Jmat.Real.erf

Percentage Accurate: 79.1% → 86.5%

Time: 23.8s

Alternatives: 14

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_0}}{t\_0}}{t\_0}}{t\_0}}{t\_0 \cdot e^{x \cdot x}}\\ t_2 := {t\_1}^{2}\\ \frac{\frac{1}{1 + t\_2} + \frac{{t\_1}^{4}}{-1 - t\_2}}{1 + t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1
         (/
          (+
           0.254829592
           (/
            (+
             -0.284496736
             (/
              (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_0)) t_0))
              t_0))
            t_0))
          (* t_0 (exp (* x x)))))
        (t_2 (pow t_1 2.0)))
   (/ (+ (/ 1.0 (+ 1.0 t_2)) (/ (pow t_1 4.0) (- -1.0 t_2))) (+ 1.0 t_1))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0)) / (t_0 * exp((x * x)));
	double t_2 = pow(t_1, 2.0);
	return ((1.0 / (1.0 + t_2)) + (pow(t_1, 4.0) / (-1.0 - t_2))) / (1.0 + t_1);
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_0)) / t_0)) / t_0)) / t_0)) / Float64(t_0 * exp(Float64(x * x))))
	t_2 = t_1 ^ 2.0
	return Float64(Float64(Float64(1.0 / Float64(1.0 + t_2)) + Float64((t_1 ^ 4.0) / Float64(-1.0 - t_2))) / Float64(1.0 + t_1))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, N[(N[(N[(1.0 / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$1, 4.0], $MachinePrecision] / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 80.0%

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

4. Applied rewrites 80.0%

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

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{1 - {\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}}}{1 + \frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}} \]

   2. flip-- N/A

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - {\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2} \cdot {\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}}{1 + {\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}}}}{1 + \frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}} \]

6. Applied rewrites 87.1%

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

7. Final simplification 87.1%

   \[\leadsto \frac{\frac{1}{1 + {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}} + \frac{{\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{4}}{-1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}}} \]
8. Add Preprocessing

Alternative 2: 79.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := 1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_0}}{t\_0}\\ t_2 := 0.3275911 \cdot \left|x\right|\\ t_3 := \frac{1}{1 + t\_2}\\ t_4 := e^{\left|x\right| \cdot \left(-\left|x\right|\right)}\\ t_5 := 0.254829592 + \frac{-0.284496736 + \frac{t\_1}{t\_0}}{t\_0}\\ t_6 := \frac{t\_5}{t\_0 \cdot e^{x \cdot x}}\\ t_7 := {t\_6}^{2}\\ \mathbf{if}\;\left(t\_3 \cdot \left(0.254829592 + t\_3 \cdot \left(-0.284496736 + t\_3 \cdot \left(1.421413741 + t\_3 \cdot \left(-1.453152027 + 1.061405429 \cdot t\_3\right)\right)\right)\right)\right) \cdot t\_4 \leq 0.999999999:\\ \;\;\;\;1 + t\_4 \cdot \left(\left(0.254829592 + \mathsf{fma}\left(t\_1 \cdot \frac{\mathsf{fma}\left(\left|x\right|, -0.3275911, 1\right)}{t\_0}, \frac{1}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, \frac{-0.284496736}{t\_0}\right)\right) \cdot \frac{1}{-1 - t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + t\_7} + \frac{{t\_6}^{4}}{-1 - t\_7}}{1 + \frac{t\_5}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1 (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_0)) t_0)))
        (t_2 (* 0.3275911 (fabs x)))
        (t_3 (/ 1.0 (+ 1.0 t_2)))
        (t_4 (exp (* (fabs x) (- (fabs x)))))
        (t_5 (+ 0.254829592 (/ (+ -0.284496736 (/ t_1 t_0)) t_0)))
        (t_6 (/ t_5 (* t_0 (exp (* x x)))))
        (t_7 (pow t_6 2.0)))
   (if (<=
        (*
         (*
          t_3
          (+
           0.254829592
           (*
            t_3
            (+
             -0.284496736
             (*
              t_3
              (+ 1.421413741 (* t_3 (+ -1.453152027 (* 1.061405429 t_3)))))))))
         t_4)
        0.999999999)
     (+
      1.0
      (*
       t_4
       (*
        (+
         0.254829592
         (fma
          (* t_1 (/ (fma (fabs x) -0.3275911 1.0) t_0))
          (/ 1.0 (fma (* x x) -0.10731592879921 1.0))
          (/ -0.284496736 t_0)))
        (/ 1.0 (- -1.0 t_2)))))
     (/
      (+ (/ 1.0 (+ 1.0 t_7)) (/ (pow t_6 4.0) (- -1.0 t_7)))
      (+ 1.0 (/ t_5 t_0))))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = 1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0);
	double t_2 = 0.3275911 * fabs(x);
	double t_3 = 1.0 / (1.0 + t_2);
	double t_4 = exp((fabs(x) * -fabs(x)));
	double t_5 = 0.254829592 + ((-0.284496736 + (t_1 / t_0)) / t_0);
	double t_6 = t_5 / (t_0 * exp((x * x)));
	double t_7 = pow(t_6, 2.0);
	double tmp;
	if (((t_3 * (0.254829592 + (t_3 * (-0.284496736 + (t_3 * (1.421413741 + (t_3 * (-1.453152027 + (1.061405429 * t_3))))))))) * t_4) <= 0.999999999) {
		tmp = 1.0 + (t_4 * ((0.254829592 + fma((t_1 * (fma(fabs(x), -0.3275911, 1.0) / t_0)), (1.0 / fma((x * x), -0.10731592879921, 1.0)), (-0.284496736 / t_0))) * (1.0 / (-1.0 - t_2))));
	} else {
		tmp = ((1.0 / (1.0 + t_7)) + (pow(t_6, 4.0) / (-1.0 - t_7))) / (1.0 + (t_5 / t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_0)) / t_0))
	t_2 = Float64(0.3275911 * abs(x))
	t_3 = Float64(1.0 / Float64(1.0 + t_2))
	t_4 = exp(Float64(abs(x) * Float64(-abs(x))))
	t_5 = Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(t_1 / t_0)) / t_0))
	t_6 = Float64(t_5 / Float64(t_0 * exp(Float64(x * x))))
	t_7 = t_6 ^ 2.0
	tmp = 0.0
	if (Float64(Float64(t_3 * Float64(0.254829592 + Float64(t_3 * Float64(-0.284496736 + Float64(t_3 * Float64(1.421413741 + Float64(t_3 * Float64(-1.453152027 + Float64(1.061405429 * t_3))))))))) * t_4) <= 0.999999999)
		tmp = Float64(1.0 + Float64(t_4 * Float64(Float64(0.254829592 + fma(Float64(t_1 * Float64(fma(abs(x), -0.3275911, 1.0) / t_0)), Float64(1.0 / fma(Float64(x * x), -0.10731592879921, 1.0)), Float64(-0.284496736 / t_0))) * Float64(1.0 / Float64(-1.0 - t_2)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(1.0 + t_7)) + Float64((t_6 ^ 4.0) / Float64(-1.0 - t_7))) / Float64(1.0 + Float64(t_5 / t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Exp[N[(N[Abs[x], $MachinePrecision] * (-N[Abs[x], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(0.254829592 + N[(N[(-0.284496736 + N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 / N[(t$95$0 * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Power[t$95$6, 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$3 * N[(0.254829592 + N[(t$95$3 * N[(-0.284496736 + N[(t$95$3 * N[(1.421413741 + N[(t$95$3 * N[(-1.453152027 + N[(1.061405429 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], 0.999999999], N[(1.0 + N[(t$95$4 * N[(N[(0.254829592 + N[(N[(t$95$1 * N[(N[(N[Abs[x], $MachinePrecision] * -0.3275911 + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(x * x), $MachinePrecision] * -0.10731592879921 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-0.284496736 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(1.0 + t$95$7), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$6, 4.0], $MachinePrecision] / N[(-1.0 - t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$5 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 0.999999999000000028

