Jmat.Real.erf

Percentage Accurate: 79.1% → 99.9%

Time: 27.4s

Alternatives: 16

Speedup: 279.5×

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: 79.1% 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, 0.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \mathsf{fma}\left(x, x \cdot -0.00011824294398844343, \mathsf{fma}\left(x, 1.128386358070218, {x}^{3} \cdot -0.37545125292247583\right)\right)\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{10^{-27} + {t_1}^{3}}{\mathsf{fma}\left(t_1, \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(x \cdot -0.00011824294398844343\right)\right) + \mathsf{fma}\left(x, 1.128386358070218, -1 \cdot 10^{-9}\right), 10^{-18}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{1.453152027}{{t_0}^{3}} + \left(\frac{0.284496736}{t_0} - \left(\left(0.254829592 + \frac{1.061405429}{{t_0}^{4}}\right) + \frac{1.421413741}{{t_0}^{2}}\right)\right)}{\frac{t_0}{e^{x \cdot \left(-x\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1
         (fma
          x
          (* x -0.00011824294398844343)
          (fma x 1.128386358070218 (* (pow x 3.0) -0.37545125292247583)))))
   (if (<= (fabs x) 5e-5)
     (/
      (+ 1e-27 (pow t_1 3.0))
      (fma
       t_1
       (+
        (fma
         (pow x 3.0)
         -0.37545125292247583
         (* x (* x -0.00011824294398844343)))
        (fma x 1.128386358070218 -1e-9))
       1e-18))
     (+
      1.0
      (/
       (+
        (/ 1.453152027 (pow t_0 3.0))
        (-
         (/ 0.284496736 t_0)
         (+
          (+ 0.254829592 (/ 1.061405429 (pow t_0 4.0)))
          (/ 1.421413741 (pow t_0 2.0)))))
       (/ t_0 (exp (* x (- x)))))))))

C

x = abs(x);
double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = fma(x, (x * -0.00011824294398844343), fma(x, 1.128386358070218, (pow(x, 3.0) * -0.37545125292247583)));
	double tmp;
	if (fabs(x) <= 5e-5) {
		tmp = (1e-27 + pow(t_1, 3.0)) / fma(t_1, (fma(pow(x, 3.0), -0.37545125292247583, (x * (x * -0.00011824294398844343))) + fma(x, 1.128386358070218, -1e-9)), 1e-18);
	} else {
		tmp = 1.0 + (((1.453152027 / pow(t_0, 3.0)) + ((0.284496736 / t_0) - ((0.254829592 + (1.061405429 / pow(t_0, 4.0))) + (1.421413741 / pow(t_0, 2.0))))) / (t_0 / exp((x * -x))));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = fma(x, Float64(x * -0.00011824294398844343), fma(x, 1.128386358070218, Float64((x ^ 3.0) * -0.37545125292247583)))
	tmp = 0.0
	if (abs(x) <= 5e-5)
		tmp = Float64(Float64(1e-27 + (t_1 ^ 3.0)) / fma(t_1, Float64(fma((x ^ 3.0), -0.37545125292247583, Float64(x * Float64(x * -0.00011824294398844343))) + fma(x, 1.128386358070218, -1e-9)), 1e-18));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(1.453152027 / (t_0 ^ 3.0)) + Float64(Float64(0.284496736 / t_0) - Float64(Float64(0.254829592 + Float64(1.061405429 / (t_0 ^ 4.0))) + Float64(1.421413741 / (t_0 ^ 2.0))))) / Float64(t_0 / exp(Float64(x * Float64(-x))))));
	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]}, Block[{t$95$1 = N[(x * N[(x * -0.00011824294398844343), $MachinePrecision] + N[(x * 1.128386358070218 + N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-5], N[(N[(1e-27 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(x * N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 1.128386358070218 + -1e-9), $MachinePrecision]), $MachinePrecision] + 1e-18), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(1.453152027 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.284496736 / t$95$0), $MachinePrecision] - N[(N[(0.254829592 + N[(1.061405429 / N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.421413741 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 / N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.00000000000000024e-5

   1. Initial program 57.7%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in x around inf 54.4%

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

   6. Simplified 54.0%

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

   7. Taylor expanded in x around 0 96.6%

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

      1. flip3-+ 96.6%

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

      2. metadata-eval 96.7%

         \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      3. fma-def 96.7%

         \[\leadsto \frac{10^{-27} + {\color{blue}{\left(\mathsf{fma}\left(-0.00011824294398844343, {x}^{2}, -0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)}}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      4. pow2 96.7%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.00011824294398844343, \color{blue}{x \cdot x}, -0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      5. fma-def 96.7%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \color{blue}{\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, 1.128386358070218 \cdot x\right)}\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      6. *-commutative 96.7%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{x \cdot 1.128386358070218}\right)\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      7. metadata-eval 96.7%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right)\right)}^{3}}{\color{blue}{10^{-18}} + \left(\left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

   9. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\frac{10^{-27} + {\left(\mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right)\right)}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right)\right)}} \]

   11. Simplified 96.7%

       \[\leadsto \color{blue}{\frac{10^{-27} + {\left(\mathsf{fma}\left(x, x \cdot -0.00011824294398844343, \mathsf{fma}\left(x, 1.128386358070218, {x}^{3} \cdot -0.37545125292247583\right)\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot -0.00011824294398844343, \mathsf{fma}\left(x, 1.128386358070218, {x}^{3} \cdot -0.37545125292247583\right)\right), \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(x \cdot -0.00011824294398844343\right)\right) + \mathsf{fma}\left(x, 1.128386358070218, -1 \cdot 10^{-9}\right), 10^{-18}\right)}} \]

   if 5.00000000000000024e-5 < (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%

