Jmat.Real.erf

Percentage Accurate: 79.0% → 99.9%

Time: 13.3s

Alternatives: 14

Speedup: 37.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + -0.284496736\\ t_1 := \frac{0.254829592 - \frac{t\_0}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}\\ \mathbf{if}\;x\_m \leq 0.00038:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{{\left(\frac{-0.254829592}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)}\right)}^{2}}{t\_1} - \frac{{\left({\left(\mathsf{fma}\left(x\_m, 0.3275911, 1\right)\right)}^{-2} \cdot t\_0\right)}^{2}}{t\_1}\right) \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (+
          (/
           (+
            (/
             (+ (/ 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))
          -0.284496736))
        (t_1
         (/
          (- 0.254829592 (/ t_0 (fma x_m 0.3275911 1.0)))
          (fma x_m 0.3275911 1.0))))
   (if (<= x_m 0.00038)
     (fma
      (fma
       (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
       x_m
       1.128386358070218)
      x_m
      1e-9)
     (-
      1.0
      (*
       (-
        (/ (pow (/ -0.254829592 (fma -0.3275911 x_m -1.0)) 2.0) t_1)
        (/ (pow (* (pow (fma x_m 0.3275911 1.0) -2.0) t_0) 2.0) t_1))
       (exp (* (- x_m) x_m)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (((((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)) + -0.284496736;
	double t_1 = (0.254829592 - (t_0 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0);
	double tmp;
	if (x_m <= 0.00038) {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = 1.0 - (((pow((-0.254829592 / fma(-0.3275911, x_m, -1.0)), 2.0) / t_1) - (pow((pow(fma(x_m, 0.3275911, 1.0), -2.0) * t_0), 2.0) / t_1)) * exp((-x_m * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(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)) + -0.284496736)
	t_1 = Float64(Float64(0.254829592 - Float64(t_0 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))
	tmp = 0.0
	if (x_m <= 0.00038)
		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((Float64(-0.254829592 / fma(-0.3275911, x_m, -1.0)) ^ 2.0) / t_1) - Float64((Float64((fma(x_m, 0.3275911, 1.0) ^ -2.0) * t_0) ^ 2.0) / t_1)) * 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[(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] + -0.284496736), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.254829592 - N[(t$95$0 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00038], 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[(N[Power[N[(-0.254829592 / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision] - N[(N[Power[N[(N[Power[N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision], -2.0], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $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 < 3.8000000000000002e-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 69.0%

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

   6. Applied rewrites 69.7%

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

   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

   9. 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 3.8000000000000002e-4 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00038:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x - 0.00011824294398844343, x, 1.128386358070218\right), x, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{{\left(\frac{-0.254829592}{\mathsf{fma}\left(-0.3275911, x, -1\right)}\right)}^{2}}{\frac{0.254829592 - \frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}} - \frac{{\left({\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{-2} \cdot \left(\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -0.284496736\right)\right)}^{2}}{\frac{0.254829592 - \frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}\right) \cdot e^{\left(-x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + -0.284496736\\ \mathbf{if}\;x\_m \leq 0.00065:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{{\left(\frac{-0.254829592}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)}\right)}^{2} - {\left({\left(\mathsf{fma}\left(x\_m, 0.3275911, 1\right)\right)}^{-2} \cdot t\_0\right)}^{2}}{\frac{0.254829592 - \frac{t\_0}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}} \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (+
          (/
           (+
            (/
             (+ (/ 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))
          -0.284496736)))
   (if (<= x_m 0.00065)
     (fma
      (fma
       (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
       x_m
       1.128386358070218)
      x_m
      1e-9)
     (-
      1.0
      (*
       (/
        (-
         (pow (/ -0.254829592 (fma -0.3275911 x_m -1.0)) 2.0)
         (pow (* (pow (fma x_m 0.3275911 1.0) -2.0) t_0) 2.0))
        (/
         (- 0.254829592 (/ t_0 (fma x_m 0.3275911 1.0)))
         (fma x_m 0.3275911 1.0)))
       (exp (* (- x_m) x_m)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (((((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)) + -0.284496736;
	double tmp;
	if (x_m <= 0.00065) {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = 1.0 - (((pow((-0.254829592 / fma(-0.3275911, x_m, -1.0)), 2.0) - pow((pow(fma(x_m, 0.3275911, 1.0), -2.0) * t_0), 2.0)) / ((0.254829592 - (t_0 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) * exp((-x_m * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(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)) + -0.284496736)
	tmp = 0.0
	if (x_m <= 0.00065)
		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((Float64(-0.254829592 / fma(-0.3275911, x_m, -1.0)) ^ 2.0) - (Float64((fma(x_m, 0.3275911, 1.0) ^ -2.0) * t_0) ^ 2.0)) / Float64(Float64(0.254829592 - Float64(t_0 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) * 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[(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] + -0.284496736), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00065], 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[(N[Power[N[(-0.254829592 / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(N[Power[N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision], -2.0], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(0.254829592 - N[(t$95$0 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $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 < 6.4999999999999997e-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 69.0%

