398844343 * pow(x, 2.0)) + (x * 1.128386358070218)));
	} else {
		tmp = 1.0 + (t_1 * (exp((x * -x)) * ((t_1 * (((1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / (1.0 + (x * 0.3275911)))))) * (-1.0 / t_0)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (abs(x) * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    if (abs(x) <= 0.0001d0) then
        tmp = 1d-9 + (((-0.37545125292247583d0) * (x ** 3.0d0)) + (((-0.00011824294398844343d0) * (x ** 2.0d0)) + (x * 1.128386358070218d0)))
    else
        tmp = 1.0d0 + (t_1 * (exp((x * -x)) * ((t_1 * (((1.421413741d0 + (t_1 * ((-1.453152027d0) + (1.061405429d0 / (1.0d0 + (x * 0.3275911d0)))))) * ((-1.0d0) / t_0)) - (-0.284496736d0))) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = 1.0 + (Math.abs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (Math.abs(x) <= 0.0001) {
		tmp = 1e-9 + ((-0.37545125292247583 * Math.pow(x, 3.0)) + ((-0.00011824294398844343 * Math.pow(x, 2.0)) + (x * 1.128386358070218)));
	} else {
		tmp = 1.0 + (t_1 * (Math.exp((x * -x)) * ((t_1 * (((1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / (1.0 + (x * 0.3275911)))))) * (-1.0 / t_0)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = 1.0 + (math.fabs(x) * 0.3275911)
	t_1 = 1.0 / t_0
	tmp = 0
	if math.fabs(x) <= 0.0001:
		tmp = 1e-9 + ((-0.37545125292247583 * math.pow(x, 3.0)) + ((-0.00011824294398844343 * math.pow(x, 2.0)) + (x * 1.128386358070218)))
	else:
		tmp = 1.0 + (t_1 * (math.exp((x * -x)) * ((t_1 * (((1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / (1.0 + (x * 0.3275911)))))) * (-1.0 / t_0)) - -0.284496736)) - 0.254829592)))
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(x * 1.128386358070218))));
	else
		tmp = Float64(1.0 + Float64(t_1 * Float64(exp(Float64(x * Float64(-x))) * Float64(Float64(t_1 * Float64(Float64(Float64(1.421413741 + Float64(t_1 * Float64(-1.453152027 + Float64(1.061405429 / Float64(1.0 + Float64(x * 0.3275911)))))) * Float64(-1.0 / t_0)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = 1.0 + (abs(x) * 0.3275911);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (abs(x) <= 0.0001)
		tmp = 1e-9 + ((-0.37545125292247583 * (x ^ 3.0)) + ((-0.00011824294398844343 * (x ^ 2.0)) + (x * 1.128386358070218)));
	else
		tmp = 1.0 + (t_1 * (exp((x * -x)) * ((t_1 * (((1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / (1.0 + (x * 0.3275911)))))) * (-1.0 / t_0)) - -0.284496736)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$1 * N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$1 * N[(N[(N[(1.421413741 + N[(t$95$1 * N[(-1.453152027 + N[(1.061405429 / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   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%

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

   4. Taylor expanded in x around 0 55.7%

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

   6. Simplified 54.1%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. pow1 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. unpow1 99.9%

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

      2. unpow1 99.9%

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

      3. sqr-pow 49.5%

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

      4. fabs-sqr 49.5%

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

      5. sqr-pow 99.3%

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

      6. unpow1 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]