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

   4. Taylor expanded in x around inf 99.9%

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

   6. Simplified 100.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{10^{-27} + {\left(\mathsf{fma}\left(x, x \cdot -0.00011824294398844343, \mathsf{fma}\left(x, 1.128386358070218, {x}^{3} \cdot -0.37545125292247583\right)\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot -0.00011824294398844343, \mathsf{fma}\left(x, 1.128386358070218, {x}^{3} \cdot -0.37545125292247583\right)\right), \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(x \cdot -0.00011824294398844343\right)\right) + \mathsf{fma}\left(x, 1.128386358070218, -1 \cdot 10^{-9}\right), 10^{-18}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}} + \left(\frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - \left(\left(0.254829592 + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}}\right) + \frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}}\right)\right)}{\frac{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}{e^{x \cdot \left(-x\right)}}}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}\\ t_1 := \mathsf{fma}\left(x, x \cdot -0.00011824294398844343, \mathsf{fma}\left(x, 1.128386358070218, {x}^{3} \cdot -0.37545125292247583\right)\right)\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{10^{-27} + {t_1}^{3}}{\mathsf{fma}\left(t_1, \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(x \cdot -0.00011824294398844343\right)\right) + \mathsf{fma}\left(x, 1.128386358070218, -1 \cdot 10^{-9}\right), 10^{-18}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{\left(\frac{1.453152027}{t_0} - \frac{\left(\frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{t_0}\right) + -0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - 0.254829592}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (fma 0.3275911 x 1.0) 3.0))
        (t_1
         (fma
          x
          (* x -0.00011824294398844343)
          (fma x 1.128386358070218 (* (pow x 3.0) -0.37545125292247583)))))
   (if (<= (fabs x) 5e-5)
     (/
      (+ 1e-27 (pow t_1 3.0))
      (fma
       t_1
       (+
        (fma
         (pow x 3.0)
         -0.37545125292247583
         (* x (* x -0.00011824294398844343)))
        (fma x 1.128386358070218 -1e-9))
       1e-18))
     (+
      1.0
      (/
       (/
        (-
         (-
          (/ 1.453152027 t_0)
          (/
           (+
            (+ (/ 1.421413741 (fma 0.3275911 x 1.0)) (/ 1.061405429 t_0))
            -0.284496736)
           (fma 0.3275911 x 1.0)))
         0.254829592)
        (pow (exp x) x))
       (fma 0.3275911 (fabs x) 1.0))))))

C

x = abs(x);
double code(double x) {
	double t_0 = pow(fma(0.3275911, x, 1.0), 3.0);
	double t_1 = fma(x, (x * -0.00011824294398844343), fma(x, 1.128386358070218, (pow(x, 3.0) * -0.37545125292247583)));
	double tmp;
	if (fabs(x) <= 5e-5) {
		tmp = (1e-27 + pow(t_1, 3.0)) / fma(t_1, (fma(pow(x, 3.0), -0.37545125292247583, (x * (x * -0.00011824294398844343))) + fma(x, 1.128386358070218, -1e-9)), 1e-18);
	} else {
		tmp = 1.0 + (((((1.453152027 / t_0) - ((((1.421413741 / fma(0.3275911, x, 1.0)) + (1.061405429 / t_0)) + -0.284496736) / fma(0.3275911, x, 1.0))) - 0.254829592) / pow(exp(x), x)) / fma(0.3275911, fabs(x), 1.0));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = fma(0.3275911, x, 1.0) ^ 3.0
	t_1 = fma(x, Float64(x * -0.00011824294398844343), fma(x, 1.128386358070218, Float64((x ^ 3.0) * -0.37545125292247583)))
	tmp = 0.0
	if (abs(x) <= 5e-5)
		tmp = Float64(Float64(1e-27 + (t_1 ^ 3.0)) / fma(t_1, Float64(fma((x ^ 3.0), -0.37545125292247583, Float64(x * Float64(x * -0.00011824294398844343))) + fma(x, 1.128386358070218, -1e-9)), 1e-18));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(Float64(1.453152027 / t_0) - Float64(Float64(Float64(Float64(1.421413741 / fma(0.3275911, x, 1.0)) + Float64(1.061405429 / t_0)) + -0.284496736) / fma(0.3275911, x, 1.0))) - 0.254829592) / (exp(x) ^ x)) / fma(0.3275911, abs(x), 1.0)));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[Power[N[(0.3275911 * x + 1.0), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * -0.00011824294398844343), $MachinePrecision] + N[(x * 1.128386358070218 + N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-5], N[(N[(1e-27 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(x * N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 1.128386358070218 + -1e-9), $MachinePrecision]), $MachinePrecision] + 1e-18), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(N[(1.453152027 / t$95$0), $MachinePrecision] - N[(N[(N[(N[(1.421413741 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision] / N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.00000000000000024e-5

   1. Initial program 57.7%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in x around inf 54.4%

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

   6. Simplified 54.0%

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

   7. Taylor expanded in x around 0 96.6%

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

      1. flip3-+ 96.6%

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

      2. metadata-eval 96.7%

         \[\leadsto \frac{\color{blue}{10^{-27}} + {\left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      3. fma-def 96.7%

         \[\leadsto \frac{10^{-27} + {\color{blue}{\left(\mathsf{fma}\left(-0.00011824294398844343, {x}^{2}, -0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)}}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      4. pow2 96.7%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.00011824294398844343, \color{blue}{x \cdot x}, -0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      5. fma-def 96.7%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \color{blue}{\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, 1.128386358070218 \cdot x\right)}\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      6. *-commutative 96.7%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{x \cdot 1.128386358070218}\right)\right)\right)}^{3}}{10^{-9} \cdot 10^{-9} + \left(\left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      7. metadata-eval 96.7%

         \[\leadsto \frac{10^{-27} + {\left(\mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right)\right)}^{3}}{\color{blue}{10^{-18}} + \left(\left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) - 10^{-9} \cdot \left(-0.00011824294398844343 \cdot {x}^{2} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right)\right)} \]

