Jmat.Real.erf

Percentage Accurate: 78.8% → 79.0%

Time: 9.5s

Alternatives: 15

Speedup: 1.3×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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: 79.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-x\right) \cdot x}\\ t_1 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_2 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_3 := \frac{\frac{\frac{\left(\frac{1.453152027}{t\_1} + 1.421413741\right) - \frac{1.061405429}{t\_1 \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{t\_2} + -0.284496736}{t\_2} + 0.254829592}{t\_1} \cdot t\_0\\ t_4 := {t\_3}^{6}\\ t_5 := \frac{\frac{\frac{1.453152027 - \frac{1.061405429}{t\_2}}{t\_1} - -1.421413741}{t\_2} + -0.284496736}{t\_2} + 0.254829592\\ \frac{\frac{\frac{1 - t\_4 \cdot t\_4}{1 + t\_4}}{1 - {t\_3}^{3}}}{1 + \left({\left(\frac{t\_5}{t\_2 \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{t\_5}{t\_1} \cdot t\_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* (- x) x)))
        (t_1 (fma -0.3275911 (fabs x) -1.0))
        (t_2 (fma (fabs x) 0.3275911 1.0))
        (t_3
         (*
          (/
           (+
            (/
             (+
              (/
               (-
                (+ (/ 1.453152027 t_1) 1.421413741)
                (/ 1.061405429 (* t_1 (fma 0.3275911 (fabs x) 1.0))))
               t_2)
              -0.284496736)
             t_2)
            0.254829592)
           t_1)
          t_0))
        (t_4 (pow t_3 6.0))
        (t_5
         (+
          (/
           (+
            (/
             (- (/ (- 1.453152027 (/ 1.061405429 t_2)) t_1) -1.421413741)
             t_2)
            -0.284496736)
           t_2)
          0.254829592)))
   (/
    (/ (/ (- 1.0 (* t_4 t_4)) (+ 1.0 t_4)) (- 1.0 (pow t_3 3.0)))
    (+
     1.0
     (-
      (pow (/ t_5 (* t_2 (exp (* x x)))) 2.0)
      (* 1.0 (* (/ t_5 t_1) t_0)))))))

C

double code(double x) {
	double t_0 = exp((-x * x));
	double t_1 = fma(-0.3275911, fabs(x), -1.0);
	double t_2 = fma(fabs(x), 0.3275911, 1.0);
	double t_3 = ((((((((1.453152027 / t_1) + 1.421413741) - (1.061405429 / (t_1 * fma(0.3275911, fabs(x), 1.0)))) / t_2) + -0.284496736) / t_2) + 0.254829592) / t_1) * t_0;
	double t_4 = pow(t_3, 6.0);
	double t_5 = ((((((1.453152027 - (1.061405429 / t_2)) / t_1) - -1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592;
	return (((1.0 - (t_4 * t_4)) / (1.0 + t_4)) / (1.0 - pow(t_3, 3.0))) / (1.0 + (pow((t_5 / (t_2 * exp((x * x)))), 2.0) - (1.0 * ((t_5 / t_1) * t_0))));
}

Julia

function code(x)
	t_0 = exp(Float64(Float64(-x) * x))
	t_1 = fma(-0.3275911, abs(x), -1.0)
	t_2 = fma(abs(x), 0.3275911, 1.0)
	t_3 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 / t_1) + 1.421413741) - Float64(1.061405429 / Float64(t_1 * fma(0.3275911, abs(x), 1.0)))) / t_2) + -0.284496736) / t_2) + 0.254829592) / t_1) * t_0)
	t_4 = t_3 ^ 6.0
	t_5 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_2)) / t_1) - -1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592)
	return Float64(Float64(Float64(Float64(1.0 - Float64(t_4 * t_4)) / Float64(1.0 + t_4)) / Float64(1.0 - (t_3 ^ 3.0))) / Float64(1.0 + Float64((Float64(t_5 / Float64(t_2 * exp(Float64(x * x)))) ^ 2.0) - Float64(1.0 * Float64(Float64(t_5 / t_1) * t_0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(N[(N[(N[(N[(1.453152027 / t$95$1), $MachinePrecision] + 1.421413741), $MachinePrecision] - N[(1.061405429 / N[(t$95$1 * N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$2), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, 6.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$2), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$2), $MachinePrecision] + 0.254829592), $MachinePrecision]}, N[(N[(N[(N[(1.0 - N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$4), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(t$95$5 / N[(t$95$2 * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[(1.0 * N[(N[(t$95$5 / t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\color{blue}{\left(\frac{1421413741}{1000000000} + \frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]
5. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}\right) - \color{blue}{\frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \color{blue}{\frac{1061405429}{1000000000}} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \color{blue}{\frac{1061405429}{1000000000}} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000} \cdot 1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   7. negate-sub N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| + \left(\mathsf{neg}\left(1\right)\right)} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   8. lift-fabs.f64 N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| + \left(\mathsf{neg}\left(1\right)\right)} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| + -1} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   10. lift-fma.f64 N/A

       \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   11. associate-*r/ N/A

       \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000} \cdot 1}{\color{blue}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000}}{\color{blue}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

6. Applied rewrites 78.8%

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

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   2. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3} \cdot {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 - {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

