Jmat.Real.erf

Percentage Accurate: 78.7% → 99.9%

Time: 31.4s

Alternatives: 11

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: 78.7% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

x = abs(x)
function code(x)
	t_0 = exp(Float64(x * x))
	t_1 = 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))) / fma(0.3275911, x, 1.0))) / Float64(t_0 * fma(0.3275911, x, 1.0)))
	tmp = 0.0
	if (abs(x) <= 1e-5)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + cbrt(Float64(Float64(x * 1.128386358070218) * Float64(Float64(x * 1.128386358070218) * Float64(x * 1.128386358070218)))))));
	else
		tmp = Float64(Float64(1.0 - Float64((Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) ^ 3.0) / (Float64(fma(x, 0.3275911, 1.0) * t_0) ^ 3.0))) / fma(Float64(1.0 + t_1), t_1, 1.0));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000008e-5

   1. Initial program 57.9%

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

   3. Simplified 58.0%

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

   5. Applied egg-rr 58.1%

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

      1. distribute-frac-neg 58.1%

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

      2. distribute-neg-in 58.1%

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

      3. metadata-eval 58.1%

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

      4. sub-neg 58.1%

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

   7. Simplified 56.0%

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

   8. Taylor expanded in x around 0 96.9%

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

      1. *-commutative 96.9%

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

      2. add-cbrt-cube 96.9%

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

   10. Applied egg-rr 96.9%

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

   if 1.00000000000000008e-5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Simplified 100.0%

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

      1. cube-div 100.0%

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

      2. pow-exp 100.0%

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

      3. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. sub-neg 100.0%

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

       2. *-commutative 100.0%

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

   11. Applied egg-rr 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 98.6%

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

Alternative 2: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000008e-5

   1. Initial program 57.9%

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

   3. Simplified 58.0%

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

   5. Applied egg-rr 58.1%

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

      1. distribute-frac-neg 58.1%

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

      2. distribute-neg-in 58.1%

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

      3. metadata-eval 58.1%

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

      4. sub-neg 58.1%

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

   7. Simplified 56.0%

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

   8. Taylor expanded in x around 0 96.9%

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

      1. *-commutative 96.9%

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

      2. add-cbrt-cube 96.9%

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

   10. Applied egg-rr 96.9%

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

   if 1.00000000000000008e-5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

   7. Simplified 100.0%

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

      1. add-log-exp 100.0%

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

      2. pow-exp 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 98.6%

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

Alternative 3: 99.9% accurate, 1.2× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1e-5)
   (+
    1e-9
    (+
     (* -0.37545125292247583 (pow x 3.0))
     (+
      (* -0.00011824294398844343 (pow x 2.0))
      (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)))
       (fma 0.3275911 x 1.0)))
     (* (exp (* x x)) (fma 0.3275911 x 1.0))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (fabs(x) <= 1e-5) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x, 3.0)) + ((-0.00011824294398844343 * pow(x, 2.0)) + 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))) / fma(0.3275911, x, 1.0))) / (exp((x * x)) * fma(0.3275911, x, 1.0)));
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1e-5], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[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[(-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[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000008e-5

   1. Initial program 57.9%

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

   3. Simplified 58.0%

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

   5. Applied egg-rr 58.1%

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

      1. distribute-frac-neg 58.1%

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

      2. distribute-neg-in 58.1%

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

      3. metadata-eval 58.1%

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

      4. sub-neg 58.1%

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

   7. Simplified 56.0%

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

   8. Taylor expanded in x around 0 96.9%

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

      1. *-commutative 96.9%

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

      2. add-cbrt-cube 96.9%

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

   10. Applied egg-rr 96.9%

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

   if 1.00000000000000008e-5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.6%

