Jmat.Real.erf

Percentage Accurate: 79.2% → 99.9%

Time: 15.4s

Alternatives: 12

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: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(e^{x\_m}\right)}^{\left(-x\_m\right)}\\ t_1 := \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right) \cdot x\_m\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m, 0.3275911, -1\right), \frac{1.421413741 + \frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)} + -1.453152027}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(0.10731592879921, x\_m \cdot x\_m, -1\right)}, -0.284496736\right)}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 0.254829592\\ t_3 := 1 - \frac{t\_2}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)} \cdot t\_0\\ \mathbf{if}\;x\_m \leq 0.0006:\\ \;\;\;\;\mathsf{fma}\left(\frac{{t\_1}^{2} - 1.2732557730789702}{t\_1 - 1.128386358070218}, x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_3} - \frac{{\left(\frac{t\_2}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} \cdot t\_0\right)}^{2}}{t\_3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (pow (exp x_m) (- x_m)))
        (t_1 (* (- (* -0.37545125292247583 x_m) 0.00011824294398844343) x_m))
        (t_2
         (+
          (/
           (fma
            (fma x_m 0.3275911 -1.0)
            (/
             (+
              1.421413741
              (/
               (+ (/ -1.061405429 (fma -0.3275911 x_m -1.0)) -1.453152027)
               (fma x_m 0.3275911 1.0)))
             (fma 0.10731592879921 (* x_m x_m) -1.0))
            -0.284496736)
           (fma x_m 0.3275911 1.0))
          0.254829592))
        (t_3 (- 1.0 (* (/ t_2 (fma -0.3275911 x_m -1.0)) t_0))))
   (if (<= x_m 0.0006)
     (fma
      (/ (- (pow t_1 2.0) 1.2732557730789702) (- t_1 1.128386358070218))
      x_m
      1e-9)
     (-
      (/ 1.0 t_3)
      (/ (pow (* (/ t_2 (fma x_m 0.3275911 1.0)) t_0) 2.0) t_3)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = pow(exp(x_m), -x_m);
	double t_1 = ((-0.37545125292247583 * x_m) - 0.00011824294398844343) * x_m;
	double t_2 = (fma(fma(x_m, 0.3275911, -1.0), ((1.421413741 + (((-1.061405429 / fma(-0.3275911, x_m, -1.0)) + -1.453152027) / fma(x_m, 0.3275911, 1.0))) / fma(0.10731592879921, (x_m * x_m), -1.0)), -0.284496736) / fma(x_m, 0.3275911, 1.0)) + 0.254829592;
	double t_3 = 1.0 - ((t_2 / fma(-0.3275911, x_m, -1.0)) * t_0);
	double tmp;
	if (x_m <= 0.0006) {
		tmp = fma(((pow(t_1, 2.0) - 1.2732557730789702) / (t_1 - 1.128386358070218)), x_m, 1e-9);
	} else {
		tmp = (1.0 / t_3) - (pow(((t_2 / fma(x_m, 0.3275911, 1.0)) * t_0), 2.0) / t_3);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = exp(x_m) ^ Float64(-x_m)
	t_1 = Float64(Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343) * x_m)
	t_2 = Float64(Float64(fma(fma(x_m, 0.3275911, -1.0), Float64(Float64(1.421413741 + Float64(Float64(Float64(-1.061405429 / fma(-0.3275911, x_m, -1.0)) + -1.453152027) / fma(x_m, 0.3275911, 1.0))) / fma(0.10731592879921, Float64(x_m * x_m), -1.0)), -0.284496736) / fma(x_m, 0.3275911, 1.0)) + 0.254829592)
	t_3 = Float64(1.0 - Float64(Float64(t_2 / fma(-0.3275911, x_m, -1.0)) * t_0))
	tmp = 0.0
	if (x_m <= 0.0006)
		tmp = fma(Float64(Float64((t_1 ^ 2.0) - 1.2732557730789702) / Float64(t_1 - 1.128386358070218)), x_m, 1e-9);
	else
		tmp = Float64(Float64(1.0 / t_3) - Float64((Float64(Float64(t_2 / fma(x_m, 0.3275911, 1.0)) * t_0) ^ 2.0) / t_3));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x$95$m * 0.3275911 + -1.0), $MachinePrecision] * N[(N[(1.421413741 + N[(N[(N[(-1.061405429 / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.10731592879921 * N[(x$95$m * x$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[(N[(t$95$2 / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0006], N[(N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] - 1.2732557730789702), $MachinePrecision] / N[(t$95$1 - 1.128386358070218), $MachinePrecision]), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], N[(N[(1.0 / t$95$3), $MachinePrecision] - N[(N[Power[N[(N[(t$95$2 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999947e-4