   1. Initial program 79.1%

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

   4. Applied rewrites 79.1%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\mathsf{fma}\left(\frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot 0.10731592879921}, \frac{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{1}{1 + -0.3275911 \cdot \left|x\right|}}, \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \color{blue}{\left(\frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}} \cdot \frac{\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\frac{1}{1 + \frac{-3275911}{10000000} \cdot \left|x\right|}} + \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \left(\color{blue}{\frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}} \cdot \frac{\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\frac{1}{1 + \frac{-3275911}{10000000} \cdot \left|x\right|}} + \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

      3. associate-*l/ N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \left(\color{blue}{\frac{\left(\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\frac{1}{1 + \frac{-3275911}{10000000} \cdot \left|x\right|}}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}} + \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

      4. div-inv N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \left(\color{blue}{\left(\left(\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\frac{1}{1 + \frac{-3275911}{10000000} \cdot \left|x\right|}}\right) \cdot \frac{1}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}} + \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \color{blue}{\mathsf{fma}\left(\left(\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\frac{1}{1 + \frac{-3275911}{10000000} \cdot \left|x\right|}}, \frac{1}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}, \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   6. Applied rewrites 79.1%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\mathsf{fma}\left(\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) \cdot \frac{\mathsf{fma}\left(\left|x\right|, -0.3275911, 1\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{1}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   if 0.999999999000000028 < (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))

   1. Initial program 79.1%

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

   4. Applied rewrites 79.1%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{1 - {\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}}}{1 + \frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}} \]

      2. flip-- N/A

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - {\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2} \cdot {\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}}{1 + {\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}}}}{1 + \frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}} \]

   6. Applied rewrites 86.5%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{1 + {\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}} - \frac{{\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{4}}{1 + {\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}}}{1 + \frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\color{blue}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{1}{1 + {\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}} - \frac{{\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{4}}{1 + {\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}}}{1 + \frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\color{blue}{\frac{3275911}{10000000} \cdot \left|x\right| + 1}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{1 + {\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}} - \frac{{\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{4}}{1 + {\left(\frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}}}{1 + \frac{\frac{31853699}{125000000} + \frac{\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\color{blue}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}} \]

      3. lower-fabs.f64 85.0

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

   9. Applied rewrites 85.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + 1.061405429 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \leq 0.999999999:\\ \;\;\;\;1 + e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(\left(0.254829592 + \mathsf{fma}\left(\left(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)}\right) \cdot \frac{\mathsf{fma}\left(\left|x\right|, -0.3275911, 1\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{1}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right) \cdot \frac{1}{-1 - 0.3275911 \cdot \left|x\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}} + \frac{{\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{4}}{-1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}}\right)}^{2}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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)))))))