   9. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\frac{10^{-27} + {\left(\mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right)\right)}^{3}}{10^{-18} + \left(\mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right) \cdot \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right) - 10^{-9} \cdot \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, x \cdot 1.128386358070218\right)\right)\right)}} \]

   11. Simplified 96.7%

       \[\leadsto \color{blue}{\frac{10^{-27} + {\left(\mathsf{fma}\left(x, x \cdot -0.00011824294398844343, \mathsf{fma}\left(x, 1.128386358070218, {x}^{3} \cdot -0.37545125292247583\right)\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot -0.00011824294398844343, \mathsf{fma}\left(x, 1.128386358070218, {x}^{3} \cdot -0.37545125292247583\right)\right), \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(x \cdot -0.00011824294398844343\right)\right) + \mathsf{fma}\left(x, 1.128386358070218, -1 \cdot 10^{-9}\right), 10^{-18}\right)}} \]

   if 5.00000000000000024e-5 < (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%

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

   4. Taylor expanded in x around 0 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{10^{-27} + {\left(\mathsf{fma}\left(x, x \cdot -0.00011824294398844343, \mathsf{fma}\left(x, 1.128386358070218, {x}^{3} \cdot -0.37545125292247583\right)\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot -0.00011824294398844343, \mathsf{fma}\left(x, 1.128386358070218, {x}^{3} \cdot -0.37545125292247583\right)\right), \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(x \cdot -0.00011824294398844343\right)\right) + \mathsf{fma}\left(x, 1.128386358070218, -1 \cdot 10^{-9}\right), 10^{-18}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{\left(\frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \frac{\left(\frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right) + -0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - 0.254829592}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-5}:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left({x}^{3} \cdot -0.37545125292247583 + \sqrt[3]{\left(x \cdot 1.128386358070218\right) \cdot \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{\left(\frac{1.453152027}{t_0} - \frac{\left(\frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{t_0}\right) + -0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - 0.254829592}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (fma 0.3275911 x 1.0) 3.0)))
   (if (<= (fabs x) 5e-5)
     (+
      1e-9
      (+
       (* -0.00011824294398844343 (pow x 2.0))
       (+
        (* (pow x 3.0) -0.37545125292247583)
        (cbrt
         (*
          (* x 1.128386358070218)
          (* (* x 1.128386358070218) (* x 1.128386358070218)))))))
     (+
      1.0
      (/
       (/
        (-
         (-
          (/ 1.453152027 t_0)
          (/
           (+
            (+ (/ 1.421413741 (fma 0.3275911 x 1.0)) (/ 1.061405429 t_0))
            -0.284496736)
           (fma 0.3275911 x 1.0)))
         0.254829592)
        (pow (exp x) x))
       (fma 0.3275911 (fabs x) 1.0))))))

C

x = abs(x);
double code(double x) {
	double t_0 = pow(fma(0.3275911, x, 1.0), 3.0);
	double tmp;
	if (fabs(x) <= 5e-5) {
		tmp = 1e-9 + ((-0.00011824294398844343 * pow(x, 2.0)) + ((pow(x, 3.0) * -0.37545125292247583) + cbrt(((x * 1.128386358070218) * ((x * 1.128386358070218) * (x * 1.128386358070218))))));
	} else {
		tmp = 1.0 + (((((1.453152027 / t_0) - ((((1.421413741 / fma(0.3275911, x, 1.0)) + (1.061405429 / t_0)) + -0.284496736) / fma(0.3275911, x, 1.0))) - 0.254829592) / pow(exp(x), x)) / fma(0.3275911, fabs(x), 1.0));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = fma(0.3275911, x, 1.0) ^ 3.0
	tmp = 0.0
	if (abs(x) <= 5e-5)
		tmp = Float64(1e-9 + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(Float64((x ^ 3.0) * -0.37545125292247583) + cbrt(Float64(Float64(x * 1.128386358070218) * Float64(Float64(x * 1.128386358070218) * Float64(x * 1.128386358070218)))))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(Float64(1.453152027 / t_0) - Float64(Float64(Float64(Float64(1.421413741 / fma(0.3275911, x, 1.0)) + Float64(1.061405429 / t_0)) + -0.284496736) / fma(0.3275911, x, 1.0))) - 0.254829592) / (exp(x) ^ x)) / fma(0.3275911, abs(x), 1.0)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.00000000000000024e-5

   1. Initial program 57.7%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in x around inf 54.4%

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

   6. Simplified 54.0%

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

   7. Taylor expanded in x around 0 96.6%

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

      1. *-commutative 96.6%

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

      2. add-cbrt-cube 96.6%

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

   9. Applied egg-rr 96.6%

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

   if 5.00000000000000024e-5 < (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%

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

   4. Taylor expanded in x around 0 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-5}:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left({x}^{3} \cdot -0.37545125292247583 + \sqrt[3]{\left(x \cdot 1.128386358070218\right) \cdot \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{\left(\frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \frac{\left(\frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}}\right) + -0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - 0.254829592}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-5}:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left({x}^{3} \cdot -0.37545125292247583 + \sqrt[3]{\left(x \cdot 1.128386358070218\right) \cdot \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{-x \cdot x} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