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

Alternative 4: 99.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x\right| \cdot 0.3275911}\\ t_1 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;x \leq 0.00052:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + \sqrt[3]{\left(x \cdot 1.128386358070218\right) \cdot \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + t_0 \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(t_0 \cdot \left(t_0 \cdot \left(\left(-1.453152027 + \frac{1.061405429}{t_1}\right) \cdot \frac{-1}{t_1} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x) 0.3275911))))
        (t_1 (+ 1.0 (* x 0.3275911))))
   (if (<= x 0.00052)
     (+
      1e-9
      (+
       (* -0.37545125292247583 (pow x 3.0))
       (+
        (* -0.00011824294398844343 (pow x 2.0))
        (cbrt
         (*
          (* x 1.128386358070218)
          (* (* x 1.128386358070218) (* x 1.128386358070218)))))))
     (+
      1.0
      (*
       t_0
       (*
        (exp (* x (- x)))
        (-
         (*
          t_0
          (-
           (*
            t_0
            (-
             (* (+ -1.453152027 (/ 1.061405429 t_1)) (/ -1.0 t_1))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 / (1.0 + (fabs(x) * 0.3275911));
	double t_1 = 1.0 + (x * 0.3275911);
	double tmp;
	if (x <= 0.00052) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x, 3.0)) + ((-0.00011824294398844343 * pow(x, 2.0)) + cbrt(((x * 1.128386358070218) * ((x * 1.128386358070218) * (x * 1.128386358070218))))));
	} else {
		tmp = 1.0 + (t_0 * (exp((x * -x)) * ((t_0 * ((t_0 * (((-1.453152027 + (1.061405429 / t_1)) * (-1.0 / t_1)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (Math.abs(x) * 0.3275911));
	double t_1 = 1.0 + (x * 0.3275911);
	double tmp;
	if (x <= 0.00052) {
		tmp = 1e-9 + ((-0.37545125292247583 * Math.pow(x, 3.0)) + ((-0.00011824294398844343 * Math.pow(x, 2.0)) + Math.cbrt(((x * 1.128386358070218) * ((x * 1.128386358070218) * (x * 1.128386358070218))))));
	} else {
		tmp = 1.0 + (t_0 * (Math.exp((x * -x)) * ((t_0 * ((t_0 * (((-1.453152027 + (1.061405429 / t_1)) * (-1.0 / t_1)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x) * 0.3275911)))
	t_1 = Float64(1.0 + Float64(x * 0.3275911))
	tmp = 0.0
	if (x <= 0.00052)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + cbrt(Float64(Float64(x * 1.128386358070218) * Float64(Float64(x * 1.128386358070218) * Float64(x * 1.128386358070218)))))));
	else
		tmp = Float64(1.0 + Float64(t_0 * Float64(exp(Float64(x * Float64(-x))) * Float64(Float64(t_0 * Float64(Float64(t_0 * Float64(Float64(Float64(-1.453152027 + Float64(1.061405429 / t_1)) * Float64(-1.0 / t_1)) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.00052], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[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[(t$95$0 * N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 * N[(N[(t$95$0 * N[(N[(N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.19999999999999954e-4

   1. Initial program 73.8%

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

   3. Simplified 73.8%

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

   5. Applied egg-rr 73.9%

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

      1. distribute-frac-neg 73.9%

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

      2. distribute-neg-in 73.9%

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

      3. metadata-eval 73.9%

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

      4. sub-neg 73.9%

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

   7. Simplified 72.6%

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

   8. Taylor expanded in x around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. add-cbrt-cube 60.8%

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

   10. Applied egg-rr 60.8%

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

   if 5.19999999999999954e-4 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. pow1 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. unpow1 99.9%