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

   6. Applied rewrites 69.7%

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

   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

   9. 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 6.4999999999999997e-4 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. sqr-abs N/A

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

      5. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00065:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x - 0.00011824294398844343, x, 1.128386358070218\right), x, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{{\left(\frac{-0.254829592}{\mathsf{fma}\left(-0.3275911, x, -1\right)}\right)}^{2} - {\left({\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{-2} \cdot \left(\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -0.284496736\right)\right)}^{2}}{\frac{0.254829592 - \frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}} \cdot e^{\left(-x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + -0.284496736\\ \mathbf{if}\;x\_m \leq 0.0007:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left({\left(\frac{-0.254829592}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)}\right)}^{2} - {t\_0}^{2} \cdot {\left(\mathsf{fma}\left(0.3275911, x\_m, 1\right)\right)}^{-4}\right) \cdot e^{\left(-x\_m\right) \cdot x\_m}}{0.254829592 - \frac{t\_0}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}} \cdot \mathsf{fma}\left(0.3275911, x\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (+
          (/
           (+
            (/
             (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
             (fma 0.3275911 x_m 1.0))
            1.421413741)
           (fma 0.3275911 x_m 1.0))
          -0.284496736)))
   (if (<= x_m 0.0007)
     (fma
      (fma
       (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
       x_m
       1.128386358070218)
      x_m
      1e-9)
     (-
      1.0
      (*
       (/
        (*
         (-
          (pow (/ -0.254829592 (fma -0.3275911 x_m -1.0)) 2.0)
          (* (pow t_0 2.0) (pow (fma 0.3275911 x_m 1.0) -4.0)))
         (exp (* (- x_m) x_m)))
        (- 0.254829592 (/ t_0 (fma 0.3275911 x_m 1.0))))
       (fma 0.3275911 x_m 1.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = ((((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) + 1.421413741) / fma(0.3275911, x_m, 1.0)) + -0.284496736;
	double tmp;
	if (x_m <= 0.0007) {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = 1.0 - ((((pow((-0.254829592 / fma(-0.3275911, x_m, -1.0)), 2.0) - (pow(t_0, 2.0) * pow(fma(0.3275911, x_m, 1.0), -4.0))) * exp((-x_m * x_m))) / (0.254829592 - (t_0 / fma(0.3275911, x_m, 1.0)))) * fma(0.3275911, x_m, 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(Float64(Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) + 1.421413741) / fma(0.3275911, x_m, 1.0)) + -0.284496736)
	tmp = 0.0
	if (x_m <= 0.0007)
		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(Float64((Float64(-0.254829592 / fma(-0.3275911, x_m, -1.0)) ^ 2.0) - Float64((t_0 ^ 2.0) * (fma(0.3275911, x_m, 1.0) ^ -4.0))) * exp(Float64(Float64(-x_m) * x_m))) / Float64(0.254829592 - Float64(t_0 / fma(0.3275911, x_m, 1.0)))) * fma(0.3275911, x_m, 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + 1.421413741), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0007], 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[(N[(N[Power[N[(-0.254829592 / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.254829592 - N[(t$95$0 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.99999999999999993e-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 69.0%

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

   6. Applied rewrites 69.7%

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

   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

   9. 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 6.99999999999999993e-4 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|}\\ \mathbf{if}\;\left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\_m\right) \cdot x\_m} \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m))))))
   (if (<=
        (*
         (*
          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)))
        0.0)
     1.0
     (fma (fma -0.00011824294398844343 x_m 1.128386358070218) x_m 1e-9))))

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 (((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))) <= 0.0) {
		tmp = 1.0;
	} else {
		tmp = fma(fma(-0.00011824294398844343, x_m, 1.128386358070218), x_m, 1e-9);
	}
	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(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))) <= 0.0)
		tmp = 1.0;
	else
		tmp = fma(fma(-0.00011824294398844343, x_m, 1.128386358070218), x_m, 1e-9);
	end
	return 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[(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], 0.0], 1.0, N[(N[(-0.00011824294398844343 * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 0.0

