Jmat.Real.erf

Percentage Accurate: 79.2% → 99.7%

Time: 27.7s

Alternatives: 10

Speedup: 279.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \sqrt[3]{\frac{1.061405429}{t_0}}\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + t_1 \cdot \left(t_1 \cdot t_1\right)}{t_0}}{t_0}}{t_0}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{t_0}, 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0)) (t_1 (cbrt (/ 1.061405429 t_0))))
   (if (<= (fabs x) 5e-8)
     (fma x 1.128386358070218 1e-9)
     (fma
      (-
       -0.254829592
       (/
        (+
         -0.284496736
         (/ (+ 1.421413741 (/ (+ -1.453152027 (* t_1 (* t_1 t_1))) t_0)) t_0))
        t_0))
      (/ (pow (exp x) (- x)) t_0)
      1.0))))

C

x = abs(x);
double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = cbrt((1.061405429 / t_0));
	double tmp;
	if (fabs(x) <= 5e-8) {
		tmp = fma(x, 1.128386358070218, 1e-9);
	} else {
		tmp = fma((-0.254829592 - ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (t_1 * (t_1 * t_1))) / t_0)) / t_0)) / t_0)), (pow(exp(x), -x) / t_0), 1.0);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = cbrt(Float64(1.061405429 / t_0))
	tmp = 0.0
	if (abs(x) <= 5e-8)
		tmp = fma(x, 1.128386358070218, 1e-9);
	else
		tmp = fma(Float64(-0.254829592 - Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(t_1 * Float64(t_1 * t_1))) / t_0)) / t_0)) / t_0)), Float64((exp(x) ^ Float64(-x)) / t_0), 1.0);
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(1.061405429 / t$95$0), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-8], N[(x * 1.128386358070218 + 1e-9), $MachinePrecision], N[(N[(-0.254829592 - N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.9999999999999998e-8

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in x around 0 55.4%

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

   6. Simplified 53.4%

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

   7. Taylor expanded in x around 0 97.1%

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

      1. +-commutative 97.1%

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

      2. *-commutative 97.1%

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

      3. fma-def 97.1%

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

   9. Simplified 97.1%

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

   if 4.9999999999999998e-8 < (fabs.f64 x)

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

      1. add-cube-cbrt 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \sqrt[3]{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \cdot \left(\sqrt[3]{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \cdot \sqrt[3]{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ \mathbf{if}\;\left|x\right| \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{1 - \frac{{\left(e^{x}\right)}^{\left(-x\right)} \cdot \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{t_0 \cdot \left(-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}}{t_0}}{t_0}\right)}{t_0}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0)))
   (if (<= (fabs x) 1.5e-6)
     (fma x 1.128386358070218 1e-9)
     (log
      (exp
       (-
        1.0
        (/
         (*
          (pow (exp x) (- x))
          (+
           0.254829592
           (/
            (+
             -0.284496736
             (/
              (+
               1.421413741
               (/
                (-
                 2.111650813574209
                 (pow (/ 1.061405429 (fma 0.3275911 x 1.0)) 2.0))
                (*
                 t_0
                 (+ -1.453152027 (/ -1.061405429 (fma 0.3275911 x 1.0))))))
              t_0))
            t_0)))
         t_0)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double tmp;
	if (fabs(x) <= 1.5e-6) {
		tmp = fma(x, 1.128386358070218, 1e-9);
	} else {
		tmp = log(exp((1.0 - ((pow(exp(x), -x) * (0.254829592 + ((-0.284496736 + ((1.421413741 + ((2.111650813574209 - pow((1.061405429 / fma(0.3275911, x, 1.0)), 2.0)) / (t_0 * (-1.453152027 + (-1.061405429 / fma(0.3275911, x, 1.0)))))) / t_0)) / t_0))) / t_0))));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	tmp = 0.0
	if (abs(x) <= 1.5e-6)
		tmp = fma(x, 1.128386358070218, 1e-9);
	else
		tmp = log(exp(Float64(1.0 - Float64(Float64((exp(x) ^ Float64(-x)) * Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(2.111650813574209 - (Float64(1.061405429 / fma(0.3275911, x, 1.0)) ^ 2.0)) / Float64(t_0 * Float64(-1.453152027 + Float64(-1.061405429 / fma(0.3275911, x, 1.0)))))) / t_0)) / t_0))) / t_0))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1.5e-6], N[(x * 1.128386358070218 + 1e-9), $MachinePrecision], N[Log[N[Exp[N[(1.0 - N[(N[(N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] * N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(2.111650813574209 - N[Power[N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(-1.453152027 + N[(-1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.5e-6