8. Applied rewrites 78.9%

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

   1. lift--.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{1 - {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3} \cdot {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}{1 - {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

10. Applied rewrites 79.0%

    \[\leadsto \frac{\frac{\color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\left(\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.421413741\right) - \frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{6} \cdot {\left(\frac{\frac{\frac{\left(\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.421413741\right) - \frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{6}}{1 + {\left(\frac{\frac{\frac{\left(\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.421413741\right) - \frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{6}}}}{1 - {\left(\frac{\frac{\frac{\left(\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.421413741\right) - \frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}{1 + \left({\left(\frac{\frac{\frac{\frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]
11. Add Preprocessing

Alternative 2: 79.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-x\right) \cdot x}\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_3 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_4 := \frac{\frac{\frac{1.453152027 - \frac{-1.061405429}{t\_2}}{t\_2} - -1.421413741}{t\_1} + -0.284496736}{t\_1}\\ t_5 := {\left(t\_0 \cdot \frac{t\_4 + 0.254829592}{t\_2}\right)}^{3}\\ t_6 := \frac{\frac{\frac{1.453152027 - \frac{1.061405429}{t\_3}}{t\_2} - -1.421413741}{t\_3} + -0.284496736}{t\_3} + 0.254829592\\ \frac{\frac{1 + {\left({\left(t\_0 \cdot \left(\frac{t\_4}{t\_2} + \frac{0.254829592}{t\_2}\right)\right)}^{3}\right)}^{3}}{1 + \left(t\_5 \cdot t\_5 - 1 \cdot t\_5\right)}}{1 + \left({\left(\frac{t\_6}{t\_3 \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{t\_6}{t\_2} \cdot t\_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* (- x) x)))
        (t_1 (fma 0.3275911 (fabs x) 1.0))
        (t_2 (fma -0.3275911 (fabs x) -1.0))
        (t_3 (fma (fabs x) 0.3275911 1.0))
        (t_4
         (/
          (+
           (/
            (- (/ (- 1.453152027 (/ -1.061405429 t_2)) t_2) -1.421413741)
            t_1)
           -0.284496736)
          t_1))
        (t_5 (pow (* t_0 (/ (+ t_4 0.254829592) t_2)) 3.0))
        (t_6
         (+
          (/
           (+
            (/
             (- (/ (- 1.453152027 (/ 1.061405429 t_3)) t_2) -1.421413741)
             t_3)
            -0.284496736)
           t_3)
          0.254829592)))
   (/
    (/
     (+ 1.0 (pow (pow (* t_0 (+ (/ t_4 t_2) (/ 0.254829592 t_2))) 3.0) 3.0))
     (+ 1.0 (- (* t_5 t_5) (* 1.0 t_5))))
    (+
     1.0
     (-
      (pow (/ t_6 (* t_3 (exp (* x x)))) 2.0)
      (* 1.0 (* (/ t_6 t_2) t_0)))))))

C

double code(double x) {
	double t_0 = exp((-x * x));
	double t_1 = fma(0.3275911, fabs(x), 1.0);
	double t_2 = fma(-0.3275911, fabs(x), -1.0);
	double t_3 = fma(fabs(x), 0.3275911, 1.0);
	double t_4 = (((((1.453152027 - (-1.061405429 / t_2)) / t_2) - -1.421413741) / t_1) + -0.284496736) / t_1;
	double t_5 = pow((t_0 * ((t_4 + 0.254829592) / t_2)), 3.0);
	double t_6 = ((((((1.453152027 - (1.061405429 / t_3)) / t_2) - -1.421413741) / t_3) + -0.284496736) / t_3) + 0.254829592;
	return ((1.0 + pow(pow((t_0 * ((t_4 / t_2) + (0.254829592 / t_2))), 3.0), 3.0)) / (1.0 + ((t_5 * t_5) - (1.0 * t_5)))) / (1.0 + (pow((t_6 / (t_3 * exp((x * x)))), 2.0) - (1.0 * ((t_6 / t_2) * t_0))));
}

Julia

function code(x)
	t_0 = exp(Float64(Float64(-x) * x))
	t_1 = fma(0.3275911, abs(x), 1.0)
	t_2 = fma(-0.3275911, abs(x), -1.0)
	t_3 = fma(abs(x), 0.3275911, 1.0)
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(-1.061405429 / t_2)) / t_2) - -1.421413741) / t_1) + -0.284496736) / t_1)
	t_5 = Float64(t_0 * Float64(Float64(t_4 + 0.254829592) / t_2)) ^ 3.0
	t_6 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_3)) / t_2) - -1.421413741) / t_3) + -0.284496736) / t_3) + 0.254829592)
	return Float64(Float64(Float64(1.0 + ((Float64(t_0 * Float64(Float64(t_4 / t_2) + Float64(0.254829592 / t_2))) ^ 3.0) ^ 3.0)) / Float64(1.0 + Float64(Float64(t_5 * t_5) - Float64(1.0 * t_5)))) / Float64(1.0 + Float64((Float64(t_6 / Float64(t_3 * exp(Float64(x * x)))) ^ 2.0) - Float64(1.0 * Float64(Float64(t_6 / t_2) * t_0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(1.453152027 - N[(-1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(t$95$0 * N[(N[(t$95$4 + 0.254829592), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$3), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$3), $MachinePrecision] + 0.254829592), $MachinePrecision]}, N[(N[(N[(1.0 + N[Power[N[Power[N[(t$95$0 * N[(N[(t$95$4 / t$95$2), $MachinePrecision] + N[(0.254829592 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(t$95$5 * t$95$5), $MachinePrecision] - N[(1.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(t$95$6 / N[(t$95$3 * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[(1.0 * N[(N[(t$95$6 / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