   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%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{1} \]

   if 0.0 < (*.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 58.6%

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

   4. Applied rewrites 55.9%

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

   6. Applied rewrites 57.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.00011824294398844343, x, 1.128386358070218\right), x, 10^{-9}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x} \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.00011824294398844343, x, 1.128386358070218\right), x, 10^{-9}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|}\\ \mathbf{if}\;\left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\_m\right) \cdot x\_m} \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.128386358070218, x\_m, 10^{-9}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m))))))
   (if (<=
        (*
         (*
          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)))
        0.0)
     1.0
     (fma 1.128386358070218 x_m 1e-9))))

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 (((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))) <= 0.0) {
		tmp = 1.0;
	} else {
		tmp = fma(1.128386358070218, x_m, 1e-9);
	}
	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(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))) <= 0.0)
		tmp = 1.0;
	else
		tmp = fma(1.128386358070218, x_m, 1e-9);
	end
	return 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[(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], 0.0], 1.0, N[(1.128386358070218 * x$95$m + 1e-9), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 0.0

   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%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{1} \]

   if 0.0 < (*.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 58.6%

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

   4. Applied rewrites 55.9%

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

   6. Applied rewrites 57.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 94.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.128386358070218, x, 10^{-9}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x} \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.128386358070218, x, 10^{-9}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|}\\ \mathbf{if}\;x\_m \leq 0.00036:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + \frac{\mathsf{fma}\left(0.4760396709921597, x\_m, 0.391746598\right)}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right) \cdot \mathsf{fma}\left(-0.3275911, x\_m, -1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m))))))
   (if (<= x_m 0.00036)
     (fma
      (fma
       (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
       x_m
       1.128386358070218)
      x_m
      1e-9)
     (-
      1.0
      (*
       (*
        t_0
        (+
         0.254829592
         (*
          t_0
          (+
           -0.284496736
           (*
            t_0
            (+
             1.421413741
             (/
              (fma 0.4760396709921597 x_m 0.391746598)
              (* (fma x_m 0.3275911 1.0) (fma -0.3275911 x_m -1.0)))))))))
       (exp (* (- x_m) x_m)))))))

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 (x_m <= 0.00036) {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (fma(0.4760396709921597, x_m, 0.391746598) / (fma(x_m, 0.3275911, 1.0) * fma(-0.3275911, x_m, -1.0))))))))) * exp((-x_m * x_m)));
	}
	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 (x_m <= 0.00036)
		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(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(fma(0.4760396709921597, x_m, 0.391746598) / Float64(fma(x_m, 0.3275911, 1.0) * fma(-0.3275911, x_m, -1.0))))))))) * 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[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00036], 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[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(N[(0.4760396709921597 * x$95$m + 0.391746598), $MachinePrecision] / N[(N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision] * N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 3.60000000000000023e-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 69.0%

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

   6. Applied rewrites 69.7%

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

   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

   9. 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 3.60000000000000023e-4 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{\color{blue}{\frac{195873299}{500000000} + \frac{4760396709921597}{10000000000000000} \cdot x}}{\mathsf{fma}\left(x, \frac{3275911}{10000000}, 1\right) \cdot \mathsf{fma}\left(\frac{-3275911}{10000000}, x, -1\right)}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{\color{blue}{\mathsf{fma}\left(0.4760396709921597, x, 0.391746598\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot \mathsf{fma}\left(-0.3275911, x, -1\right)}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00036:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x - 0.00011824294398844343, x, 1.128386358070218\right), x, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{\mathsf{fma}\left(0.4760396709921597, x, 0.391746598\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot \mathsf{fma}\left(-0.3275911, x, -1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0006:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{e^{\left(-x\_m\right) \cdot x\_m} \cdot \left(\frac{\frac{\mathsf{fma}\left(\frac{\frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.10731592879921, x\_m \cdot x\_m, -1\right)}, \mathsf{fma}\left(x\_m, 0.3275911, -1\right), 1.421413741\right)}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + 0.254829592\right)}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\\ \end{array} \end{array} \]