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) 5e-5)
     (+
      1e-9
      (+
       (* -0.00011824294398844343 (pow x 2.0))
       (+
        (* (pow x 3.0) -0.37545125292247583)
        (cbrt
         (*
          (* x 1.128386358070218)
          (* (* x 1.128386358070218) (* x 1.128386358070218)))))))
     (+
      1.0
      (*
       (*
        (exp (- (* x x)))
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (+
              (/ 1.061405429 (pow (fma 0.3275911 x 1.0) 2.0))
              (/ -1.453152027 (fma 0.3275911 x 1.0)))))))))
       (/ -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 (fabs(x) <= 5e-5) {
		tmp = 1e-9 + ((-0.00011824294398844343 * pow(x, 2.0)) + ((pow(x, 3.0) * -0.37545125292247583) + cbrt(((x * 1.128386358070218) * ((x * 1.128386358070218) * (x * 1.128386358070218))))));
	} else {
		tmp = 1.0 + ((exp(-(x * x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + ((1.061405429 / pow(fma(0.3275911, x, 1.0), 2.0)) + (-1.453152027 / fma(0.3275911, x, 1.0))))))))) * (-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 (abs(x) <= 5e-5)
		tmp = Float64(1e-9 + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(Float64((x ^ 3.0) * -0.37545125292247583) + cbrt(Float64(Float64(x * 1.128386358070218) * Float64(Float64(x * 1.128386358070218) * Float64(x * 1.128386358070218)))))));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(-Float64(x * x))) * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(Float64(1.061405429 / (fma(0.3275911, x, 1.0) ^ 2.0)) + Float64(-1.453152027 / fma(0.3275911, x, 1.0))))))))) * Float64(-1.0 / t_0)));
	end
	return 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], 5e-5], N[(1e-9 + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision] + N[Power[N[(N[(x * 1.128386358070218), $MachinePrecision] * N[(N[(x * 1.128386358070218), $MachinePrecision] * N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(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[(N[(1.061405429 / N[Power[N[(0.3275911 * x + 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.453152027 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.00000000000000024e-5

   1. Initial program 57.7%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in x around inf 54.4%

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

   6. Simplified 54.0%

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

   7. Taylor expanded in x around 0 96.6%

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

      1. *-commutative 96.6%

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

      2. add-cbrt-cube 96.6%

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

   9. Applied egg-rr 96.6%

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

   if 5.00000000000000024e-5 < (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. Taylor expanded in x around 0 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. unpow2 99.9%

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

      5. associate-/r* 99.9%

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

      6. fma-def 99.9%

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

      7. associate-*r/ 99.9%

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

      8. metadata-eval 99.9%

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

      9. div-sub 99.9%

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

      10. sub-neg 99.9%

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

      11. metadata-eval 99.9%

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

      12. +-commutative 99.9%

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

      13. fma-def 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 99.9%

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

   8. Taylor expanded in x around 0 99.9%

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

      1. sub-neg 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. associate-*r/ 99.9%

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

      7. +-commutative 99.9%

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

      8. metadata-eval 99.9%

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

      9. fma-def 99.9%

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

      10. unpow1 99.9%

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

      11. sqr-pow 49.9%

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

      12. fabs-sqr 49.9%

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

      13. sqr-pow 99.9%

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

      14. unpow1 99.9%

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

      15. distribute-neg-frac 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-5}:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left({x}^{3} \cdot -0.37545125292247583 + \sqrt[3]{\left(x \cdot 1.128386358070218\right) \cdot \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00052:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left({x}^{3} \cdot -0.37545125292247583 + \sqrt[3]{\left(x \cdot 1.128386358070218\right) \cdot \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{1 + x \cdot 0.3275911}}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.00052)
   (+
    1e-9
    (+
     (* -0.00011824294398844343 (pow x 2.0))
     (+
      (* (pow x 3.0) -0.37545125292247583)
      (cbrt
       (*
        (* x 1.128386358070218)
        (* (* x 1.128386358070218) (* x 1.128386358070218)))))))
   (-
    1.0
    (/
     (/
      (+
       0.254829592
       (/
        (+
         -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)))
        (+ 1.0 (* x 0.3275911))))
      (pow (exp x) x))
     (fma 0.3275911 (fabs x) 1.0)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.00052) {
		tmp = 1e-9 + ((-0.00011824294398844343 * pow(x, 2.0)) + ((pow(x, 3.0) * -0.37545125292247583) + cbrt(((x * 1.128386358070218) * ((x * 1.128386358070218) * (x * 1.128386358070218))))));
	} else {
		tmp = 1.0 - (((0.254829592 + ((-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))) / (1.0 + (x * 0.3275911)))) / pow(exp(x), x)) / fma(0.3275911, fabs(x), 1.0));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.00052)
		tmp = Float64(1e-9 + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(Float64((x ^ 3.0) * -0.37545125292247583) + cbrt(Float64(Float64(x * 1.128386358070218) * Float64(Float64(x * 1.128386358070218) * Float64(x * 1.128386358070218)))))));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.254829592 + 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))) / Float64(1.0 + Float64(x * 0.3275911)))) / (exp(x) ^ x)) / fma(0.3275911, abs(x), 1.0)));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.00052], N[(1e-9 + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision] + N[Power[N[(N[(x * 1.128386358070218), $MachinePrecision] * N[(N[(x * 1.128386358070218), $MachinePrecision] * N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(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] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.19999999999999954e-4

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in x around inf 69.6%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in x around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. add-cbrt-cube 65.0%

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

   9. Applied egg-rr 65.0%

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

   if 5.19999999999999954e-4 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. fma-def 99.8%