Alternative 7: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - t_1 \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + 1.453152027 \cdot \frac{-1}{t_0}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= x 1.0)
     (+
      1e-9
      (+
       (* -0.37545125292247583 (pow x 3.0))
       (+ (* -0.00011824294398844343 (pow x 2.0)) (* x 1.128386358070218))))
     (-
      1.0
      (*
       t_1
       (*
        (exp (* x (- x)))
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (* t_1 (+ 1.421413741 (* 1.453152027 (/ -1.0 t_0)))))))))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x <= 1.0) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x, 3.0)) + ((-0.00011824294398844343 * pow(x, 2.0)) + (x * 1.128386358070218)));
	} else {
		tmp = 1.0 - (t_1 * (exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (1.453152027 * (-1.0 / t_0)))))))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (abs(x) * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    if (x <= 1.0d0) then
        tmp = 1d-9 + (((-0.37545125292247583d0) * (x ** 3.0d0)) + (((-0.00011824294398844343d0) * (x ** 2.0d0)) + (x * 1.128386358070218d0)))
    else
        tmp = 1.0d0 - (t_1 * (exp((x * -x)) * (0.254829592d0 + (t_1 * ((-0.284496736d0) + (t_1 * (1.421413741d0 + (1.453152027d0 * ((-1.0d0) / t_0)))))))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = 1.0 + (Math.abs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x <= 1.0) {
		tmp = 1e-9 + ((-0.37545125292247583 * Math.pow(x, 3.0)) + ((-0.00011824294398844343 * Math.pow(x, 2.0)) + (x * 1.128386358070218)));
	} else {
		tmp = 1.0 - (t_1 * (Math.exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (1.453152027 * (-1.0 / t_0)))))))));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = 1.0 + (math.fabs(x) * 0.3275911)
	t_1 = 1.0 / t_0
	tmp = 0
	if x <= 1.0:
		tmp = 1e-9 + ((-0.37545125292247583 * math.pow(x, 3.0)) + ((-0.00011824294398844343 * math.pow(x, 2.0)) + (x * 1.128386358070218)))
	else:
		tmp = 1.0 - (t_1 * (math.exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (1.453152027 * (-1.0 / t_0)))))))))
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(x * 1.128386358070218))));
	else
		tmp = Float64(1.0 - Float64(t_1 * Float64(exp(Float64(x * Float64(-x))) * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(1.453152027 * Float64(-1.0 / t_0))))))))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = 1.0 + (abs(x) * 0.3275911);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1e-9 + ((-0.37545125292247583 * (x ^ 3.0)) + ((-0.00011824294398844343 * (x ^ 2.0)) + (x * 1.128386358070218)));
	else
		tmp = 1.0 - (t_1 * (exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (1.453152027 * (-1.0 / t_0)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x, 1.0], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(t$95$1 * N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(1.453152027 * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in x around 0 41.4%

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

   6. Simplified 38.9%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 71.4%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]

   if 1 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. unpow1 100.0%

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

      3. sqr-pow 100.0%

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

      4. fabs-sqr 100.0%

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

      5. sqr-pow 100.0%

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

      6. unpow1 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + 1.453152027 \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 55.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.7)
   (+
    1e-9
    (+
     (* -0.37545125292247583 (pow x 3.0))
     (+ (* -0.00011824294398844343 (pow x 2.0)) (* x 1.128386358070218))))
   1e-9))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x, 3.0)) + ((-0.00011824294398844343 * pow(x, 2.0)) + (x * 1.128386358070218)));
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.7d0) then
        tmp = 1d-9 + (((-0.37545125292247583d0) * (x ** 3.0d0)) + (((-0.00011824294398844343d0) * (x ** 2.0d0)) + (x * 1.128386358070218d0)))
    else
        tmp = 1d-9
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = 1e-9 + ((-0.37545125292247583 * Math.pow(x, 3.0)) + ((-0.00011824294398844343 * Math.pow(x, 2.0)) + (x * 1.128386358070218)));
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.7:
		tmp = 1e-9 + ((-0.37545125292247583 * math.pow(x, 3.0)) + ((-0.00011824294398844343 * math.pow(x, 2.0)) + (x * 1.128386358070218)))
	else:
		tmp = 1e-9
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(x * 1.128386358070218))));
	else
		tmp = 1e-9;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.7)
		tmp = 1e-9 + ((-0.37545125292247583 * (x ^ 3.0)) + ((-0.00011824294398844343 * (x ^ 2.0)) + (x * 1.128386358070218)));
	else
		tmp = 1e-9;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.7], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-9]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in x around 0 41.4%

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

   6. Simplified 38.9%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 71.4%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]

   if 1.69999999999999996 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 4.5%

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

   6. Simplified 4.5%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 11.1%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \]