   1. Initial program 73.8%

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

   4. Applied rewrites 72.9%

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

   6. Applied rewrites 73.0%

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

   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 63.2%

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

   11. Applied rewrites 62.5%

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x\right)}^{2} - 1.2732557730789702}{\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x - 1.128386358070218}, x, 10^{-9}\right) \]

   if 5.99999999999999947e-4 < x

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

   8. Applied rewrites 99.9%

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

Alternative 2: 99.1% 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 0.0%

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

   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 59.0%

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 96.7%

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

4. Final simplification 98.4%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right) \cdot x\_m\\ \mathbf{if}\;x\_m \leq 0.0006:\\ \;\;\;\;\mathsf{fma}\left(\frac{{t\_0}^{2} - 1.2732557730789702}{t\_0 - 1.128386358070218}, x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\mathsf{fma}\left(\frac{\left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right) + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.10731592879921, x\_m \cdot x\_m, -1\right)}, \mathsf{fma}\left(0.3275911, x\_m, -1\right), -0.284496736\right)}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (- (* -0.37545125292247583 x_m) 0.00011824294398844343) x_m)))
   (if (<= x_m 0.0006)
     (fma
      (/ (- (pow t_0 2.0) 1.2732557730789702) (- t_0 1.128386358070218))
      x_m
      1e-9)
     (-
      1.0
      (*
       (/
        (+
         (/
          (fma
           (/
            (+
             (+ 1.421413741 (/ -1.453152027 (fma 0.3275911 x_m 1.0)))
             (/
              (/ 1.061405429 (fma 0.3275911 x_m 1.0))
              (fma 0.3275911 x_m 1.0)))
            (fma 0.10731592879921 (* x_m x_m) -1.0))
           (fma 0.3275911 x_m -1.0)
           -0.284496736)
          (fma x_m 0.3275911 1.0))
         0.254829592)
        (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 = ((-0.37545125292247583 * x_m) - 0.00011824294398844343) * x_m;
	double tmp;
	if (x_m <= 0.0006) {
		tmp = fma(((pow(t_0, 2.0) - 1.2732557730789702) / (t_0 - 1.128386358070218)), x_m, 1e-9);
	} else {
		tmp = 1.0 - ((((fma((((1.421413741 + (-1.453152027 / fma(0.3275911, x_m, 1.0))) + ((1.061405429 / fma(0.3275911, x_m, 1.0)) / fma(0.3275911, x_m, 1.0))) / fma(0.10731592879921, (x_m * x_m), -1.0)), fma(0.3275911, x_m, -1.0), -0.284496736) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / 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(-0.37545125292247583 * x_m) - 0.00011824294398844343) * x_m)
	tmp = 0.0
	if (x_m <= 0.0006)
		tmp = fma(Float64(Float64((t_0 ^ 2.0) - 1.2732557730789702) / Float64(t_0 - 1.128386358070218)), x_m, 1e-9);
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(Float64(fma(Float64(Float64(Float64(1.421413741 + Float64(-1.453152027 / fma(0.3275911, x_m, 1.0))) + Float64(Float64(1.061405429 / fma(0.3275911, x_m, 1.0)) / fma(0.3275911, x_m, 1.0))) / fma(0.10731592879921, Float64(x_m * x_m), -1.0)), fma(0.3275911, x_m, -1.0), -0.284496736) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / 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[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0006], N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - 1.2732557730789702), $MachinePrecision] / N[(t$95$0 - 1.128386358070218), $MachinePrecision]), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(1.421413741 + N[(-1.453152027 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.10731592879921 * N[(x$95$m * x$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(0.3275911 * x$95$m + -1.0), $MachinePrecision] + -0.284496736), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $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 < 5.99999999999999947e-4