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

      3. associate-*l/ 99.8%

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

   7. Simplified 99.8%

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

      1. fma-udef 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00052:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left({x}^{3} \cdot -0.37545125292247583 + \sqrt[3]{\left(x \cdot 1.128386358070218\right) \cdot \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{1 + x \cdot 0.3275911}}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ t_2 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left({x}^{3} \cdot -0.37545125292247583 + \sqrt[3]{\left(x \cdot 1.128386358070218\right) \cdot \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{-x \cdot x} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + \frac{1}{t_2} \cdot \left(-1.453152027 + \frac{1.061405429}{t_2}\right)\right)\right)\right)\right) \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

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))
        (t_2 (+ 1.0 (* x 0.3275911))))
   (if (<= x 0.0005)
     (+
      1e-9
      (+
       (* -0.00011824294398844343 (pow x 2.0))
       (+
        (* (pow x 3.0) -0.37545125292247583)
        (cbrt
         (*
          (* x 1.128386358070218)
          (* (* x 1.128386358070218) (* x 1.128386358070218)))))))
     (+
      1.0
      (*
       (*
        (exp (- (* x x)))
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (* (/ 1.0 t_2) (+ -1.453152027 (/ 1.061405429 t_2)))))))))
       (/ -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 t_2 = 1.0 + (x * 0.3275911);
	double tmp;
	if (x <= 0.0005) {
		tmp = 1e-9 + ((-0.00011824294398844343 * pow(x, 2.0)) + ((pow(x, 3.0) * -0.37545125292247583) + cbrt(((x * 1.128386358070218) * ((x * 1.128386358070218) * (x * 1.128386358070218))))));
	} else {
		tmp = 1.0 + ((exp(-(x * x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + ((1.0 / t_2) * (-1.453152027 + (1.061405429 / t_2))))))))) * (-1.0 / t_0));
	}
	return tmp;
}

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 t_2 = 1.0 + (x * 0.3275911);
	double tmp;
	if (x <= 0.0005) {
		tmp = 1e-9 + ((-0.00011824294398844343 * Math.pow(x, 2.0)) + ((Math.pow(x, 3.0) * -0.37545125292247583) + Math.cbrt(((x * 1.128386358070218) * ((x * 1.128386358070218) * (x * 1.128386358070218))))));
	} else {
		tmp = 1.0 + ((Math.exp(-(x * x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + ((1.0 / t_2) * (-1.453152027 + (1.061405429 / t_2))))))))) * (-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)
	t_2 = Float64(1.0 + Float64(x * 0.3275911))
	tmp = 0.0
	if (x <= 0.0005)
		tmp = Float64(1e-9 + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(Float64((x ^ 3.0) * -0.37545125292247583) + cbrt(Float64(Float64(x * 1.128386358070218) * Float64(Float64(x * 1.128386358070218) * Float64(x * 1.128386358070218)))))));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(-Float64(x * x))) * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(Float64(1.0 / t_2) * Float64(-1.453152027 + Float64(1.061405429 / t_2))))))))) * Float64(-1.0 / t_0)));
	end
	return 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]}, Block[{t$95$2 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.0005], N[(1e-9 + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision] + N[Power[N[(N[(x * 1.128386358070218), $MachinePrecision] * N[(N[(x * 1.128386358070218), $MachinePrecision] * N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(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[(N[(1.0 / t$95$2), $MachinePrecision] * N[(-1.453152027 + N[(1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-4

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in x around inf 69.6%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in x around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. add-cbrt-cube 65.0%

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

   9. Applied egg-rr 65.0%

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

   if 5.0000000000000001e-4 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. unpow1 99.8%

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

      3. sqr-pow 99.8%

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

      4. fabs-sqr 99.8%

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

      5. sqr-pow 99.8%

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

      6. unpow1 99.8%

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

   7. Simplified 99.8%

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

      1. pow1 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. unpow1 99.8%

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

       2. unpow1 99.8%

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

       3. sqr-pow 99.8%

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

       4. fabs-sqr 99.8%

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

       5. sqr-pow 99.8%

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

       6. unpow1 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left({x}^{3} \cdot -0.37545125292247583 + \sqrt[3]{\left(x \cdot 1.128386358070218\right) \cdot \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 7: 99.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left({x}^{3} \cdot -0.37545125292247583 + \sqrt[3]{\left(x \cdot 1.128386358070218\right) \cdot \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592}{{\left(e^{x}\right)}^{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.96)
   (+
    1e-9
    (+
     (* -0.00011824294398844343 (pow x 2.0))
     (+
      (* (pow x 3.0) -0.37545125292247583)
      (cbrt
       (*
        (* x 1.128386358070218)
        (* (* x 1.128386358070218) (* x 1.128386358070218)))))))
   (+ 1.0 (/ -0.254829592 (pow (exp x) x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.96) {
		tmp = 1e-9 + ((-0.00011824294398844343 * pow(x, 2.0)) + ((pow(x, 3.0) * -0.37545125292247583) + cbrt(((x * 1.128386358070218) * ((x * 1.128386358070218) * (x * 1.128386358070218))))));
	} else {
		tmp = 1.0 + (-0.254829592 / pow(exp(x), x));
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.96) {
		tmp = 1e-9 + ((-0.00011824294398844343 * Math.pow(x, 2.0)) + ((Math.pow(x, 3.0) * -0.37545125292247583) + Math.cbrt(((x * 1.128386358070218) * ((x * 1.128386358070218) * (x * 1.128386358070218))))));
	} else {
		tmp = 1.0 + (-0.254829592 / Math.pow(Math.exp(x), x));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.96)
		tmp = Float64(1e-9 + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(Float64((x ^ 3.0) * -0.37545125292247583) + cbrt(Float64(Float64(x * 1.128386358070218) * Float64(Float64(x * 1.128386358070218) * Float64(x * 1.128386358070218)))))));
	else
		tmp = Float64(1.0 + Float64(-0.254829592 / (exp(x) ^ x)));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.96], N[(1e-9 + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision] + N[Power[N[(N[(x * 1.128386358070218), $MachinePrecision] * N[(N[(x * 1.128386358070218), $MachinePrecision] * N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.254829592 / N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.95999999999999996