5. Applied rewrites 79.0%

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

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \color{blue}{\frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}}\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\color{blue}{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   3. div-add N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \color{blue}{\left(\frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{\frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{\color{blue}{\frac{31853699}{125000000} \cdot 1}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\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{31853699}{125000000} \cdot \frac{1}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}}\right)\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   6. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{31853699}{125000000} \cdot \frac{1}{\color{blue}{\frac{-3275911}{10000000} \cdot \left|x\right| + -1}}\right)\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   7. lift-fabs.f64 N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{31853699}{125000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \color{blue}{\left|x\right|} + -1}\right)\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{31853699}{125000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}\right)\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   9. negate-sub N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \left(\frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{31853699}{125000000} \cdot \frac{1}{\color{blue}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}\right)\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

7. Applied rewrites 79.0%

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

Alternative 3: 79.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-x\right) \cdot x}\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_3 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_4 := {\left(t\_0 \cdot \frac{\frac{\frac{\frac{1.453152027 - \frac{-1.061405429}{t\_3}}{t\_3} - -1.421413741}{t\_1} + -0.284496736}{t\_1} + 0.254829592}{t\_3}\right)}^{3}\\ t_5 := \frac{\frac{\frac{1.453152027 - \frac{1.061405429}{t\_2}}{t\_3} - -1.421413741}{t\_2} + -0.284496736}{t\_2} + 0.254829592\\ \frac{\frac{1 + {\left({\left(t\_0 \cdot \frac{\frac{\frac{1.421413741 + \left(\frac{1.061405429}{t\_3 \cdot t\_3} + \frac{1.453152027}{t\_3}\right)}{t\_1} + -0.284496736}{t\_1} + 0.254829592}{t\_3}\right)}^{3}\right)}^{3}}{1 + \left(t\_4 \cdot t\_4 - 1 \cdot t\_4\right)}}{1 + \left({\left(\frac{t\_5}{t\_2 \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{t\_5}{t\_3} \cdot t\_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* (- x) x)))
        (t_1 (fma 0.3275911 (fabs x) 1.0))
        (t_2 (fma (fabs x) 0.3275911 1.0))
        (t_3 (fma -0.3275911 (fabs x) -1.0))
        (t_4
         (pow
          (*
           t_0
           (/
            (+
             (/
              (+
               (/
                (- (/ (- 1.453152027 (/ -1.061405429 t_3)) t_3) -1.421413741)
                t_1)
               -0.284496736)
              t_1)
             0.254829592)
            t_3))
          3.0))
        (t_5
         (+
          (/
           (+
            (/
             (- (/ (- 1.453152027 (/ 1.061405429 t_2)) t_3) -1.421413741)
             t_2)
            -0.284496736)
           t_2)
          0.254829592)))
   (/
    (/
     (+
      1.0
      (pow
       (pow
        (*
         t_0
         (/
          (+
           (/
            (+
             (/
              (+
               1.421413741
               (+ (/ 1.061405429 (* t_3 t_3)) (/ 1.453152027 t_3)))
              t_1)
             -0.284496736)
            t_1)
           0.254829592)
          t_3))
        3.0)
       3.0))
     (+ 1.0 (- (* t_4 t_4) (* 1.0 t_4))))
    (+
     1.0
     (-
      (pow (/ t_5 (* t_2 (exp (* x x)))) 2.0)
      (* 1.0 (* (/ t_5 t_3) t_0)))))))

C

double code(double x) {
	double t_0 = exp((-x * x));
	double t_1 = fma(0.3275911, fabs(x), 1.0);
	double t_2 = fma(fabs(x), 0.3275911, 1.0);
	double t_3 = fma(-0.3275911, fabs(x), -1.0);
	double t_4 = pow((t_0 * ((((((((1.453152027 - (-1.061405429 / t_3)) / t_3) - -1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592) / t_3)), 3.0);
	double t_5 = ((((((1.453152027 - (1.061405429 / t_2)) / t_3) - -1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592;
	return ((1.0 + pow(pow((t_0 * ((((((1.421413741 + ((1.061405429 / (t_3 * t_3)) + (1.453152027 / t_3))) / t_1) + -0.284496736) / t_1) + 0.254829592) / t_3)), 3.0), 3.0)) / (1.0 + ((t_4 * t_4) - (1.0 * t_4)))) / (1.0 + (pow((t_5 / (t_2 * exp((x * x)))), 2.0) - (1.0 * ((t_5 / t_3) * t_0))));
}