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in x around 0 55.4%

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

   6. Simplified 53.4%

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

   7. Taylor expanded in x around 0 96.9%

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

      1. +-commutative 96.9%

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

      2. *-commutative 96.9%

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

      3. fma-def 96.9%

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

   9. Simplified 96.9%

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

   if 1.5e-6 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. flip-+ 99.8%

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

      2. metadata-eval 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{\color{blue}{2.111650813574209} - \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{-1.453152027 - \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. +-commutative 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - \frac{1.061405429}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \cdot \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{-1.453152027 - \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. fma-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \cdot \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{-1.453152027 - \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. +-commutative 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}}}{-1.453152027 - \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. fma-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}}{-1.453152027 - \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      7. +-commutative 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{-1.453152027 - \frac{1.061405429}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      8. fma-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{-1.453152027 - \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   5. Applied egg-rr 99.8%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\frac{2.111650813574209 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{-1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   6. Step-by-step derivation

      1. unpow2 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - \color{blue}{{\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2}}}{-1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. unpow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{1}}\right|, 1\right)}\right)}^{2}}{-1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. sqr-pow 54.2%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|, 1\right)}\right)}^{2}}{-1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. fabs-sqr 54.2%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}, 1\right)}\right)}^{2}}{-1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. sqr-pow 98.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{1}}, 1\right)}\right)}^{2}}{-1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. unpow1 98.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right)}\right)}^{2}}{-1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      7. sub-neg 98.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{\color{blue}{-1.453152027 + \left(-\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      8. fma-udef 98.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \left(-\frac{1.061405429}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      9. +-commutative 98.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \left(-\frac{1.061405429}{\color{blue}{1 + 0.3275911 \cdot \left|x\right|}}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      10. distribute-neg-frac 98.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \color{blue}{\frac{-1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      11. metadata-eval 98.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{\color{blue}{-1.061405429}}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      12. +-commutative 98.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      13. fma-udef 98.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      14. unpow1 98.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{1}}\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      15. sqr-pow 54.2%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      16. fabs-sqr 54.2%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      17. sqr-pow 98.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{1}}, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      18. unpow1 98.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   7. Simplified 98.8%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   9. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\log \left(e^{1 - \frac{{\left(e^{x}\right)}^{\left(-x\right)} \cdot \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot \left(-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{1 - \frac{{\left(e^{x}\right)}^{\left(-x\right)} \cdot \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot \left(-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ \mathbf{if}\;\left|x\right| \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)} \cdot \left(\frac{\frac{\frac{{\left(\frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} - 2.111650813574209}{t_0 \cdot \left(-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)} - 1.421413741}{t_0} - -0.284496736}{t_0} - 0.254829592\right)}{t_0}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0)))
   (if (<= (fabs x) 1.5e-6)
     (fma x 1.128386358070218 1e-9)
     (exp
      (log1p
       (/
        (*
         (pow (exp x) (- x))
         (-
          (/
           (-
            (/
             (-
              (/
               (-
                (pow (/ 1.061405429 (fma x 0.3275911 1.0)) 2.0)
                2.111650813574209)
               (* t_0 (+ -1.453152027 (/ -1.061405429 (fma x 0.3275911 1.0)))))
              1.421413741)
             t_0)
            -0.284496736)
           t_0)
          0.254829592))
        t_0))))))

