Jmat.Real.erf

Percentage Accurate: 79.0% → 99.9%

Time: 9.9s

Alternatives: 8

Speedup: 20.1×

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: 79.0% 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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ \mathbf{if}\;x\_m \leq 0.00062:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|} \cdot \left(0.254829592 + \left(\frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)} - \frac{-1}{t\_0} \cdot \frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0}\right)\right)\right) \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (if (<= x_m 0.00062)
     (fma
      (fma
       (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
       x_m
       1.128386358070218)
      x_m
      1e-9)
     (-
      1.0
      (*
       (*
        (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m))))
        (+
         0.254829592
         (-
          (/ 0.284496736 (fma -0.3275911 (fabs x_m) -1.0))
          (*
           (/ -1.0 t_0)
           (/
            (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741)
            t_0)))))
       (exp (* (- x_m) x_m)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double tmp;
	if (x_m <= 0.00062) {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = 1.0 - (((1.0 / (1.0 + (0.3275911 * fabs(x_m)))) * (0.254829592 + ((0.284496736 / fma(-0.3275911, fabs(x_m), -1.0)) - ((-1.0 / t_0) * (((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0))))) * exp((-x_m * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	tmp = 0.0
	if (x_m <= 0.00062)
		tmp = fma(fma(Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x_m)))) * Float64(0.254829592 + Float64(Float64(0.284496736 / fma(-0.3275911, abs(x_m), -1.0)) - Float64(Float64(-1.0 / t_0) * Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0))))) * exp(Float64(Float64(-x_m) * x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00062], N[(N[(N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], N[(1.0 - N[(N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.254829592 + N[(N[(0.284496736 / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[(N[(N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.2e-4

   1. Initial program 70.9%

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

   4. Applied rewrites 71.0%

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

   6. Applied rewrites 70.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

   9. Applied rewrites 67.9%

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

       1. lift-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

   11. Applied rewrites 67.9%

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

   if 6.2e-4 < x

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00062:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x - 0.00011824294398844343, x, 1.128386358070218\right), x, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \left(\frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \cdot \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|}\\ \mathbf{if}\;1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\_m\right) \cdot x\_m} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m))))))
   (if (<=
        (-
         1.0
         (*
          (*
           t_0
           (+
            0.254829592
            (*
             t_0
             (+
              -0.284496736
              (*
               t_0
               (+
                1.421413741
                (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
          (exp (* (- x_m) x_m))))
        2e-7)
     1e-9
     1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x_m)));
	double tmp;
	if ((1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp((-x_m * x_m)))) <= 2e-7) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (1.0d0 + (0.3275911d0 * abs(x_m)))
    if ((1.0d0 - ((t_0 * (0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (t_0 * 1.061405429d0))))))))) * exp((-x_m * x_m)))) <= 2d-7) then
        tmp = 1d-9
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * Math.abs(x_m)));
	double tmp;
	if ((1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * Math.exp((-x_m * x_m)))) <= 2e-7) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 / (1.0 + (0.3275911 * math.fabs(x_m)))
	tmp = 0
	if (1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * math.exp((-x_m * x_m)))) <= 2e-7:
		tmp = 1e-9
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x_m))))
	tmp = 0.0
	if (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(-x_m) * x_m)))) <= 2e-7)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 / (1.0 + (0.3275911 * abs(x_m)));
	tmp = 0.0;
	if ((1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp((-x_m * x_m)))) <= 2e-7)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-7], 1e-9, 1.0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))) < 1.9999999999999999e-7

   1. Initial program 57.6%

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

   4. Applied rewrites 57.8%

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

   6. Applied rewrites 56.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000}} \]
   8. Step-by-step derivation

   9. Applied rewrites 95.3%

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

   if 1.9999999999999999e-7 < (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00054:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-0.284496736 + \left(\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, x\_m, 1\right) \cdot \mathsf{fma}\left(0.3275911, x\_m, 1\right)} - \frac{-1.421413741}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}\right)}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)}, e^{\left(-x\_m\right) \cdot x\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00054)
   (fma
    (fma
     (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
     x_m
     1.128386358070218)
    x_m
    1e-9)
   (fma
    (/
     (+
      (/
       (+
        -0.284496736
        (-
         (/
          (- (/ 1.061405429 (fma 0.3275911 x_m 1.0)) 1.453152027)
          (* (fma 0.3275911 x_m 1.0) (fma 0.3275911 x_m 1.0)))
         (/ -1.421413741 (fma x_m 0.3275911 1.0))))
       (fma x_m 0.3275911 1.0))
      0.254829592)
     (fma -0.3275911 x_m -1.0))
    (exp (* (- x_m) x_m))
    1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00054) {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = fma(((((-0.284496736 + ((((1.061405429 / fma(0.3275911, x_m, 1.0)) - 1.453152027) / (fma(0.3275911, x_m, 1.0) * fma(0.3275911, x_m, 1.0))) - (-1.421413741 / fma(x_m, 0.3275911, 1.0)))) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / fma(-0.3275911, x_m, -1.0)), exp((-x_m * x_m)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00054)
		tmp = fma(fma(Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	else
		tmp = fma(Float64(Float64(Float64(Float64(-0.284496736 + Float64(Float64(Float64(Float64(1.061405429 / fma(0.3275911, x_m, 1.0)) - 1.453152027) / Float64(fma(0.3275911, x_m, 1.0) * fma(0.3275911, x_m, 1.0))) - Float64(-1.421413741 / fma(x_m, 0.3275911, 1.0)))) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / fma(-0.3275911, x_m, -1.0)), exp(Float64(Float64(-x_m) * x_m)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.00054], N[(N[(N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], N[(N[(N[(N[(N[(-0.284496736 + N[(N[(N[(N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - 1.453152027), $MachinePrecision] / N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.421413741 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.40000000000000007e-4