Julia

function code(x)
	t_0 = exp(Float64(Float64(-x) * x))
	t_1 = fma(0.3275911, abs(x), 1.0)
	t_2 = fma(abs(x), 0.3275911, 1.0)
	t_3 = fma(-0.3275911, abs(x), -1.0)
	t_4 = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(-1.061405429 / t_3)) / t_3) - -1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592) / t_3)) ^ 3.0
	t_5 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_2)) / t_3) - -1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592)
	return Float64(Float64(Float64(1.0 + ((Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(1.421413741 + Float64(Float64(1.061405429 / Float64(t_3 * t_3)) + Float64(1.453152027 / t_3))) / t_1) + -0.284496736) / t_1) + 0.254829592) / t_3)) ^ 3.0) ^ 3.0)) / Float64(1.0 + Float64(Float64(t_4 * t_4) - Float64(1.0 * t_4)))) / Float64(1.0 + Float64((Float64(t_5 / Float64(t_2 * exp(Float64(x * x)))) ^ 2.0) - Float64(1.0 * Float64(Float64(t_5 / t_3) * t_0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(-1.061405429 / t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$2), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$2), $MachinePrecision] + 0.254829592), $MachinePrecision]}, N[(N[(N[(1.0 + N[Power[N[Power[N[(t$95$0 * N[(N[(N[(N[(N[(N[(1.421413741 + N[(N[(1.061405429 / N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(1.453152027 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(t$95$4 * t$95$4), $MachinePrecision] - N[(1.0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(t$95$5 / N[(t$95$2 * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[(1.0 * N[(N[(t$95$5 / t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

5. Applied rewrites 79.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\color{blue}{\frac{1421413741}{1000000000} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}^{2}} + \frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]
7. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\frac{1421413741}{1000000000} + \color{blue}{\left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}^{2}} + \frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   2. negate-sub N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\frac{1421413741}{1000000000} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}^{2}} + \frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}\right)}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   3. lift-fabs.f64 N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\frac{1421413741}{1000000000} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}^{2}} + \frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| + \left(\mathsf{neg}\left(1\right)\right)}\right)}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\frac{1421413741}{1000000000} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}^{2}} + \frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| + -1}\right)}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   5. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\frac{1421413741}{1000000000} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}^{2}} + \frac{1453152027}{1000000000} \cdot \frac{1}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \color{blue}{\left|x\right|}, -1\right)}\right)}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\frac{1421413741}{1000000000} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}^{2}} + \frac{\frac{1453152027}{1000000000} \cdot 1}{\color{blue}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}}\right)}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\frac{1421413741}{1000000000} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}^{2}} + \frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\color{blue}{\frac{-3275911}{10000000}}, \left|x\right|, -1\right)}\right)}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   8. lift-/.f64 N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\frac{1421413741}{1000000000} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}^{2}} + \frac{\frac{1453152027}{1000000000}}{\color{blue}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}}\right)}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   9. lower-+.f64 N/A

      \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\frac{1421413741}{1000000000} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}^{2}} + \color{blue}{\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}}\right)}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}^{3}}{1 + \left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3} - 1 \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}\right)}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

8. Applied rewrites 79.0%

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

Alternative 4: 79.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := e^{\left(-x\right) \cdot x}\\ t_2 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_3 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_4 := \frac{\frac{\frac{1.453152027 - \frac{1.061405429}{t\_0}}{t\_3} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592\\ t_5 := t\_1 \cdot \frac{\frac{\frac{\frac{1.453152027 - \frac{-1.061405429}{t\_3}}{t\_3} - -1.421413741}{t\_2} + -0.284496736}{t\_2} + 0.254829592}{t\_3}\\ t_6 := {t\_5}^{3}\\ \frac{\frac{1 + {t\_6}^{3}}{1 + \left({t\_5}^{6} - t\_6\right)}}{1 + \left({\left(\frac{t\_4}{t\_0 \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{t\_4}{t\_3} \cdot t\_1\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1 (exp (* (- x) x)))
        (t_2 (fma 0.3275911 (fabs x) 1.0))
        (t_3 (fma -0.3275911 (fabs x) -1.0))
        (t_4
         (+
          (/
           (+
            (/
             (- (/ (- 1.453152027 (/ 1.061405429 t_0)) t_3) -1.421413741)
             t_0)
            -0.284496736)
           t_0)
          0.254829592))
        (t_5
         (*
          t_1
          (/
           (+
            (/
             (+
              (/
               (- (/ (- 1.453152027 (/ -1.061405429 t_3)) t_3) -1.421413741)
               t_2)
              -0.284496736)
             t_2)
            0.254829592)
           t_3)))
        (t_6 (pow t_5 3.0)))
   (/
    (/ (+ 1.0 (pow t_6 3.0)) (+ 1.0 (- (pow t_5 6.0) t_6)))
    (+
     1.0
     (-
      (pow (/ t_4 (* t_0 (exp (* x x)))) 2.0)
      (* 1.0 (* (/ t_4 t_3) t_1)))))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = exp((-x * x));
	double t_2 = fma(0.3275911, fabs(x), 1.0);
	double t_3 = fma(-0.3275911, fabs(x), -1.0);
	double t_4 = ((((((1.453152027 - (1.061405429 / t_0)) / t_3) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592;
	double t_5 = t_1 * ((((((((1.453152027 - (-1.061405429 / t_3)) / t_3) - -1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592) / t_3);
	double t_6 = pow(t_5, 3.0);
	return ((1.0 + pow(t_6, 3.0)) / (1.0 + (pow(t_5, 6.0) - t_6))) / (1.0 + (pow((t_4 / (t_0 * exp((x * x)))), 2.0) - (1.0 * ((t_4 / t_3) * t_1))));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = exp(Float64(Float64(-x) * x))
	t_2 = fma(0.3275911, abs(x), 1.0)
	t_3 = fma(-0.3275911, abs(x), -1.0)
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_0)) / t_3) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592)
	t_5 = Float64(t_1 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(-1.061405429 / t_3)) / t_3) - -1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592) / t_3))
	t_6 = t_5 ^ 3.0
	return Float64(Float64(Float64(1.0 + (t_6 ^ 3.0)) / Float64(1.0 + Float64((t_5 ^ 6.0) - t_6))) / Float64(1.0 + Float64((Float64(t_4 / Float64(t_0 * exp(Float64(x * x)))) ^ 2.0) - Float64(1.0 * Float64(Float64(t_4 / t_3) * t_1)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 * N[(N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(-1.061405429 / t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$2), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$2), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$5, 3.0], $MachinePrecision]}, N[(N[(N[(1.0 + N[Power[t$95$6, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[t$95$5, 6.0], $MachinePrecision] - t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(t$95$4 / N[(t$95$0 * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[(1.0 * N[(N[(t$95$4 / t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