   1. Initial program 73.8%

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

   4. Applied rewrites 72.9%

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

   6. Applied rewrites 73.0%

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

   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 63.2%

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

   11. Applied rewrites 62.5%

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x\right)}^{2} - 1.2732557730789702}{\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x - 1.128386358070218}, x, 10^{-9}\right) \]

   if 5.99999999999999947e-4 < x

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. sqr-abs-rev N/A

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

      5. lift-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. div-add N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-fma.f64 N/A

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

      18. lower-/.f64 99.8

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

      19. lift-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-fma.f64 99.8

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

   10. Applied rewrites 99.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0006:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x\right)}^{2} - 1.2732557730789702}{\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x - 1.128386358070218}, x, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\mathsf{fma}\left(\frac{\left(1.421413741 + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}, \mathsf{fma}\left(0.3275911, x, -1\right), -0.284496736\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{\left(-x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right) \cdot x\_m\\ \mathbf{if}\;x\_m \leq 0.00055:\\ \;\;\;\;\mathsf{fma}\left(\frac{{t\_0}^{2} - 1.2732557730789702}{t\_0 - 1.128386358070218}, x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\frac{\frac{\mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(0.10731592879921, x\_m \cdot x\_m, -1\right)}, \mathsf{fma}\left(0.3275911, x\_m, -1\right), -1.453152027\right)}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (- (* -0.37545125292247583 x_m) 0.00011824294398844343) x_m)))
   (if (<= x_m 0.00055)
     (fma
      (/ (- (pow t_0 2.0) 1.2732557730789702) (- t_0 1.128386358070218))
      x_m
      1e-9)
     (-
      1.0
      (*
       (/
        (+
         (/
          (+
           (/
            (+
             (/
              (fma
               (/ 1.061405429 (fma 0.10731592879921 (* x_m x_m) -1.0))
               (fma 0.3275911 x_m -1.0)
               -1.453152027)
              (fma x_m 0.3275911 1.0))
             1.421413741)
            (fma x_m 0.3275911 1.0))
           -0.284496736)
          (fma x_m 0.3275911 1.0))
         0.254829592)
        (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 = ((-0.37545125292247583 * x_m) - 0.00011824294398844343) * x_m;
	double tmp;
	if (x_m <= 0.00055) {
		tmp = fma(((pow(t_0, 2.0) - 1.2732557730789702) / (t_0 - 1.128386358070218)), x_m, 1e-9);
	} else {
		tmp = 1.0 - ((((((((fma((1.061405429 / fma(0.10731592879921, (x_m * x_m), -1.0)), fma(0.3275911, x_m, -1.0), -1.453152027) / fma(x_m, 0.3275911, 1.0)) + 1.421413741) / fma(x_m, 0.3275911, 1.0)) + -0.284496736) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / 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(-0.37545125292247583 * x_m) - 0.00011824294398844343) * x_m)
	tmp = 0.0
	if (x_m <= 0.00055)
		tmp = fma(Float64(Float64((t_0 ^ 2.0) - 1.2732557730789702) / Float64(t_0 - 1.128386358070218)), x_m, 1e-9);
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(fma(Float64(1.061405429 / fma(0.10731592879921, Float64(x_m * x_m), -1.0)), fma(0.3275911, x_m, -1.0), -1.453152027) / fma(x_m, 0.3275911, 1.0)) + 1.421413741) / fma(x_m, 0.3275911, 1.0)) + -0.284496736) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / 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[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00055], N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - 1.2732557730789702), $MachinePrecision] / N[(t$95$0 - 1.128386358070218), $MachinePrecision]), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / N[(0.10731592879921 * N[(x$95$m * x$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(0.3275911 * x$95$m + -1.0), $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] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $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 < 5.50000000000000033e-4