Alternative 9: 55.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.7)
   (+
    1e-9
    (fma
     (pow x 3.0)
     -0.37545125292247583
     (* x (+ 1.128386358070218 (* x -0.00011824294398844343)))))
   1e-9))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = 1e-9 + fma(pow(x, 3.0), -0.37545125292247583, (x * (1.128386358070218 + (x * -0.00011824294398844343))));
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = Float64(1e-9 + fma((x ^ 3.0), -0.37545125292247583, Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343)))));
	else
		tmp = 1e-9;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.7], N[(1e-9 + N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-9]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in x around 0 41.4%

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

   6. Simplified 38.9%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 41.8%

      \[\leadsto 1 + \color{blue}{\left(\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) - 0.999999999\right)} \]

   8. Taylor expanded in x around 0 71.4%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 71.4%

         \[\leadsto 10^{-9} + \left(\color{blue}{{x}^{3} \cdot -0.37545125292247583} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) \]

      2. fma-def 71.4%

         \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, -0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]

      3. +-commutative 71.4%

         \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}}\right) \]

      4. *-commutative 71.4%

         \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{x \cdot 1.128386358070218} + -0.00011824294398844343 \cdot {x}^{2}\right) \]

      5. *-commutative 71.4%

         \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot 1.128386358070218 + \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) \]

      6. unpow2 71.4%

         \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot 1.128386358070218 + \color{blue}{\left(x \cdot x\right)} \cdot -0.00011824294398844343\right) \]

      7. associate-*r* 71.4%

         \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot 1.128386358070218 + \color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)}\right) \]

      8. distribute-lft-out 71.4%

         \[\leadsto 10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)}\right) \]

   10. Simplified 71.4%

       \[\leadsto \color{blue}{10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)} \]

   if 1.69999999999999996 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 4.5%

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

   6. Simplified 4.5%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 11.1%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \]

Alternative 10: 54.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 9500:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 9500.0)
   (+ 1e-9 (fma (* x x) -0.00011824294398844343 (* x 1.128386358070218)))
   1e-9))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 9500.0) {
		tmp = 1e-9 + fma((x * x), -0.00011824294398844343, (x * 1.128386358070218));
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 9500.0)
		tmp = Float64(1e-9 + fma(Float64(x * x), -0.00011824294398844343, Float64(x * 1.128386358070218)));
	else
		tmp = 1e-9;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 9500.0], N[(1e-9 + N[(N[(x * x), $MachinePrecision] * -0.00011824294398844343 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-9]

Derivation

1. Split input into 2 regimes

2. if x < 9500

   1. Initial program 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in x around 0 41.4%

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

   6. Simplified 38.9%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 70.7%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   8. Step-by-step derivation

      1. *-commutative 70.7%

         \[\leadsto 10^{-9} + \left(\color{blue}{{x}^{2} \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right) \]

      2. fma-def 70.7%

         \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left({x}^{2}, -0.00011824294398844343, 1.128386358070218 \cdot x\right)} \]

      3. unpow2 70.7%

         \[\leadsto 10^{-9} + \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.00011824294398844343, 1.128386358070218 \cdot x\right) \]

      4. *-commutative 70.7%

         \[\leadsto 10^{-9} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, \color{blue}{x \cdot 1.128386358070218}\right) \]

   9. Simplified 70.7%

      \[\leadsto \color{blue}{10^{-9} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)} \]

   if 9500 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 4.5%

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

   6. Simplified 4.5%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 11.1%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9500:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left(x \cdot x, -0.00011824294398844343, x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \]

Alternative 11: 54.9% accurate, 77.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 9500:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 9500.0)
   (+ 1e-9 (* x (+ 1.128386358070218 (* x -0.00011824294398844343))))
   1e-9))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 9500.0) {
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)));
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 9500.0d0) then
        tmp = 1d-9 + (x * (1.128386358070218d0 + (x * (-0.00011824294398844343d0))))
    else
        tmp = 1d-9
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 9500.0) {
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)));
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 9500.0:
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)))
	else:
		tmp = 1e-9
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 9500.0)
		tmp = Float64(1e-9 + Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343))));
	else
		tmp = 1e-9;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 9500.0)
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)));
	else
		tmp = 1e-9;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 9500.0], N[(1e-9 + N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-9]