   1. Initial program 70.9%

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

   4. Applied rewrites 71.0%

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

   6. Applied rewrites 70.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

   9. Applied rewrites 67.9%

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

       1. lift-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

   11. Applied rewrites 67.9%

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

   if 5.40000000000000007e-4 < x

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

   8. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 100.0

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

   10. Applied rewrites 100.0%

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

Alternative 4: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0006:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-0.284496736 + \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)}, e^{\left(-x\_m\right) \cdot x\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0006)
   (fma
    (fma
     (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
     x_m
     1.128386358070218)
    x_m
    1e-9)
   (fma
    (/
     (+
      (/
       (+
        -0.284496736
        (/
         (-
          (/
           (- (/ 1.061405429 (fma x_m 0.3275911 1.0)) 1.453152027)
           (fma x_m 0.3275911 1.0))
          -1.421413741)
         (fma x_m 0.3275911 1.0)))
       (fma x_m 0.3275911 1.0))
      0.254829592)
     (fma -0.3275911 x_m -1.0))
    (exp (* (- x_m) x_m))
    1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0006) {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = fma(((((-0.284496736 + (((((1.061405429 / fma(x_m, 0.3275911, 1.0)) - 1.453152027) / fma(x_m, 0.3275911, 1.0)) - -1.421413741) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / fma(-0.3275911, x_m, -1.0)), exp((-x_m * x_m)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0006)
		tmp = fma(fma(Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	else
		tmp = fma(Float64(Float64(Float64(Float64(-0.284496736 + Float64(Float64(Float64(Float64(Float64(1.061405429 / fma(x_m, 0.3275911, 1.0)) - 1.453152027) / fma(x_m, 0.3275911, 1.0)) - -1.421413741) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / fma(-0.3275911, x_m, -1.0)), exp(Float64(Float64(-x_m) * x_m)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0006], N[(N[(N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], N[(N[(N[(N[(N[(-0.284496736 + N[(N[(N[(N[(N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] - 1.453152027), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999947e-4

   1. Initial program 70.9%

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

   4. Applied rewrites 71.0%

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

   6. Applied rewrites 70.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

   9. Applied rewrites 67.9%

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

       1. lift-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

   11. Applied rewrites 67.9%

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

   if 5.99999999999999947e-4 < x

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.9%

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

Alternative 5: 99.7% accurate, 9.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (fma
    (fma
     (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
     x_m
     1.128386358070218)
    x_m
    1e-9)
   1.0))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 71.0%

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

   4. Applied rewrites 71.1%

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

   6. Applied rewrites 70.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

   9. Applied rewrites 67.8%

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

       1. lift-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

   11. Applied rewrites 67.8%

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

   if 1.1000000000000001 < x

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

Alternative 6: 99.5% accurate, 13.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88)
   (fma (fma -0.00011824294398844343 x_m 1.128386358070218) x_m 1e-9)
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = fma(fma(-0.00011824294398844343, x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.88)
		tmp = fma(fma(-0.00011824294398844343, x_m, 1.128386358070218), x_m, 1e-9);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.88], N[(N[(-0.00011824294398844343 * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.0%

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

   4. Applied rewrites 71.1%

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

   6. Applied rewrites 70.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000}} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.4%

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

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \frac{-2364858879768868679}{20000000000000000000000} \cdot x\right)} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto x \cdot \left(\frac{564193179035109}{500000000000000} + \frac{-2364858879768868679}{20000000000000000000000} \cdot x\right) + \color{blue}{\frac{1}{1000000000}} \]

       2. *-commutative N/A

          \[\leadsto \left(\frac{564193179035109}{500000000000000} + \frac{-2364858879768868679}{20000000000000000000000} \cdot x\right) \cdot x + \frac{1}{1000000000} \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{564193179035109}{500000000000000} + \frac{-2364858879768868679}{20000000000000000000000} \cdot x, \color{blue}{x}, \frac{1}{1000000000}\right) \]

   12. Applied rewrites 66.7%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

Alternative 7: 99.4% accurate, 20.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88) (fma x_m 1.128386358070218 1e-9) 1.0))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.88], N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.0%

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

   4. Applied rewrites 71.1%

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

   6. Applied rewrites 70.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + \frac{564193179035109}{500000000000000} \cdot x} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{1000000000} + \color{blue}{\frac{564193179035109}{500000000000000} \cdot x} \]

      2. lower-*.f64 66.7

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

   9. Applied rewrites 66.7%

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

       1. lift-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{\frac{564193179035109}{500000000000000} \cdot x} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \frac{564193179035109}{500000000000000} \cdot \color{blue}{x} \]

       3. +-commutative N/A

          \[\leadsto \frac{564193179035109}{500000000000000} \cdot x + \color{blue}{\frac{1}{1000000000}} \]

       4. *-commutative N/A

          \[\leadsto x \cdot \frac{564193179035109}{500000000000000} + \frac{1}{1000000000} \]

       5. lower-fma.f64 66.7

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

   11. Applied rewrites 66.7%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

Alternative 8: 53.4% accurate, 262.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 10^{-9} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1e-9)

C

x_m = fabs(x);
double code(double x_m) {
	return 1e-9;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = 1d-9
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1e-9;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1e-9

Julia

x_m = abs(x)
function code(x_m)
	return 1e-9
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1e-9;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1e-9

Derivation

1. Initial program 79.0%

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

4. Applied rewrites 79.1%

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

6. Applied rewrites 78.6%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{1000000000}} \]
8. Step-by-step derivation

9. Applied rewrites 52.5%

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

Reproduce

herbie shell --seed 2025073 
(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)))))))