   1. Initial program 73.8%

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

   4. Applied rewrites 72.9%

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

   6. Applied rewrites 73.0%

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

   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 63.2%

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

   11. Applied rewrites 62.5%

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x\right)}^{2} - 1.2732557730789702}{\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x - 1.128386358070218}, x, 10^{-9}\right) \]

   if 5.50000000000000033e-4 < x

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00055:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x\right)}^{2} - 1.2732557730789702}{\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x - 1.128386358070218}, x, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\frac{\frac{\mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}, \mathsf{fma}\left(0.3275911, x, -1\right), -1.453152027\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{\left(-x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right) \cdot x\_m\\ \mathbf{if}\;x\_m \leq 0.0006:\\ \;\;\;\;\mathsf{fma}\left(\frac{{t\_0}^{2} - 1.2732557730789702}{t\_0 - 1.128386358070218}, x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\mathsf{fma}\left(\frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.10731592879921, x\_m \cdot x\_m, -1\right)}, \mathsf{fma}\left(0.3275911, x\_m, -1\right), -0.284496736\right)}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (- (* -0.37545125292247583 x_m) 0.00011824294398844343) x_m)))
   (if (<= x_m 0.0006)
     (fma
      (/ (- (pow t_0 2.0) 1.2732557730789702) (- t_0 1.128386358070218))
      x_m
      1e-9)
     (-
      1.0
      (*
       (/
        (+
         (/
          (fma
           (/
            (+
             1.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
              (fma 0.3275911 x_m 1.0)))
            (fma 0.10731592879921 (* x_m x_m) -1.0))
           (fma 0.3275911 x_m -1.0)
           -0.284496736)
          (fma x_m 0.3275911 1.0))
         0.254829592)
        (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 = ((-0.37545125292247583 * x_m) - 0.00011824294398844343) * x_m;
	double tmp;
	if (x_m <= 0.0006) {
		tmp = fma(((pow(t_0, 2.0) - 1.2732557730789702) / (t_0 - 1.128386358070218)), x_m, 1e-9);
	} else {
		tmp = 1.0 - ((((fma(((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.10731592879921, (x_m * x_m), -1.0)), fma(0.3275911, x_m, -1.0), -0.284496736) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / 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(-0.37545125292247583 * x_m) - 0.00011824294398844343) * x_m)
	tmp = 0.0
	if (x_m <= 0.0006)
		tmp = fma(Float64(Float64((t_0 ^ 2.0) - 1.2732557730789702) / Float64(t_0 - 1.128386358070218)), x_m, 1e-9);
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(Float64(fma(Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.10731592879921, Float64(x_m * x_m), -1.0)), fma(0.3275911, x_m, -1.0), -0.284496736) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / 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[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0006], N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - 1.2732557730789702), $MachinePrecision] / N[(t$95$0 - 1.128386358070218), $MachinePrecision]), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], N[(1.0 - N[(N[(N[(N[(N[(N[(N[(1.421413741 + 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]), $MachinePrecision] / N[(0.10731592879921 * N[(x$95$m * x$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(0.3275911 * x$95$m + -1.0), $MachinePrecision] + -0.284496736), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $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 < 5.99999999999999947e-4

   1. Initial program 73.8%

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

   4. Applied rewrites 72.9%

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

   6. Applied rewrites 73.0%

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

   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 63.2%

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

   11. Applied rewrites 62.5%

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x\right)}^{2} - 1.2732557730789702}{\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x - 1.128386358070218}, x, 10^{-9}\right) \]

   if 5.99999999999999947e-4 < x

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. sqr-abs-rev N/A

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

      5. lift-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0006:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x\right)}^{2} - 1.2732557730789702}{\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x - 1.128386358070218}, x, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\mathsf{fma}\left(\frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}, \mathsf{fma}\left(0.3275911, x, -1\right), -0.284496736\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{\left(-x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right) \cdot x\_m\\ \mathbf{if}\;x\_m \leq 0.0007:\\ \;\;\;\;\mathsf{fma}\left(\frac{{t\_0}^{2} - 1.2732557730789702}{t\_0 - 1.128386358070218}, x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (- (* -0.37545125292247583 x_m) 0.00011824294398844343) x_m)))
   (if (<= x_m 0.0007)
     (fma
      (/ (- (pow t_0 2.0) 1.2732557730789702) (- t_0 1.128386358070218))
      x_m
      1e-9)
     (-
      1.0
      (*
       (/
        (+
         (/
          (+
           (/
            (+
             (/
              (+ (/ 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)
          (fma x_m 0.3275911 1.0))
         0.254829592)
        (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 = ((-0.37545125292247583 * x_m) - 0.00011824294398844343) * x_m;
	double tmp;
	if (x_m <= 0.0007) {
		tmp = fma(((pow(t_0, 2.0) - 1.2732557730789702) / (t_0 - 1.128386358070218)), x_m, 1e-9);
	} else {
		tmp = 1.0 - ((((((((((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) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / 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(-0.37545125292247583 * x_m) - 0.00011824294398844343) * x_m)
	tmp = 0.0
	if (x_m <= 0.0007)
		tmp = fma(Float64(Float64((t_0 ^ 2.0) - 1.2732557730789702) / Float64(t_0 - 1.128386358070218)), x_m, 1e-9);
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(Float64(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) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / 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[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0007], N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - 1.2732557730789702), $MachinePrecision] / N[(t$95$0 - 1.128386358070218), $MachinePrecision]), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], N[(1.0 - N[(N[(N[(N[(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] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $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 < 6.99999999999999993e-4