5. Applied rewrites 79.0%

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

7. Applied rewrites 79.0%

   \[\leadsto \frac{\frac{1 + {\left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}^{3}\right)}^{3}}{1 + \color{blue}{\left({\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}^{6} - {\left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\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)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}^{3}\right)}}}{1 + \left({\left(\frac{\frac{\frac{\frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]
8. Add Preprocessing

Alternative 5: 78.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := \frac{\frac{\frac{1.453152027 - \frac{1.061405429}{t\_1}}{t\_0} - -1.421413741}{t\_1} + -0.284496736}{t\_1} + 0.254829592\\ t_3 := e^{\left(-x\right) \cdot x}\\ t_4 := \frac{\frac{\frac{\left(\frac{1.453152027}{t\_0} + 1.421413741\right) - \frac{1.061405429}{t\_0 \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{t\_1} + -0.284496736}{t\_1} + 0.254829592}{t\_0} \cdot t\_3\\ \frac{\frac{1 - {t\_4}^{6}}{1 - {t\_4}^{3}}}{1 + \left({\left(\frac{t\_2}{t\_1 \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{t\_2}{t\_0} \cdot t\_3\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.3275911 (fabs x) -1.0))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2
         (+
          (/
           (+
            (/
             (- (/ (- 1.453152027 (/ 1.061405429 t_1)) t_0) -1.421413741)
             t_1)
            -0.284496736)
           t_1)
          0.254829592))
        (t_3 (exp (* (- x) x)))
        (t_4
         (*
          (/
           (+
            (/
             (+
              (/
               (-
                (+ (/ 1.453152027 t_0) 1.421413741)
                (/ 1.061405429 (* t_0 (fma 0.3275911 (fabs x) 1.0))))
               t_1)
              -0.284496736)
             t_1)
            0.254829592)
           t_0)
          t_3)))
   (/
    (/ (- 1.0 (pow t_4 6.0)) (- 1.0 (pow t_4 3.0)))
    (+
     1.0
     (-
      (pow (/ t_2 (* t_1 (exp (* x x)))) 2.0)
      (* 1.0 (* (/ t_2 t_0) t_3)))))))

C

double code(double x) {
	double t_0 = fma(-0.3275911, fabs(x), -1.0);
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = ((((((1.453152027 - (1.061405429 / t_1)) / t_0) - -1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592;
	double t_3 = exp((-x * x));
	double t_4 = ((((((((1.453152027 / t_0) + 1.421413741) - (1.061405429 / (t_0 * fma(0.3275911, fabs(x), 1.0)))) / t_1) + -0.284496736) / t_1) + 0.254829592) / t_0) * t_3;
	return ((1.0 - pow(t_4, 6.0)) / (1.0 - pow(t_4, 3.0))) / (1.0 + (pow((t_2 / (t_1 * exp((x * x)))), 2.0) - (1.0 * ((t_2 / t_0) * t_3))));
}

Julia

function code(x)
	t_0 = fma(-0.3275911, abs(x), -1.0)
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_1)) / t_0) - -1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592)
	t_3 = exp(Float64(Float64(-x) * x))
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 / t_0) + 1.421413741) - Float64(1.061405429 / Float64(t_0 * fma(0.3275911, abs(x), 1.0)))) / t_1) + -0.284496736) / t_1) + 0.254829592) / t_0) * t_3)
	return Float64(Float64(Float64(1.0 - (t_4 ^ 6.0)) / Float64(1.0 - (t_4 ^ 3.0))) / Float64(1.0 + Float64((Float64(t_2 / Float64(t_1 * exp(Float64(x * x)))) ^ 2.0) - Float64(1.0 * Float64(Float64(t_2 / t_0) * t_3)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(N[(N[(N[(1.453152027 / t$95$0), $MachinePrecision] + 1.421413741), $MachinePrecision] - N[(1.061405429 / N[(t$95$0 * N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[t$95$4, 6.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[t$95$4, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(t$95$2 / N[(t$95$1 * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[(1.0 * N[(N[(t$95$2 / t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\color{blue}{\left(\frac{1421413741}{1000000000} + \frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]
5. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}\right) - \color{blue}{\frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \color{blue}{\frac{1061405429}{1000000000}} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \color{blue}{\frac{1061405429}{1000000000}} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000} \cdot 1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   7. negate-sub N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| + \left(\mathsf{neg}\left(1\right)\right)} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   8. lift-fabs.f64 N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| + \left(\mathsf{neg}\left(1\right)\right)} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| + -1} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   10. lift-fma.f64 N/A