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

      2. unpow1 99.9%

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

      3. sqr-pow 99.9%

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

      4. fabs-sqr 99.9%

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

      5. sqr-pow 99.9%

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

      6. unpow1 99.9%

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

   7. Simplified 99.9%

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

      1. pow1 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. unpow1 99.9%

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

       2. unpow1 99.9%

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

       3. sqr-pow 99.9%

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

       4. fabs-sqr 99.9%

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

       5. sqr-pow 99.9%

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

       6. unpow1 99.9%

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

   11. Simplified 99.9%

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

4. Final simplification 71.2%

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

Alternative 5: 99.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + \sqrt[3]{\left(x \cdot 1.128386358070218\right) \cdot \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.1)
   (+
    1e-9
    (+
     (* -0.37545125292247583 (pow x 3.0))
     (+
      (* -0.00011824294398844343 (pow x 2.0))
      (cbrt
       (*
        (* x 1.128386358070218)
        (* (* x 1.128386358070218) (* x 1.128386358070218)))))))
   1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.1) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x, 3.0)) + ((-0.00011824294398844343 * pow(x, 2.0)) + cbrt(((x * 1.128386358070218) * ((x * 1.128386358070218) * (x * 1.128386358070218))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.1) {
		tmp = 1e-9 + ((-0.37545125292247583 * Math.pow(x, 3.0)) + ((-0.00011824294398844343 * Math.pow(x, 2.0)) + Math.cbrt(((x * 1.128386358070218) * ((x * 1.128386358070218) * (x * 1.128386358070218))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.1)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + cbrt(Float64(Float64(x * 1.128386358070218) * Float64(Float64(x * 1.128386358070218) * Float64(x * 1.128386358070218)))))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.1], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[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], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 73.9%

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

   3. Simplified 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. distribute-frac-neg 74.0%

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

      2. distribute-neg-in 74.0%

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

      3. metadata-eval 74.0%

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

      4. sub-neg 74.0%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in x around 0 61.0%

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

      1. *-commutative 61.0%

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

      2. add-cbrt-cube 60.7%

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

   10. Applied egg-rr 60.7%

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

   if 1.1000000000000001 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. associate--l+ 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + \sqrt[3]{\left(x \cdot 1.128386358070218\right) \cdot \left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 99.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.1)
   (+
    1e-9
    (fma
     (pow x 3.0)
     -0.37545125292247583
     (* x (+ 1.128386358070218 (* x -0.00011824294398844343)))))
   1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.1) {
		tmp = 1e-9 + fma(pow(x, 3.0), -0.37545125292247583, (x * (1.128386358070218 + (x * -0.00011824294398844343))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.1)
		tmp = Float64(1e-9 + fma((x ^ 3.0), -0.37545125292247583, Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343)))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.1], N[(1e-9 + N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 73.9%