   1. Initial program 73.8%

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

   4. Applied rewrites 72.9%

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

   6. Applied rewrites 73.0%

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

   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 63.2%

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

   11. Applied rewrites 62.5%

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x\right)}^{2} - 1.2732557730789702}{\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x - 1.128386358070218}, x, 10^{-9}\right) \]

   if 6.99999999999999993e-4 < x

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. sqr-abs N/A

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

      5. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0007:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x\right)}^{2} - 1.2732557730789702}{\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x - 1.128386358070218}, x, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{\left(-x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right) \cdot x\_m\\ \mathbf{if}\;x\_m \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(\frac{{t\_0}^{2} - 1.2732557730789702}{t\_0 - 1.128386358070218}, x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (- (* -0.37545125292247583 x_m) 0.00011824294398844343) x_m)))
   (if (<= x_m 1.05)
     (fma
      (/ (- (pow t_0 2.0) 1.2732557730789702) (- t_0 1.128386358070218))
      x_m
      1e-9)
     1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = ((-0.37545125292247583 * x_m) - 0.00011824294398844343) * x_m;
	double tmp;
	if (x_m <= 1.05) {
		tmp = fma(((pow(t_0, 2.0) - 1.2732557730789702) / (t_0 - 1.128386358070218)), x_m, 1e-9);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343) * x_m)
	tmp = 0.0
	if (x_m <= 1.05)
		tmp = fma(Float64(Float64((t_0 ^ 2.0) - 1.2732557730789702) / Float64(t_0 - 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_] := Block[{t$95$0 = N[(N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 1.05], N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - 1.2732557730789702), $MachinePrecision] / N[(t$95$0 - 1.128386358070218), $MachinePrecision]), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 74.0%

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

   4. Applied rewrites 73.1%

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

   6. Applied rewrites 73.2%

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

   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 63.1%

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

   11. Applied rewrites 62.4%

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x\right)}^{2} - 1.2732557730789702}{\left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right) \cdot x - 1.128386358070218}, x, 10^{-9}\right) \]

   if 1.05000000000000004 < 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 0.0%

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

   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 8: 99.7% accurate, 10.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583, x\_m, -0.00011824294398844343\right), x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.05)
   (fma
    (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 <= 1.05) {
		tmp = fma(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 <= 1.05)
		tmp = fma(fma(fma(-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, 1.05], N[(N[(N[(-0.37545125292247583 * x$95$m + -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 < 1.05000000000000004