   1. Initial program 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in x around inf 69.7%

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

   6. Simplified 69.4%

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

   7. Taylor expanded in x around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. add-cbrt-cube 65.0%

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

   9. Applied egg-rr 65.0%

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

   if 0.95999999999999996 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 98.8%

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

   8. Taylor expanded in x around 0 98.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + \left({x}^{3} \cdot -0.37545125292247583 + \sqrt[3]{\left(x \cdot 1.128386358070218\right) \cdot \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592}{{\left(e^{x}\right)}^{x}}\\ \end{array} \]

Alternative 8: 99.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;10^{-9} + \left(x \cdot \left(x \cdot -0.00011824294398844343\right) + \left({x}^{3} \cdot -0.37545125292247583 + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592}{{\left(e^{x}\right)}^{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.96)
   (+
    1e-9
    (+
     (* x (* x -0.00011824294398844343))
     (+ (* (pow x 3.0) -0.37545125292247583) (* x 1.128386358070218))))
   (+ 1.0 (/ -0.254829592 (pow (exp x) x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.96) {
		tmp = 1e-9 + ((x * (x * -0.00011824294398844343)) + ((pow(x, 3.0) * -0.37545125292247583) + (x * 1.128386358070218)));
	} else {
		tmp = 1.0 + (-0.254829592 / pow(exp(x), x));
	}
	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 <= 0.96d0) then
        tmp = 1d-9 + ((x * (x * (-0.00011824294398844343d0))) + (((x ** 3.0d0) * (-0.37545125292247583d0)) + (x * 1.128386358070218d0)))
    else
        tmp = 1.0d0 + ((-0.254829592d0) / (exp(x) ** x))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.96) {
		tmp = 1e-9 + ((x * (x * -0.00011824294398844343)) + ((Math.pow(x, 3.0) * -0.37545125292247583) + (x * 1.128386358070218)));
	} else {
		tmp = 1.0 + (-0.254829592 / Math.pow(Math.exp(x), x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.96:
		tmp = 1e-9 + ((x * (x * -0.00011824294398844343)) + ((math.pow(x, 3.0) * -0.37545125292247583) + (x * 1.128386358070218)))
	else:
		tmp = 1.0 + (-0.254829592 / math.pow(math.exp(x), x))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.96)
		tmp = Float64(1e-9 + Float64(Float64(x * Float64(x * -0.00011824294398844343)) + Float64(Float64((x ^ 3.0) * -0.37545125292247583) + Float64(x * 1.128386358070218))));
	else
		tmp = Float64(1.0 + Float64(-0.254829592 / (exp(x) ^ x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.96)
		tmp = 1e-9 + ((x * (x * -0.00011824294398844343)) + (((x ^ 3.0) * -0.37545125292247583) + (x * 1.128386358070218)));
	else
		tmp = 1.0 + (-0.254829592 / (exp(x) ^ x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.96], N[(1e-9 + N[(N[(x * N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision] + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.254829592 / N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.95999999999999996

   1. Initial program 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in x around inf 69.7%

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

   6. Simplified 69.4%

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

   7. Taylor expanded in x around 0 65.1%

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

      1. pow1 65.1%

         \[\leadsto 10^{-9} + \left(\color{blue}{{\left(-0.00011824294398844343 \cdot {x}^{2}\right)}^{1}} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]

      2. pow2 65.1%

         \[\leadsto 10^{-9} + \left({\left(-0.00011824294398844343 \cdot \color{blue}{\left(x \cdot x\right)}\right)}^{1} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]

      3. *-commutative 65.1%

         \[\leadsto 10^{-9} + \left({\color{blue}{\left(\left(x \cdot x\right) \cdot -0.00011824294398844343\right)}}^{1} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]

   9. Applied egg-rr 65.1%

      \[\leadsto 10^{-9} + \left(\color{blue}{{\left(\left(x \cdot x\right) \cdot -0.00011824294398844343\right)}^{1}} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]
   10. Step-by-step derivation

       1. unpow1 65.1%

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

       2. associate-*l* 65.2%

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

   11. Simplified 65.2%

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

   if 0.95999999999999996 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 98.8%

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

   8. Taylor expanded in x around 0 98.8%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;10^{-9} + \left(x \cdot \left(x \cdot -0.00011824294398844343\right) + \left({x}^{3} \cdot -0.37545125292247583 + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592}{{\left(e^{x}\right)}^{x}}\\ \end{array} \]