Derivation

1. Split input into 2 regimes

2. if x < 9500

   1. Initial program 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in x around 0 41.4%

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

   6. Simplified 38.9%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 41.3%

      \[\leadsto 1 + \color{blue}{\left(\left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right) - 0.999999999\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 41.3%

         \[\leadsto 1 + \color{blue}{\left(-0.00011824294398844343 \cdot {x}^{2} + \left(1.128386358070218 \cdot x - 0.999999999\right)\right)} \]

      2. *-commutative 41.3%

         \[\leadsto 1 + \left(\color{blue}{{x}^{2} \cdot -0.00011824294398844343} + \left(1.128386358070218 \cdot x - 0.999999999\right)\right) \]

      3. unpow2 41.3%

         \[\leadsto 1 + \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.00011824294398844343 + \left(1.128386358070218 \cdot x - 0.999999999\right)\right) \]

      4. fma-neg 41.3%

         \[\leadsto 1 + \left(\left(x \cdot x\right) \cdot -0.00011824294398844343 + \color{blue}{\mathsf{fma}\left(1.128386358070218, x, -0.999999999\right)}\right) \]

      5. metadata-eval 41.3%

         \[\leadsto 1 + \left(\left(x \cdot x\right) \cdot -0.00011824294398844343 + \mathsf{fma}\left(1.128386358070218, x, \color{blue}{-0.999999999}\right)\right) \]

   9. Simplified 41.3%

      \[\leadsto 1 + \color{blue}{\left(\left(x \cdot x\right) \cdot -0.00011824294398844343 + \mathsf{fma}\left(1.128386358070218, x, -0.999999999\right)\right)} \]

   10. Taylor expanded in x around 0 70.7%

       \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   11. Step-by-step derivation

       1. +-commutative 70.7%

          \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

       2. *-commutative 70.7%

          \[\leadsto 10^{-9} + \left(\color{blue}{x \cdot 1.128386358070218} + -0.00011824294398844343 \cdot {x}^{2}\right) \]

       3. *-commutative 70.7%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) \]

       4. unpow2 70.7%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{\left(x \cdot x\right)} \cdot -0.00011824294398844343\right) \]

       5. associate-*r* 70.7%

          \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)}\right) \]

       6. distribute-lft-out 70.7%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]

   12. Simplified 70.7%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]

   if 9500 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 4.5%

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

   6. Simplified 4.5%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 11.1%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9500:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \]

Alternative 12: 54.8% accurate, 121.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 920000000:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 920000000.0) (+ 1e-9 (* x 1.128386358070218)) 1e-9))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 920000000.0) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 920000000.0d0) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 1d-9
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 920000000.0) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 920000000.0:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1e-9
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 920000000.0)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = 1e-9;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 920000000.0)
		tmp = 1e-9 + (x * 1.128386358070218);
	else
		tmp = 1e-9;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 920000000.0], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1e-9]

Derivation

1. Split input into 2 regimes

2. if x < 9.2e8

   1. Initial program 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in x around 0 41.4%

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

   6. Simplified 38.9%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 70.6%

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

      1. *-commutative 70.6%

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

   9. Simplified 70.6%

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

   if 9.2e8 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 4.5%

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

   6. Simplified 4.5%

      \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

   7. Taylor expanded in x around 0 11.1%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 920000000:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \]

Alternative 13: 52.9% accurate, 856.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 1e-9)

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 1d-9
end function

Java

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

Python

x = abs(x)
def code(x):
	return 1e-9

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 1e-9

Derivation

1. Initial program 76.5%

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

3. Simplified 76.5%

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

4. Taylor expanded in x around 0 33.5%

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

6. Simplified 31.5%

   \[\leadsto \color{blue}{1 + \left(\left(\frac{0.284496736}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}}\right) + \mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right) + \left(\left(0.254829592 + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right), \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{5}}\right)\right)\right)} \]

7. Taylor expanded in x around 0 58.5%

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

8. Final simplification 58.5%

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

Reproduce

herbie shell --seed 2023277 
(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)))))))