   1. Initial program 74.0%

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

   4. Applied rewrites 73.1%

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

   6. Applied rewrites 73.2%

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

   8. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3275911, x, -1\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{-1.421413741 + \frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -1.453152027}{\mathsf{fma}\left(-0.3275911, x, -1\right)}}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -0.284496736, \mathsf{fma}\left(-0.0834799063558312, x, -0.254829592\right)\right)}{\mathsf{fma}\left(-0.3275911, x, -1\right)}, 1\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3275911, x, -1\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{-1.421413741 + \frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -1.453152027}{\mathsf{fma}\left(-0.3275911, x, -1\right)}}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -0.284496736, \mathsf{fma}\left(-0.0834799063558312, x, -0.254829592\right)\right)}{\mathsf{fma}\left(-0.3275911, x, -1\right)}, 1\right) - \mathsf{fma}\left(\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3275911, x, -1\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{-1.421413741 + \frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -1.453152027}{\mathsf{fma}\left(-0.3275911, x, -1\right)}}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -0.284496736, \mathsf{fma}\left(-0.0834799063558312, x, -0.254829592\right)\right)}{\mathsf{fma}\left(-0.3275911, x, -1\right)}, 1\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3275911, x, -1\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{-1.421413741 + \frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -1.453152027}{\mathsf{fma}\left(-0.3275911, x, -1\right)}}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -0.284496736, \mathsf{fma}\left(-0.0834799063558312, x, -0.254829592\right)\right)}{\mathsf{fma}\left(-0.3275911, x, -1\right)}, 1\right) \cdot {\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3275911, x, -1\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{-1.421413741 + \frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -1.453152027}{\mathsf{fma}\left(-0.3275911, x, -1\right)}}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -0.284496736, \mathsf{fma}\left(-0.0834799063558312, x, -0.254829592\right)\right)}{\mathsf{fma}\left(-0.3275911, x, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3275911, x, -1\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{-1.421413741 + \frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -1.453152027}{\mathsf{fma}\left(-0.3275911, x, -1\right)}}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -0.284496736, \mathsf{fma}\left(-0.0834799063558312, x, -0.254829592\right)\right)}{\mathsf{fma}\left(-0.3275911, x, -1\right)}, 1\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3275911, x, -1\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{-1.421413741 + \frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -1.453152027}{\mathsf{fma}\left(-0.3275911, x, -1\right)}}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -0.284496736, \mathsf{fma}\left(-0.0834799063558312, x, -0.254829592\right)\right)}{\mathsf{fma}\left(-0.3275911, x, -1\right)}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3275911, x, -1\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{-1.421413741 + \frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -1.453152027}{\mathsf{fma}\left(-0.3275911, x, -1\right)}}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -0.284496736, \mathsf{fma}\left(-0.0834799063558312, x, -0.254829592\right)\right)}{\mathsf{fma}\left(-0.3275911, x, -1\right)}, 1\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3275911, x, -1\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{-1.421413741 + \frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -1.453152027}{\mathsf{fma}\left(-0.3275911, x, -1\right)}}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -0.284496736, \mathsf{fma}\left(-0.0834799063558312, x, -0.254829592\right)\right)}{\mathsf{fma}\left(-0.3275911, x, -1\right)}, 1\right)}} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.1%

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

   if 1.05000000000000004 < 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 0.0%

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

   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 9: 99.3% accurate, 10.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.05)
   (fma (fma (* -0.37545125292247583 x_m) 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 <= 1.05) {
		tmp = fma(fma((-0.37545125292247583 * x_m), 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 <= 1.05)
		tmp = fma(fma(Float64(-0.37545125292247583 * x_m), 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, 1.05], N[(N[(N[(-0.37545125292247583 * x$95$m), $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 < 1.05000000000000004

   1. Initial program 74.0%

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

   4. Applied rewrites 73.1%

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

   6. Applied rewrites 73.2%

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

   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 63.1%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x, x, \frac{564193179035109}{500000000000000}\right), x, \frac{1}{1000000000}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 62.8%

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

   if 1.05000000000000004 < 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 0.0%

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

   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 10: 99.4% accurate, 13.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.88:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.00011824361065510943, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88)
   (fma (fma -0.00011824361065510943 x_m 1.128386358070218) x_m 1e-9)
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = fma(fma(-0.00011824361065510943, x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.88)
		tmp = fma(fma(-0.00011824361065510943, x_m, 1.128386358070218), x_m, 1e-9);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.88], N[(N[(-0.00011824361065510943 * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 74.0%

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.00011824361065510943, x, 1.128386358070218\right), x, 10^{-9}\right)} \]

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   4. Applied rewrites 0.0%

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

   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 11: 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 73.7%

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

   4. Applied rewrites 35.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 65.4%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 99.4%

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

   4. Applied rewrites 2.1%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 95.9%

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

Alternative 12: 53.1% accurate, 262.0× speedup

Math

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

FPCore

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

C

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

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

4. Applied rewrites 27.6%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 52.3%

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

Reproduce

herbie shell --seed 2025019 
(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)))))))