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

   3. Simplified 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. distribute-frac-neg 74.0%

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

      2. distribute-neg-in 74.0%

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

      3. metadata-eval 74.0%

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

      4. sub-neg 74.0%

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

   7. Simplified 72.7%

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

      1. add-log-exp 71.4%

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

      2. pow-exp 71.4%

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

   9. Applied egg-rr 71.4%

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

   10. Taylor expanded in x around 0 61.0%

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

       1. fma-def 61.0%

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

       2. *-commutative 61.0%

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

       3. *-commutative 61.0%

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

       4. fma-def 61.0%

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

       5. unpow2 61.0%

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

       6. fma-udef 61.0%

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

       7. *-commutative 61.0%

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

       8. fma-def 61.0%

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

       9. fma-def 61.0%

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

       10. +-commutative 61.0%

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

       11. associate-*l* 61.0%

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

       12. distribute-lft-out 61.0%

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

   12. Simplified 61.0%

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

   if 1.1000000000000001 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. associate--l+ 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 99.4% accurate, 6.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + \frac{{x}^{4} \cdot 1.3981393803054172 \cdot 10^{-8} - \left(x \cdot x\right) \cdot 1.2732557730789702}{x \cdot \left(x \cdot -0.00011824294398844343\right) - x \cdot 1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.88)
   (+
    1e-9
    (/
     (- (* (pow x 4.0) 1.3981393803054172e-8) (* (* x x) 1.2732557730789702))
     (- (* x (* x -0.00011824294398844343)) (* x 1.128386358070218))))
   1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.88) {
		tmp = 1e-9 + (((pow(x, 4.0) * 1.3981393803054172e-8) - ((x * x) * 1.2732557730789702)) / ((x * (x * -0.00011824294398844343)) - (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 <= 0.88d0) then
        tmp = 1d-9 + ((((x ** 4.0d0) * 1.3981393803054172d-8) - ((x * x) * 1.2732557730789702d0)) / ((x * (x * (-0.00011824294398844343d0))) - (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 <= 0.88) {
		tmp = 1e-9 + (((Math.pow(x, 4.0) * 1.3981393803054172e-8) - ((x * x) * 1.2732557730789702)) / ((x * (x * -0.00011824294398844343)) - (x * 1.128386358070218)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.88:
		tmp = 1e-9 + (((math.pow(x, 4.0) * 1.3981393803054172e-8) - ((x * x) * 1.2732557730789702)) / ((x * (x * -0.00011824294398844343)) - (x * 1.128386358070218)))
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(1e-9 + Float64(Float64(Float64((x ^ 4.0) * 1.3981393803054172e-8) - Float64(Float64(x * x) * 1.2732557730789702)) / Float64(Float64(x * Float64(x * -0.00011824294398844343)) - 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 <= 0.88)
		tmp = 1e-9 + ((((x ^ 4.0) * 1.3981393803054172e-8) - ((x * x) * 1.2732557730789702)) / ((x * (x * -0.00011824294398844343)) - (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, 0.88], N[(1e-9 + N[(N[(N[(N[Power[x, 4.0], $MachinePrecision] * 1.3981393803054172e-8), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision] - N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 73.9%

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

   3. Simplified 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. distribute-frac-neg 74.0%

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

      2. distribute-neg-in 74.0%

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

      3. metadata-eval 74.0%

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

      4. sub-neg 74.0%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in x around 0 60.4%

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

      1. *-commutative 60.4%

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

      2. *-commutative 60.4%

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

      3. fma-def 60.4%

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

      4. unpow2 60.4%

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

   10. Simplified 60.4%

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

       1. fma-udef 60.4%

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

       2. *-commutative 60.4%

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

       3. pow2 60.4%

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

       4. *-commutative 60.4%

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

       5. flip-+ 60.3%

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

       6. pow2 60.3%

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

       7. *-commutative 60.3%

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

       8. pow2 60.3%

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

       9. *-commutative 60.3%

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

       10. *-commutative 60.3%

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

       11. *-commutative 60.3%

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

       12. pow2 60.3%

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

       13. *-commutative 60.3%

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

       14. *-commutative 60.3%

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

   12. Applied egg-rr 60.3%

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

       1. unpow2 60.3%

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

       2. unpow2 60.3%

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

       3. swap-sqr 60.3%

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

       4. pow-sqr 60.3%

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

       5. metadata-eval 60.3%

          \[\leadsto 10^{-9} + \frac{{x}^{\color{blue}{4}} \cdot \left(-0.00011824294398844343 \cdot -0.00011824294398844343\right) - \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)}{\left(x \cdot x\right) \cdot -0.00011824294398844343 - x \cdot 1.128386358070218} \]

       6. metadata-eval 60.3%

          \[\leadsto 10^{-9} + \frac{{x}^{4} \cdot \color{blue}{1.3981393803054172 \cdot 10^{-8}} - \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)}{\left(x \cdot x\right) \cdot -0.00011824294398844343 - x \cdot 1.128386358070218} \]

       7. swap-sqr 60.3%

          \[\leadsto 10^{-9} + \frac{{x}^{4} \cdot 1.3981393803054172 \cdot 10^{-8} - \color{blue}{\left(x \cdot x\right) \cdot \left(1.128386358070218 \cdot 1.128386358070218\right)}}{\left(x \cdot x\right) \cdot -0.00011824294398844343 - x \cdot 1.128386358070218} \]