       \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   11. associate-*r/ N/A

       \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000} \cdot 1}{\color{blue}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000}}{\color{blue}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

6. Applied rewrites 78.8%

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

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   2. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3} \cdot {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 - {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

8. Applied rewrites 78.9%

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

10. Applied rewrites 78.9%

    \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\left(\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.421413741\right) - \frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{6}}{1 - {\left(\frac{\frac{\frac{\left(\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + 1.421413741\right) - \frac{1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}}{1 + \left({\left(\frac{\frac{\frac{\frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]
11. Add Preprocessing

Alternative 6: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := e^{\left(-x\right) \cdot x}\\ t_3 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_4 := \frac{\frac{\frac{1.453152027 - \frac{1.061405429}{t\_3}}{t\_0} - -1.421413741}{t\_3} + -0.284496736}{t\_3} + 0.254829592\\ \frac{1 + {\left(\left(\frac{\frac{\frac{\frac{1.453152027 - \frac{-1.061405429}{t\_0}}{t\_0} - -1.421413741}{t\_1} + -0.284496736}{t\_1}}{t\_0} + \frac{0.254829592}{t\_0}\right) \cdot t\_2\right)}^{3}}{1 + \left({\left(\frac{t\_4}{t\_3 \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{t\_4}{t\_0} \cdot t\_2\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.3275911 (fabs x) -1.0))
        (t_1 (fma 0.3275911 (fabs x) 1.0))
        (t_2 (exp (* (- x) x)))
        (t_3 (fma (fabs x) 0.3275911 1.0))
        (t_4
         (+
          (/
           (+
            (/
             (- (/ (- 1.453152027 (/ 1.061405429 t_3)) t_0) -1.421413741)
             t_3)
            -0.284496736)
           t_3)
          0.254829592)))
   (/
    (+
     1.0
     (pow
      (*
       (+
        (/
         (/
          (+
           (/
            (- (/ (- 1.453152027 (/ -1.061405429 t_0)) t_0) -1.421413741)
            t_1)
           -0.284496736)
          t_1)
         t_0)
        (/ 0.254829592 t_0))
       t_2)
      3.0))
    (+
     1.0
     (-
      (pow (/ t_4 (* t_3 (exp (* x x)))) 2.0)
      (* 1.0 (* (/ t_4 t_0) t_2)))))))

C

double code(double x) {
	double t_0 = fma(-0.3275911, fabs(x), -1.0);
	double t_1 = fma(0.3275911, fabs(x), 1.0);
	double t_2 = exp((-x * x));
	double t_3 = fma(fabs(x), 0.3275911, 1.0);
	double t_4 = ((((((1.453152027 - (1.061405429 / t_3)) / t_0) - -1.421413741) / t_3) + -0.284496736) / t_3) + 0.254829592;
	return (1.0 + pow((((((((((1.453152027 - (-1.061405429 / t_0)) / t_0) - -1.421413741) / t_1) + -0.284496736) / t_1) / t_0) + (0.254829592 / t_0)) * t_2), 3.0)) / (1.0 + (pow((t_4 / (t_3 * exp((x * x)))), 2.0) - (1.0 * ((t_4 / t_0) * t_2))));
}

Julia

function code(x)
	t_0 = fma(-0.3275911, abs(x), -1.0)
	t_1 = fma(0.3275911, abs(x), 1.0)
	t_2 = exp(Float64(Float64(-x) * x))
	t_3 = fma(abs(x), 0.3275911, 1.0)
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_3)) / t_0) - -1.421413741) / t_3) + -0.284496736) / t_3) + 0.254829592)
	return Float64(Float64(1.0 + (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(-1.061405429 / t_0)) / t_0) - -1.421413741) / t_1) + -0.284496736) / t_1) / t_0) + Float64(0.254829592 / t_0)) * t_2) ^ 3.0)) / Float64(1.0 + Float64((Float64(t_4 / Float64(t_3 * exp(Float64(x * x)))) ^ 2.0) - Float64(1.0 * Float64(Float64(t_4 / t_0) * t_2)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$3), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$3), $MachinePrecision] + 0.254829592), $MachinePrecision]}, N[(N[(1.0 + N[Power[N[(N[(N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(-1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(0.254829592 / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(t$95$4 / N[(t$95$3 * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[(1.0 * N[(N[(t$95$4 / t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