Alternative 9: 99.7% accurate, 7.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05:\\ \;\;\;\;10^{-9} + \left(x \cdot \left(x \cdot -0.00011824294398844343\right) + \left({x}^{3} \cdot -0.37545125292247583 + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.05)
   (+
    1e-9
    (+
     (* x (* x -0.00011824294398844343))
     (+ (* (pow x 3.0) -0.37545125292247583) (* x 1.128386358070218))))
   (- 1.0 (/ 0.7778892405807117 (* x (exp (* x x)))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.05) {
		tmp = 1e-9 + ((x * (x * -0.00011824294398844343)) + ((pow(x, 3.0) * -0.37545125292247583) + (x * 1.128386358070218)));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x * exp((x * x))));
	}
	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 <= 1.05d0) then
        tmp = 1d-9 + ((x * (x * (-0.00011824294398844343d0))) + (((x ** 3.0d0) * (-0.37545125292247583d0)) + (x * 1.128386358070218d0)))
    else
        tmp = 1.0d0 - (0.7778892405807117d0 / (x * exp((x * x))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.05) {
		tmp = 1e-9 + ((x * (x * -0.00011824294398844343)) + ((Math.pow(x, 3.0) * -0.37545125292247583) + (x * 1.128386358070218)));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x * Math.exp((x * x))));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.05:
		tmp = 1e-9 + ((x * (x * -0.00011824294398844343)) + ((math.pow(x, 3.0) * -0.37545125292247583) + (x * 1.128386358070218)))
	else:
		tmp = 1.0 - (0.7778892405807117 / (x * math.exp((x * x))))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.05)
		tmp = Float64(1e-9 + Float64(Float64(x * Float64(x * -0.00011824294398844343)) + Float64(Float64((x ^ 3.0) * -0.37545125292247583) + Float64(x * 1.128386358070218))));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x * exp(Float64(x * x)))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.05)
		tmp = 1e-9 + ((x * (x * -0.00011824294398844343)) + (((x ^ 3.0) * -0.37545125292247583) + (x * 1.128386358070218)));
	else
		tmp = 1.0 - (0.7778892405807117 / (x * exp((x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.05], N[(1e-9 + N[(N[(x * N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision] + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in x around inf 69.7%

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

   6. Simplified 69.4%

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

   7. Taylor expanded in x around 0 65.1%

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

      1. pow1 65.1%

         \[\leadsto 10^{-9} + \left(\color{blue}{{\left(-0.00011824294398844343 \cdot {x}^{2}\right)}^{1}} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]

      2. pow2 65.1%

         \[\leadsto 10^{-9} + \left({\left(-0.00011824294398844343 \cdot \color{blue}{\left(x \cdot x\right)}\right)}^{1} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]

      3. *-commutative 65.1%

         \[\leadsto 10^{-9} + \left({\color{blue}{\left(\left(x \cdot x\right) \cdot -0.00011824294398844343\right)}}^{1} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]

   9. Applied egg-rr 65.1%

      \[\leadsto 10^{-9} + \left(\color{blue}{{\left(\left(x \cdot x\right) \cdot -0.00011824294398844343\right)}^{1}} + \left(-0.37545125292247583 \cdot {x}^{3} + 1.128386358070218 \cdot x\right)\right) \]
   10. Step-by-step derivation

       1. unpow1 65.1%

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

       2. associate-*l* 65.2%

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

   11. Simplified 65.2%

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

   if 1.05000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 98.8%

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

      1. associate-*r/ 98.8%

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

      2. metadata-eval 98.8%

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

      3. *-commutative 98.8%

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

      4. unpow2 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05:\\ \;\;\;\;10^{-9} + \left(x \cdot \left(x \cdot -0.00011824294398844343\right) + \left({x}^{3} \cdot -0.37545125292247583 + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 10: 99.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + \left(x \cdot 1.128386358070218 + -0.00011824294398844343 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.88)
   (+ 1e-9 (+ (* x 1.128386358070218) (* -0.00011824294398844343 (* x x))))
   (- 1.0 (/ 0.7778892405807117 (* x (exp (* x x)))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.88) {
		tmp = 1e-9 + ((x * 1.128386358070218) + (-0.00011824294398844343 * (x * x)));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x * exp((x * x))));
	}
	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 <= 0.88d0) then
        tmp = 1d-9 + ((x * 1.128386358070218d0) + ((-0.00011824294398844343d0) * (x * x)))
    else
        tmp = 1.0d0 - (0.7778892405807117d0 / (x * exp((x * x))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.88) {
		tmp = 1e-9 + ((x * 1.128386358070218) + (-0.00011824294398844343 * (x * x)));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x * Math.exp((x * x))));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.88:
		tmp = 1e-9 + ((x * 1.128386358070218) + (-0.00011824294398844343 * (x * x)))
	else:
		tmp = 1.0 - (0.7778892405807117 / (x * math.exp((x * x))))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(1e-9 + Float64(Float64(x * 1.128386358070218) + Float64(-0.00011824294398844343 * Float64(x * x))));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x * exp(Float64(x * x)))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.88)
		tmp = 1e-9 + ((x * 1.128386358070218) + (-0.00011824294398844343 * (x * x)));
	else
		tmp = 1.0 - (0.7778892405807117 / (x * exp((x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.88], N[(1e-9 + N[(N[(x * 1.128386358070218), $MachinePrecision] + N[(-0.00011824294398844343 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in x around inf 69.7%