C

x = abs(x);
double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double tmp;
	if (fabs(x) <= 1.5e-6) {
		tmp = fma(x, 1.128386358070218, 1e-9);
	} else {
		tmp = exp(log1p(((pow(exp(x), -x) * (((((((pow((1.061405429 / fma(x, 0.3275911, 1.0)), 2.0) - 2.111650813574209) / (t_0 * (-1.453152027 + (-1.061405429 / fma(x, 0.3275911, 1.0))))) - 1.421413741) / t_0) - -0.284496736) / t_0) - 0.254829592)) / t_0)));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	tmp = 0.0
	if (abs(x) <= 1.5e-6)
		tmp = fma(x, 1.128386358070218, 1e-9);
	else
		tmp = exp(log1p(Float64(Float64((exp(x) ^ Float64(-x)) * Float64(Float64(Float64(Float64(Float64(Float64(Float64((Float64(1.061405429 / fma(x, 0.3275911, 1.0)) ^ 2.0) - 2.111650813574209) / Float64(t_0 * Float64(-1.453152027 + Float64(-1.061405429 / fma(x, 0.3275911, 1.0))))) - 1.421413741) / t_0) - -0.284496736) / t_0) - 0.254829592)) / t_0)));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1.5e-6], N[(x * 1.128386358070218 + 1e-9), $MachinePrecision], N[Exp[N[Log[1 + N[(N[(N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[Power[N[(1.061405429 / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - 2.111650813574209), $MachinePrecision] / N[(t$95$0 * N[(-1.453152027 + N[(-1.061405429 / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.5e-6

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in x around 0 55.4%

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

   6. Simplified 53.4%

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

   7. Taylor expanded in x around 0 96.9%

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

      1. +-commutative 96.9%

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

      2. *-commutative 96.9%

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

      3. fma-def 96.9%

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

   9. Simplified 96.9%

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

   if 1.5e-6 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. flip-+ 99.8%

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

      2. metadata-eval 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{\color{blue}{2.111650813574209} - \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{-1.453152027 - \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. +-commutative 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - \frac{1.061405429}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \cdot \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{-1.453152027 - \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. fma-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \cdot \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{-1.453152027 - \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. +-commutative 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}}}{-1.453152027 - \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. fma-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}}{-1.453152027 - \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      7. +-commutative 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{-1.453152027 - \frac{1.061405429}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      8. fma-udef 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{-1.453152027 - \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   5. Applied egg-rr 99.8%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\frac{2.111650813574209 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{-1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   6. Step-by-step derivation

      1. unpow2 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - \color{blue}{{\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2}}}{-1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. unpow1 99.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{1}}\right|, 1\right)}\right)}^{2}}{-1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. sqr-pow 54.2%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|, 1\right)}\right)}^{2}}{-1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. fabs-sqr 54.2%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}, 1\right)}\right)}^{2}}{-1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. sqr-pow 98.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{1}}, 1\right)}\right)}^{2}}{-1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. unpow1 98.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right)}\right)}^{2}}{-1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      7. sub-neg 98.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{\color{blue}{-1.453152027 + \left(-\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      8. fma-udef 98.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \left(-\frac{1.061405429}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      9. +-commutative 98.8%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \left(-\frac{1.061405429}{\color{blue}{1 + 0.3275911 \cdot \left|x\right|}}\right)}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      10. distribute-neg-frac 98.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \color{blue}{\frac{-1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      11. metadata-eval 98.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{\color{blue}{-1.061405429}}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      12. +-commutative 98.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      13. fma-udef 98.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      14. unpow1 98.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{1}}\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      15. sqr-pow 54.2%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      16. fabs-sqr 54.2%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      17. sqr-pow 98.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{1}}, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      18. unpow1 98.8%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right)}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   7. Simplified 98.8%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   9. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{{\left(e^{x}\right)}^{\left(-x\right)} \cdot \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot \left(-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}} \]

   11. Simplified 98.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{2.111650813574209 - {\left(\frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot \left(-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) \cdot \left(-{\left(e^{x}\right)}^{\left(-x\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)} \cdot \left(\frac{\frac{\frac{{\left(\frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)}^{2} - 2.111650813574209}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot \left(-1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)} - 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 0.254829592\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1.5e-6)
   (fma x 1.128386358070218 1e-9)
   (fma
    (/ (pow (exp x) (- x)) (fma 0.3275911 x 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)))
    1.0)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (fabs(x) <= 1.5e-6) {
		tmp = fma(x, 1.128386358070218, 1e-9);
	} else {
		tmp = fma((pow(exp(x), -x) / fma(0.3275911, x, 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))), 1.0);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (abs(x) <= 1.5e-6)
		tmp = fma(x, 1.128386358070218, 1e-9);
	else
		tmp = fma(Float64((exp(x) ^ Float64(-x)) / fma(0.3275911, x, 1.0)), Float64(-0.254829592 - Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))), 1.0);
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1.5e-6], N[(x * 1.128386358070218 + 1e-9), $MachinePrecision], N[(N[(N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(-0.254829592 - N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.5e-6