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)
   (-
    1.0
    (/
     (*
      (exp (* (- x_m) x_m))
      (+
       (/
        (+
         (/
          (fma
           (/
            (+ (/ 1.061405429 (fma x_m 0.3275911 1.0)) -1.453152027)
            (fma 0.10731592879921 (* x_m x_m) -1.0))
           (fma x_m 0.3275911 -1.0)
           1.421413741)
          (fma 0.3275911 x_m 1.0))
         -0.284496736)
        (fma 0.3275911 x_m 1.0))
       0.254829592))
     (fma 0.3275911 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 = 1.0 - ((exp((-x_m * x_m)) * ((((fma((((1.061405429 / fma(x_m, 0.3275911, 1.0)) + -1.453152027) / fma(0.10731592879921, (x_m * x_m), -1.0)), fma(x_m, 0.3275911, -1.0), 1.421413741) / fma(0.3275911, x_m, 1.0)) + -0.284496736) / fma(0.3275911, x_m, 1.0)) + 0.254829592)) / fma(0.3275911, 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 = Float64(1.0 - Float64(Float64(exp(Float64(Float64(-x_m) * x_m)) * Float64(Float64(Float64(Float64(fma(Float64(Float64(Float64(1.061405429 / fma(x_m, 0.3275911, 1.0)) + -1.453152027) / fma(0.10731592879921, Float64(x_m * x_m), -1.0)), fma(x_m, 0.3275911, -1.0), 1.421413741) / fma(0.3275911, x_m, 1.0)) + -0.284496736) / fma(0.3275911, x_m, 1.0)) + 0.254829592)) / fma(0.3275911, 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[(1.0 - N[(N[(N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / N[(0.10731592879921 * N[(x$95$m * x$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * 0.3275911 + -1.0), $MachinePrecision] + 1.421413741), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $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 69.0%

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

   6. Applied rewrites 69.7%

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

   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

   9. 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 99.7%

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

   8. Applied rewrites 99.7%

      \[\leadsto 1 - \frac{e^{\left(-x\right) \cdot x} \cdot \left(\frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}, \mathsf{fma}\left(x, 0.3275911, -1\right), 1.421413741\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)} + 0.254829592\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00067:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{\frac{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, x\_m, 1\right) \cdot \mathsf{fma}\left(0.3275911, x\_m, 1\right)} + \frac{-0.254829592}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)}\right) \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00067)
   (fma
    (fma
     (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
     x_m
     1.128386358070218)
    x_m
    1e-9)
   (-
    1.0
    (*
     (+
      (/
       (+
        (/
         (+
          (/
           (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
           (fma 0.3275911 x_m 1.0))
          1.421413741)
         (fma 0.3275911 x_m 1.0))
        -0.284496736)
       (* (fma 0.3275911 x_m 1.0) (fma 0.3275911 x_m 1.0)))
      (/ -0.254829592 (fma -0.3275911 x_m -1.0)))
     (exp (* (- x_m) x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00067) {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = 1.0 - ((((((((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) + 1.421413741) / fma(0.3275911, x_m, 1.0)) + -0.284496736) / (fma(0.3275911, x_m, 1.0) * fma(0.3275911, x_m, 1.0))) + (-0.254829592 / fma(-0.3275911, x_m, -1.0))) * exp((-x_m * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00067)
		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(Float64(Float64(Float64(Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) + 1.421413741) / fma(0.3275911, x_m, 1.0)) + -0.284496736) / Float64(fma(0.3275911, x_m, 1.0) * fma(0.3275911, x_m, 1.0))) + Float64(-0.254829592 / fma(-0.3275911, x_m, -1.0))) * exp(Float64(Float64(-x_m) * x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.00067], 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[(N[(N[(N[(N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + 1.421413741), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.254829592 / N[(-0.3275911 * x$95$m + -1.0), $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 < 6.7000000000000002e-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 69.0%

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

   6. Applied rewrites 69.7%

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

   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

   9. 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 6.7000000000000002e-4 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-neg-in N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-out N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-+.f64 N/A

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

   6. Applied rewrites 99.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00067:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x - 0.00011824294398844343, x, 1.128386358070218\right), x, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{\frac{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot \mathsf{fma}\left(0.3275911, x, 1\right)} + \frac{-0.254829592}{\mathsf{fma}\left(-0.3275911, x, -1\right)}\right) \cdot e^{\left(-x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.9% accurate, 1.2× speedup

Math

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

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)
   (-
    1.0
    (/
     (*
      (exp (* (- x_m) x_m))
      (+
       (/
        (+
         (/
          (+
           (/
            (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
            (fma 0.3275911 x_m 1.0))
           1.421413741)
          (fma 0.3275911 x_m 1.0))
         -0.284496736)
        (fma 0.3275911 x_m 1.0))
       0.254829592))
     (fma 0.3275911 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 = 1.0 - ((exp((-x_m * x_m)) * (((((((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) + 1.421413741) / fma(0.3275911, x_m, 1.0)) + -0.284496736) / fma(0.3275911, x_m, 1.0)) + 0.254829592)) / fma(0.3275911, 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 = Float64(1.0 - Float64(Float64(exp(Float64(Float64(-x_m) * x_m)) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) + 1.421413741) / fma(0.3275911, x_m, 1.0)) + -0.284496736) / fma(0.3275911, x_m, 1.0)) + 0.254829592)) / fma(0.3275911, 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[(1.0 - N[(N[(N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + 1.421413741), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $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 69.0%