       8. metadata-eval 60.3%

          \[\leadsto 10^{-9} + \frac{{x}^{4} \cdot 1.3981393803054172 \cdot 10^{-8} - \left(x \cdot x\right) \cdot \color{blue}{1.2732557730789702}}{\left(x \cdot x\right) \cdot -0.00011824294398844343 - x \cdot 1.128386358070218} \]

       9. associate-*l* 60.3%

          \[\leadsto 10^{-9} + \frac{{x}^{4} \cdot 1.3981393803054172 \cdot 10^{-8} - \left(x \cdot x\right) \cdot 1.2732557730789702}{\color{blue}{x \cdot \left(x \cdot -0.00011824294398844343\right)} - x \cdot 1.128386358070218} \]

   14. Simplified 60.3%

       \[\leadsto 10^{-9} + \color{blue}{\frac{{x}^{4} \cdot 1.3981393803054172 \cdot 10^{-8} - \left(x \cdot x\right) \cdot 1.2732557730789702}{x \cdot \left(x \cdot -0.00011824294398844343\right) - x \cdot 1.128386358070218}} \]

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. associate--l+ 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + \frac{{x}^{4} \cdot 1.3981393803054172 \cdot 10^{-8} - \left(x \cdot x\right) \cdot 1.2732557730789702}{x \cdot \left(x \cdot -0.00011824294398844343\right) - x \cdot 1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 99.4% accurate, 77.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(1e-9 + Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343))));
	else
		tmp = 1.0;
	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 + (x * -0.00011824294398844343)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 73.9%

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

   3. Simplified 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. distribute-frac-neg 74.0%

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

      2. distribute-neg-in 74.0%

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

      3. metadata-eval 74.0%

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

      4. sub-neg 74.0%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in x around 0 60.4%

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

      1. *-commutative 60.4%

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

      2. *-commutative 60.4%

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

      3. fma-def 60.4%

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

      4. unpow2 60.4%

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

   10. Simplified 60.4%

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

   11. Taylor expanded in x around 0 60.4%

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

       1. +-commutative 60.4%

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

       2. *-commutative 60.4%

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

       3. *-commutative 60.4%

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

       4. unpow2 60.4%

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

       5. associate-*l* 60.4%

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

       6. distribute-lft-out 60.4%

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

   13. Simplified 60.4%

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

   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%

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

   4. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. associate--l+ 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 70.8%

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

Alternative 9: 99.3% accurate, 121.2× speedup

Math

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

FPCore

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

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.88) {
		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 <= 0.88d0) 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 <= 0.88) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.88:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.88)
		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 <= 0.88)
		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, 0.88], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 73.9%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in x around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. associate--l+ 71.1%

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

   6. Simplified 70.2%

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

   7. Taylor expanded in x around 0 60.4%

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

      1. *-commutative 60.4%

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

   9. Simplified 60.4%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. associate--l+ 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 70.8%

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

Alternative 10: 97.5% 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 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in x around 0 72.0%

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

      1. +-commutative 72.0%

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

      2. associate--l+ 71.3%

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

   6. Simplified 70.5%

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

   7. Taylor expanded in x around 0 62.8%

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

   if 2.79999999999999996e-5 < 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 97.5%

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

      1. +-commutative 97.5%

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

      2. associate--l+ 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in x around inf 98.8%

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

4. Final simplification 72.3%

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

Alternative 11: 54.0% 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 80.7%

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

3. Simplified 80.8%

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

4. Taylor expanded in x around 0 78.7%

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

   1. +-commutative 78.7%

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

   2. associate--l+ 78.3%

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

6. Simplified 77.7%

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

7. Taylor expanded in x around 0 49.0%

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

8. Final simplification 49.0%

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

Reproduce

herbie shell --seed 2023283 
(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)))))))