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in x around 0 55.4%

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

   6. Simplified 53.4%

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

   7. Taylor expanded in x around 0 96.9%

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

      1. +-commutative 96.9%

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

      2. *-commutative 96.9%

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

      3. fma-def 96.9%

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

   9. Simplified 96.9%

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

   if 1.5e-6 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. fma-def 99.8%

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

   7. Simplified 98.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, 1\right)\\ \end{array} \]

Alternative 5: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ \mathbf{if}\;x \leq 1.58 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(0.254829592 + \frac{1}{t_0} \cdot \left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot {\left(\sqrt{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}\right)}^{2}\right)\right) \cdot e^{-x \cdot x}\right) \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911))))
   (if (<= x 1.58e-6)
     (fma x 1.128386358070218 1e-9)
     (+
      1.0
      (*
       (*
        (+
         0.254829592
         (*
          (/ 1.0 t_0)
          (+
           -0.284496736
           (*
            (/ 1.0 (+ 1.0 (* x 0.3275911)))
            (pow
             (sqrt
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma x 0.3275911 1.0)))
                (fma x 0.3275911 1.0))))
             2.0)))))
        (exp (- (* x x))))
       (/ -1.0 t_0))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double tmp;
	if (x <= 1.58e-6) {
		tmp = fma(x, 1.128386358070218, 1e-9);
	} else {
		tmp = 1.0 + (((0.254829592 + ((1.0 / t_0) * (-0.284496736 + ((1.0 / (1.0 + (x * 0.3275911))) * pow(sqrt((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0)))), 2.0))))) * exp(-(x * x))) * (-1.0 / t_0));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	tmp = 0.0
	if (x <= 1.58e-6)
		tmp = fma(x, 1.128386358070218, 1e-9);
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(0.254829592 + Float64(Float64(1.0 / t_0) * Float64(-0.284496736 + Float64(Float64(1.0 / Float64(1.0 + Float64(x * 0.3275911))) * (sqrt(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0)))) ^ 2.0))))) * exp(Float64(-Float64(x * x)))) * Float64(-1.0 / t_0)));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.58e-6], N[(x * 1.128386358070218 + 1e-9), $MachinePrecision], N[(1.0 + N[(N[(N[(0.254829592 + N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(-0.284496736 + N[(N[(1.0 / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[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]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.57999999999999991e-6

   1. Initial program 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in x around 0 68.3%

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

   6. Simplified 66.7%

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

   7. Taylor expanded in x around 0 65.8%

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

      1. +-commutative 65.8%

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

      2. *-commutative 65.8%

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

      3. fma-def 65.8%

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

   9. Simplified 65.8%

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

   if 1.57999999999999991e-6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. unpow1 100.0%