5. Applied rewrites 78.8%

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

Alternative 7: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := e^{\left(-x\right) \cdot x}\\ t_2 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_3 := \frac{\frac{\frac{1.453152027 - \frac{1.061405429}{t\_0}}{t\_2} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592\\ \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{1.453152027}{t\_2} + 1.421413741\right) - \frac{1.061405429}{t\_2 \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{t\_2} \cdot t\_1\right)}^{3}}{1 + \left({\left(\frac{t\_3}{t\_0 \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{t\_3}{t\_2} \cdot t\_1\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1 (exp (* (- x) x)))
        (t_2 (fma -0.3275911 (fabs x) -1.0))
        (t_3
         (+
          (/
           (+
            (/
             (- (/ (- 1.453152027 (/ 1.061405429 t_0)) t_2) -1.421413741)
             t_0)
            -0.284496736)
           t_0)
          0.254829592)))
   (/
    (+
     1.0
     (pow
      (*
       (/
        (+
         (/
          (+
           (/
            (-
             (+ (/ 1.453152027 t_2) 1.421413741)
             (/ 1.061405429 (* t_2 (fma 0.3275911 (fabs x) 1.0))))
            t_0)
           -0.284496736)
          t_0)
         0.254829592)
        t_2)
       t_1)
      3.0))
    (+
     1.0
     (-
      (pow (/ t_3 (* t_0 (exp (* x x)))) 2.0)
      (* 1.0 (* (/ t_3 t_2) t_1)))))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = exp((-x * x));
	double t_2 = fma(-0.3275911, fabs(x), -1.0);
	double t_3 = ((((((1.453152027 - (1.061405429 / t_0)) / t_2) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592;
	return (1.0 + pow((((((((((1.453152027 / t_2) + 1.421413741) - (1.061405429 / (t_2 * fma(0.3275911, fabs(x), 1.0)))) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_2) * t_1), 3.0)) / (1.0 + (pow((t_3 / (t_0 * exp((x * x)))), 2.0) - (1.0 * ((t_3 / t_2) * t_1))));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = exp(Float64(Float64(-x) * x))
	t_2 = fma(-0.3275911, abs(x), -1.0)
	t_3 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_0)) / t_2) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592)
	return Float64(Float64(1.0 + (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 / t_2) + 1.421413741) - Float64(1.061405429 / Float64(t_2 * fma(0.3275911, abs(x), 1.0)))) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_2) * t_1) ^ 3.0)) / Float64(1.0 + Float64((Float64(t_3 / Float64(t_0 * exp(Float64(x * x)))) ^ 2.0) - Float64(1.0 * Float64(Float64(t_3 / t_2) * t_1)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision]}, N[(N[(1.0 + N[Power[N[(N[(N[(N[(N[(N[(N[(N[(N[(1.453152027 / t$95$2), $MachinePrecision] + 1.421413741), $MachinePrecision] - N[(1.061405429 / N[(t$95$2 * N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(t$95$3 / N[(t$95$0 * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[(1.0 * N[(N[(t$95$3 / t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\color{blue}{\left(\frac{1421413741}{1000000000} + \frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]
5. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{1421413741}{1000000000} + \frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}\right) - \color{blue}{\frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \color{blue}{\frac{1061405429}{1000000000}} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{1453152027}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \color{blue}{\frac{1061405429}{1000000000}} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000} \cdot 1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   7. negate-sub N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| + \left(\mathsf{neg}\left(1\right)\right)} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   8. lift-fabs.f64 N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| + \left(\mathsf{neg}\left(1\right)\right)} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| + -1} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   10. lift-fma.f64 N/A

       \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{1061405429}{1000000000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   11. associate-*r/ N/A

       \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000} \cdot 1}{\color{blue}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{1 + {\left(\frac{\frac{\frac{\left(\frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{1421413741}{1000000000}\right) - \frac{\frac{1061405429}{1000000000}}{\color{blue}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot \left(\frac{\frac{\frac{\frac{\frac{1453152027}{1000000000} - \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

6. Applied rewrites 78.8%

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

Alternative 8: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_2 := \frac{\frac{\frac{1.453152027 - \frac{1.061405429}{t\_0}}{t\_1} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592\\ t_3 := \frac{t\_2}{t\_1} \cdot e^{\left(-x\right) \cdot x}\\ \frac{1 + {t\_3}^{3}}{1 + \left({\left(\frac{t\_2}{t\_0 \cdot e^{x \cdot x}}\right)}^{2} - 1 \cdot t\_3\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1 (fma -0.3275911 (fabs x) -1.0))
        (t_2
         (+
          (/
           (+
            (/
             (- (/ (- 1.453152027 (/ 1.061405429 t_0)) t_1) -1.421413741)
             t_0)
            -0.284496736)
           t_0)
          0.254829592))
        (t_3 (* (/ t_2 t_1) (exp (* (- x) x)))))
   (/
    (+ 1.0 (pow t_3 3.0))
    (+ 1.0 (- (pow (/ t_2 (* t_0 (exp (* x x)))) 2.0) (* 1.0 t_3))))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = fma(-0.3275911, fabs(x), -1.0);
	double t_2 = ((((((1.453152027 - (1.061405429 / t_0)) / t_1) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592;
	double t_3 = (t_2 / t_1) * exp((-x * x));
	return (1.0 + pow(t_3, 3.0)) / (1.0 + (pow((t_2 / (t_0 * exp((x * x)))), 2.0) - (1.0 * t_3)));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = fma(-0.3275911, abs(x), -1.0)
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_0)) / t_1) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592)
	t_3 = Float64(Float64(t_2 / t_1) * exp(Float64(Float64(-x) * x)))
	return Float64(Float64(1.0 + (t_3 ^ 3.0)) / Float64(1.0 + Float64((Float64(t_2 / Float64(t_0 * exp(Float64(x * x)))) ^ 2.0) - Float64(1.0 * t_3))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 / t$95$1), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(t$95$2 / N[(t$95$0 * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[(1.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