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

   6. Applied rewrites 69.7%

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

   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

   9. 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 99.7%

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

Alternative 10: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00067:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\frac{\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)}, e^{\left(-x\_m\right) \cdot x\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00067)
   (fma
    (fma
     (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
     x_m
     1.128386358070218)
    x_m
    1e-9)
   (fma
    (/
     (+
      (/
       (+
        (/
         (+
          (/
           (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
           (fma 0.3275911 x_m 1.0))
          1.421413741)
         (fma 0.3275911 x_m 1.0))
        -0.284496736)
       (fma 0.3275911 x_m 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 <= 0.00067) {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = fma(((((((((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) + 1.421413741) / fma(0.3275911, x_m, 1.0)) + -0.284496736) / fma(0.3275911, x_m, 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 <= 0.00067)
		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(Float64(Float64(Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) + 1.421413741) / fma(0.3275911, x_m, 1.0)) + -0.284496736) / fma(0.3275911, x_m, 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, 0.00067], 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[(N[(N[(N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + 1.421413741), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / N[(0.3275911 * x$95$m + 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 < 6.7000000000000002e-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 69.0%

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

   6. Applied rewrites 69.7%

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

   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

   9. 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 6.7000000000000002e-4 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

Alternative 11: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.58:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|} \cdot \left(0.254829592 + \frac{0.745170407}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right) \cdot \left(\mathsf{fma}\left(x\_m, 0.3275911, 1\right) \cdot \mathsf{fma}\left(-0.3275911, x\_m, -1\right)\right)}\right)\right) \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.58)
   (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
       (/
        0.745170407
        (*
         (fma -0.3275911 x_m -1.0)
         (* (fma x_m 0.3275911 1.0) (fma -0.3275911 x_m -1.0))))))
     (exp (* (- x_m) x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.58) {
		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 + (0.745170407 / (fma(-0.3275911, x_m, -1.0) * (fma(x_m, 0.3275911, 1.0) * fma(-0.3275911, x_m, -1.0)))))) * exp((-x_m * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.58)
		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(0.745170407 / Float64(fma(-0.3275911, x_m, -1.0) * Float64(fma(x_m, 0.3275911, 1.0) * fma(-0.3275911, x_m, -1.0)))))) * exp(Float64(Float64(-x_m) * x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.58], 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[(0.745170407 / N[(N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision] * N[(N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision] * N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $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 < 0.57999999999999996

   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 69.1%

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

   6. Applied rewrites 69.8%

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

   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

   9. Applied rewrites 68.0%

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

   if 0.57999999999999996 < 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 43.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{\color{blue}{\frac{745170407}{1000000000}}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, x, -1\right) \cdot \left(\mathsf{fma}\left(x, \frac{3275911}{10000000}, 1\right) \cdot \mathsf{fma}\left(\frac{-3275911}{10000000}, x, -1\right)\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{\color{blue}{0.745170407}}{\mathsf{fma}\left(-0.3275911, x, -1\right) \cdot \left(\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot \mathsf{fma}\left(-0.3275911, x, -1\right)\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x - 0.00011824294398844343, x, 1.128386358070218\right), x, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{0.745170407}{\mathsf{fma}\left(-0.3275911, x, -1\right) \cdot \left(\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot \mathsf{fma}\left(-0.3275911, x, -1\right)\right)}\right)\right) \cdot e^{\left(-x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.06:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.06)
   (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 <= 1.06) {
		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 <= 1.06)
		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, 1.06], 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 < 1.0600000000000001

   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 69.1%

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

   6. Applied rewrites 69.8%

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

   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

   9. Applied rewrites 68.0%

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

   if 1.0600000000000001 < 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%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 100.0%

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

Alternative 13: 97.7% accurate, 37.3× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.8e-5) 1e-9 1.0))

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.8e-5], 1e-9, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   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 69.0%

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

   6. Applied rewrites 69.7%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 69.4%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 98.5%

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

Alternative 14: 55.3% accurate, 262.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0;
}

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 = 1.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0

Julia

x_m = abs(x)
function code(x_m)
	return 1.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1.0

Derivation

1. Initial program 77.5%

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

4. Applied rewrites 76.1%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 51.8%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025021 
(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)))))))