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

   6. Simplified 69.4%

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

   7. Taylor expanded in x around 0 64.6%

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

      1. +-commutative 64.6%

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

      2. *-commutative 64.6%

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

      3. fma-def 64.6%

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

      4. *-commutative 64.6%

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

      5. unpow2 64.6%

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

   9. Simplified 64.6%

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

       1. fma-udef 64.6%

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

   11. Applied egg-rr 64.6%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 98.8%

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

      1. associate-*r/ 98.8%

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

      2. metadata-eval 98.8%

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

      3. *-commutative 98.8%

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

      4. unpow2 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + \left(x \cdot 1.128386358070218 + -0.00011824294398844343 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 11: 99.5% accurate, 65.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;10^{-9} + \left(x \cdot 1.128386358070218 + -0.00011824294398844343 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.89)
   (+ 1e-9 (+ (* x 1.128386358070218) (* -0.00011824294398844343 (* x x))))
   1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.89) {
		tmp = 1e-9 + ((x * 1.128386358070218) + (-0.00011824294398844343 * (x * x)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.89d0) then
        tmp = 1d-9 + ((x * 1.128386358070218d0) + ((-0.00011824294398844343d0) * (x * x)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.89) {
		tmp = 1e-9 + ((x * 1.128386358070218) + (-0.00011824294398844343 * (x * x)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.89:
		tmp = 1e-9 + ((x * 1.128386358070218) + (-0.00011824294398844343 * (x * x)))
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.89)
		tmp = Float64(1e-9 + Float64(Float64(x * 1.128386358070218) + Float64(-0.00011824294398844343 * Float64(x * x))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.89)
		tmp = 1e-9 + ((x * 1.128386358070218) + (-0.00011824294398844343 * (x * x)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.89], N[(1e-9 + N[(N[(x * 1.128386358070218), $MachinePrecision] + N[(-0.00011824294398844343 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.890000000000000013

   1. Initial program 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in x around inf 69.7%

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

   6. Simplified 69.4%

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

   7. Taylor expanded in x around 0 64.6%

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

      1. +-commutative 64.6%

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

      2. *-commutative 64.6%

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

      3. fma-def 64.6%

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

      4. *-commutative 64.6%

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

      5. unpow2 64.6%

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

   9. Simplified 64.6%

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

       1. fma-udef 64.6%

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

   11. Applied egg-rr 64.6%

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

   if 0.890000000000000013 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 98.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;10^{-9} + \left(x \cdot 1.128386358070218 + -0.00011824294398844343 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 99.5% accurate, 77.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;10^{-9} + x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.89)
   (+ 1e-9 (* x (+ (* x -0.00011824294398844343) 1.128386358070218)))
   1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.89) {
		tmp = 1e-9 + (x * ((x * -0.00011824294398844343) + 1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.89d0) then
        tmp = 1d-9 + (x * ((x * (-0.00011824294398844343d0)) + 1.128386358070218d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.89) {
		tmp = 1e-9 + (x * ((x * -0.00011824294398844343) + 1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.89:
		tmp = 1e-9 + (x * ((x * -0.00011824294398844343) + 1.128386358070218))
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.89)
		tmp = Float64(1e-9 + Float64(x * Float64(Float64(x * -0.00011824294398844343) + 1.128386358070218)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.89)
		tmp = 1e-9 + (x * ((x * -0.00011824294398844343) + 1.128386358070218));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.89], N[(1e-9 + N[(x * N[(N[(x * -0.00011824294398844343), $MachinePrecision] + 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.890000000000000013

   1. Initial program 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in x around inf 69.7%

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

   6. Simplified 69.4%

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

   7. Taylor expanded in x around 0 64.6%

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

      1. +-commutative 64.6%

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

      2. *-commutative 64.6%

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

      3. fma-def 64.6%

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

      4. *-commutative 64.6%

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

      5. unpow2 64.6%

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

   9. Simplified 64.6%

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

   10. Taylor expanded in x around 0 64.6%

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

       1. +-commutative 64.6%

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

       2. *-commutative 64.6%

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

       3. *-commutative 64.6%

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

       4. unpow2 64.6%

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

       5. associate-*l* 64.6%

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

       6. distribute-lft-out 64.6%

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

   12. Simplified 64.6%

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

   if 0.890000000000000013 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 98.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;10^{-9} + x \cdot \left(x \cdot -0.00011824294398844343 + 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 99.4% accurate, 121.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.89) (+ 1e-9 (* x 1.128386358070218)) 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.89) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.89d0) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.89) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.89:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.89)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.89)
		tmp = 1e-9 + (x * 1.128386358070218);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.89], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.890000000000000013

   1. Initial program 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in x around inf 69.7%

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

   6. Simplified 69.4%

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

   7. Taylor expanded in x around 0 64.6%

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

      1. *-commutative 64.6%

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

   9. Simplified 64.6%

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

   if 0.890000000000000013 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 98.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 58.0% accurate, 279.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;0.745170408\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 2.4e-5) 1e-9 0.745170408))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.4e-5) {
		tmp = 1e-9;
	} else {
		tmp = 0.745170408;
	}
	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 <= 2.4d-5) then
        tmp = 1d-9
    else
        tmp = 0.745170408d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.4e-5) {
		tmp = 1e-9;
	} else {
		tmp = 0.745170408;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.4e-5:
		tmp = 1e-9
	else:
		tmp = 0.745170408
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.4e-5)
		tmp = 1e-9;
	else
		tmp = 0.745170408;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4e-5)
		tmp = 1e-9;
	else
		tmp = 0.745170408;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.4e-5], 1e-9, 0.745170408]

Derivation

1. Split input into 2 regimes

2. if x < 2.4000000000000001e-5

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in x around inf 69.6%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in x around 0 66.8%

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

   if 2.4000000000000001e-5 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 99.7%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 97.5%

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

   8. Taylor expanded in x around 0 20.2%

      \[\leadsto \color{blue}{0.745170408} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;0.745170408\\ \end{array} \]

Alternative 15: 97.8% accurate, 279.5× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 2.8e-5) 1e-9 1.0))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.8d-5) then
        tmp = 1d-9
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.8e-5], 1e-9, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in x around inf 69.6%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in x around 0 66.8%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 99.7%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 97.5%

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

4. Final simplification 74.4%

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

Alternative 16: 53.4% 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 78.8%

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

3. Simplified 78.8%

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

4. Taylor expanded in x around inf 77.2%

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

6. Simplified 76.9%

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

7. Taylor expanded in x around 0 52.9%

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

8. Final simplification 52.9%

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

Reproduce

herbie shell --seed 2023279 
(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)))))))