Alternative 9: 78.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-fabs.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-exp.f64 N/A

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

   17. pow2 N/A

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

   18. lift-*.f64 N/A

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

   19. lift-fabs.f64 N/A

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

   20. lift-*.f64 78.8

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

5. Applied rewrites 78.8%

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

Alternative 10: 78.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

Alternative 11: 78.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

Alternative 12: 78.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. lower-fma.f64 78.2

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

6. Applied rewrites 78.2%

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

Alternative 13: 77.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lift-fabs.f64 N/A

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

   3. lower-fma.f64 77.2

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

6. Applied rewrites 77.2%

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

Alternative 14: 54.7% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0)))
   (- 1.0 (/ (* (exp (* (- x) x)) (- 0.254829592 (/ 0.284496736 t_0))) t_0))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	return 1.0 - ((exp((-x * x)) * (0.254829592 - (0.284496736 / t_0))) / t_0);
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	return Float64(1.0 - Float64(Float64(exp(Float64(Float64(-x) * x)) * Float64(0.254829592 - Float64(0.284496736 / t_0))) / t_0))
end

Wolfram

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

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

4. Taylor expanded in x around inf

   \[\leadsto 1 - \color{blue}{\frac{e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)} \cdot \left(\frac{31853699}{125000000} - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

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

6. Applied rewrites 54.7%

   \[\leadsto 1 - \color{blue}{\frac{e^{\left(-x\right) \cdot x} \cdot \left(0.254829592 - \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]
7. Add Preprocessing

Alternative 15: 53.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{0.254829592 - \frac{-0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  1.0
  (/
   (- 0.254829592 (/ -0.284496736 (fma -0.3275911 (fabs x) -1.0)))
   (fma 0.3275911 (fabs x) 1.0))))

C

double code(double x) {
	return 1.0 - ((0.254829592 - (-0.284496736 / fma(-0.3275911, fabs(x), -1.0))) / fma(0.3275911, fabs(x), 1.0));
}

Julia

function code(x)
	return Float64(1.0 - Float64(Float64(0.254829592 - Float64(-0.284496736 / fma(-0.3275911, abs(x), -1.0))) / fma(0.3275911, abs(x), 1.0)))
end

Wolfram

code[x_] := N[(1.0 - N[(N[(0.254829592 - N[(-0.284496736 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

4. Taylor expanded in x around inf

   \[\leadsto 1 - \color{blue}{\frac{e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)} \cdot \left(\frac{31853699}{125000000} - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

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

6. Applied rewrites 54.7%

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

7. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   2. associate-*r/ N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{\frac{8890523}{31250000} \cdot 1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   3. metadata-eval N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{\frac{8890523}{31250000}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   4. frac-2neg N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{\mathsf{neg}\left(\frac{8890523}{31250000}\right)}{\mathsf{neg}\left(\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   5. metadata-eval N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{\frac{-8890523}{31250000}}{\mathsf{neg}\left(\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   6. lift-fabs.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{\frac{-8890523}{31250000}}{\mathsf{neg}\left(\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   7. fp-cancel-sign-sub-inv N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{\frac{-8890523}{31250000}}{\mathsf{neg}\left(\left(1 - \left(\mathsf{neg}\left(\frac{3275911}{10000000}\right)\right) \cdot \left|x\right|\right)\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   8. metadata-eval N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{\frac{-8890523}{31250000}}{\mathsf{neg}\left(\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   9. lift-fabs.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{\frac{-8890523}{31250000}}{\mathsf{neg}\left(\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   10. negate-sub2 N/A

       \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{\frac{-8890523}{31250000}}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   11. negate-sub N/A

       \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{\frac{-8890523}{31250000}}{\frac{-3275911}{10000000} \cdot \left|x\right| + \left(\mathsf{neg}\left(1\right)\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   12. lift-fabs.f64 N/A

       \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{\frac{-8890523}{31250000}}{\frac{-3275911}{10000000} \cdot \left|x\right| + \left(\mathsf{neg}\left(1\right)\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   13. metadata-eval N/A

       \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{\frac{-8890523}{31250000}}{\frac{-3275911}{10000000} \cdot \left|x\right| + -1}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   14. lift-fma.f64 N/A

       \[\leadsto 1 - \frac{\frac{31853699}{125000000} - \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \]

   15. lower-/.f64 53.9

       \[\leadsto 1 - \frac{0.254829592 - \frac{-0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]

9. Applied rewrites 53.9%

   \[\leadsto 1 - \frac{0.254829592 - \frac{-0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\color{blue}{0.3275911}, \left|x\right|, 1\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025110 
(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)))))))