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

      3. sqr-pow 100.0%

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

      4. fabs-sqr 100.0%

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

      5. sqr-pow 100.0%

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

      6. unpow1 100.0%

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

   7. Simplified 100.0%

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

      1. pow1 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow1 100.0%

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

       2. unpow1 100.0%

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

       3. sqr-pow 100.0%

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

       4. fabs-sqr 100.0%

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

       5. sqr-pow 100.0%

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

       6. unpow1 100.0%

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

   11. Simplified 100.0%

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

       1. pow1 100.0%

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

   13. Applied egg-rr 100.0%

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

       1. unpow1 100.0%

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

       2. unpow1 100.0%

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

       3. sqr-pow 100.0%

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

       4. fabs-sqr 100.0%

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

       5. sqr-pow 100.0%

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

       6. unpow1 100.0%

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

   15. Simplified 100.0%

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

       1. add-sqr-sqrt 100.0%

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

       2. pow2 100.0%

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

       3. associate-*l/ 100.0%

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

       4. *-un-lft-identity 100.0%

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

       5. +-commutative 100.0%

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

       6. *-commutative 100.0%

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

       7. fma-udef 100.0%

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

       8. +-commutative 100.0%

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

       9. *-commutative 100.0%

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

       10. fma-udef 100.0%

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

   17. Applied egg-rr 100.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.58 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot {\left(\sqrt{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}\right)}^{2}\right)\right) \cdot e^{-x \cdot x}\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 6: 99.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := 1 + x \cdot 0.3275911\\ t_2 := \frac{1}{t_1}\\ \mathbf{if}\;x \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{t_0} \cdot \left(-0.284496736 + t_2 \cdot \left(1.421413741 + t_2 \cdot \left(-1.453152027 + \frac{1.061405429}{t_1}\right)\right)\right)\right)\right) \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_1 (+ 1.0 (* x 0.3275911)))
        (t_2 (/ 1.0 t_1)))
   (if (<= x 1.5e-6)
     (fma x 1.128386358070218 1e-9)
     (+
      1.0
      (*
       (*
        (exp (- (* x x)))
        (+
         0.254829592
         (*
          (/ 1.0 t_0)
          (+
           -0.284496736
           (*
            t_2
            (+ 1.421413741 (* t_2 (+ -1.453152027 (/ 1.061405429 t_1)))))))))
       (/ -1.0 t_0))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 + (x * 0.3275911);
	double t_2 = 1.0 / t_1;
	double tmp;
	if (x <= 1.5e-6) {
		tmp = fma(x, 1.128386358070218, 1e-9);
	} else {
		tmp = 1.0 + ((exp(-(x * x)) * (0.254829592 + ((1.0 / t_0) * (-0.284496736 + (t_2 * (1.421413741 + (t_2 * (-1.453152027 + (1.061405429 / t_1))))))))) * (-1.0 / t_0));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 + Float64(x * 0.3275911))
	t_2 = Float64(1.0 / t_1)
	tmp = 0.0
	if (x <= 1.5e-6)
		tmp = fma(x, 1.128386358070218, 1e-9);
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(-Float64(x * x))) * Float64(0.254829592 + Float64(Float64(1.0 / t_0) * Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(t_2 * Float64(-1.453152027 + Float64(1.061405429 / t_1))))))))) * Float64(-1.0 / t_0)));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, If[LessEqual[x, 1.5e-6], N[(x * 1.128386358070218 + 1e-9), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision] * N[(0.254829592 + N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(t$95$2 * N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e-6

   1. Initial program 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in x around 0 68.3%

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

   6. Simplified 66.7%

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

   7. Taylor expanded in x around 0 65.8%

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

      1. +-commutative 65.8%

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

      2. *-commutative 65.8%

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

      3. fma-def 65.8%

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

   9. Simplified 65.8%

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

   if 1.5e-6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. unpow1 100.0%

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

      3. sqr-pow 100.0%

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

      4. fabs-sqr 100.0%

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

      5. sqr-pow 100.0%

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

      6. unpow1 100.0%

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

   7. Simplified 100.0%

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

      1. pow1 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow1 100.0%

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

       2. unpow1 100.0%

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

       3. sqr-pow 100.0%

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

       4. fabs-sqr 100.0%

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

       5. sqr-pow 100.0%

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

       6. unpow1 100.0%

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

   11. Simplified 100.0%

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

       1. pow1 100.0%

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

   13. Applied egg-rr 100.0%

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

       1. unpow1 100.0%

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

       2. unpow1 100.0%

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

       3. sqr-pow 100.0%

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

       4. fabs-sqr 100.0%

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

       5. sqr-pow 100.0%

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

       6. unpow1 100.0%

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

   15. Simplified 100.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 7: 99.2% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.88)
		tmp = fma(x, 1.128386358070218, 1e-9);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in x around 0 68.3%

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

   6. Simplified 66.7%

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

   7. Taylor expanded in x around 0 65.8%

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

      1. +-commutative 65.8%

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

      2. *-commutative 65.8%

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

      3. fma-def 65.8%

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

   9. Simplified 65.8%

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

   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 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 75.3%

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

Alternative 8: 99.2% 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 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in x around 0 68.3%

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

   6. Simplified 66.7%

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

   7. Taylor expanded in x around 0 65.8%

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

      1. *-commutative 65.8%

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

   9. Simplified 65.8%

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

   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 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 75.3%

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

Alternative 9: 97.7% accurate, 279.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in x around 0 68.3%

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

   6. Simplified 66.7%

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

   7. Taylor expanded in x around 0 68.0%

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

   if 2.79999999999999996e-5 < 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 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 76.9%

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

Alternative 10: 53.2% 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 79.3%

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

3. Simplified 79.3%

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

4. Taylor expanded in x around 0 76.8%

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

6. Simplified 75.6%

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

7. Taylor expanded in x around 0 52.2%

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

8. Final simplification 52.2%

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

Reproduce

herbie shell --seed 2023264 
(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)))))))