Jmat.Real.erf

Percentage Accurate: 78.9% → 99.8%

Time: 13.7s

Alternatives: 11

Speedup: 20.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.9% 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: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ t_1 := \frac{0.254829592 + \frac{0.284496736 - \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)}}{t\_0}\\ t_2 := {\left(e^{x\_m}\right)}^{x\_m}\\ \mathbf{if}\;x\_m \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.00011824361065510943, x\_m, 1.128386358070218\right) \cdot x\_m + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\frac{t\_1}{t\_2}\right)}^{4}}{\left({\left(\frac{\frac{0.254829592 + \frac{0.284496736 - \frac{\left(1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}\right) + \frac{-1.453152027}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)}}{t\_0}}{t\_2}\right)}^{2} + 1\right) \cdot \mathsf{fma}\left({\left(e^{x\_m}\right)}^{\left(-x\_m\right)}, t\_1, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0))
        (t_1
         (/
          (+
           0.254829592
           (/
            (-
             0.284496736
             (/
              (+
               (/
                (+ (/ 1.061405429 (fma 0.3275911 x_m 1.0)) -1.453152027)
                (fma 0.3275911 x_m 1.0))
               1.421413741)
              (fma 0.3275911 x_m 1.0)))
            (fma -0.3275911 x_m -1.0)))
          t_0))
        (t_2 (pow (exp x_m) x_m)))
   (if (<= x_m 7.6e-6)
     (+ (* (fma -0.00011824361065510943 x_m 1.128386358070218) x_m) 1e-9)
     (/
      (- 1.0 (pow (/ t_1 t_2) 4.0))
      (*
       (+
        (pow
         (/
          (/
           (+
            0.254829592
            (/
             (-
              0.284496736
              (/
               (+
                (+
                 1.421413741
                 (/
                  (/ 1.061405429 (fma x_m 0.3275911 1.0))
                  (fma x_m 0.3275911 1.0)))
                (/ -1.453152027 (fma x_m 0.3275911 1.0)))
               (fma 0.3275911 x_m 1.0)))
             (fma -0.3275911 x_m -1.0)))
           t_0)
          t_2)
         2.0)
        1.0)
       (fma (pow (exp x_m) (- x_m)) t_1 1.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double t_1 = (0.254829592 + ((0.284496736 - (((((1.061405429 / fma(0.3275911, x_m, 1.0)) + -1.453152027) / fma(0.3275911, x_m, 1.0)) + 1.421413741) / fma(0.3275911, x_m, 1.0))) / fma(-0.3275911, x_m, -1.0))) / t_0;
	double t_2 = pow(exp(x_m), x_m);
	double tmp;
	if (x_m <= 7.6e-6) {
		tmp = (fma(-0.00011824361065510943, x_m, 1.128386358070218) * x_m) + 1e-9;
	} else {
		tmp = (1.0 - pow((t_1 / t_2), 4.0)) / ((pow((((0.254829592 + ((0.284496736 - (((1.421413741 + ((1.061405429 / fma(x_m, 0.3275911, 1.0)) / fma(x_m, 0.3275911, 1.0))) + (-1.453152027 / fma(x_m, 0.3275911, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(-0.3275911, x_m, -1.0))) / t_0) / t_2), 2.0) + 1.0) * fma(pow(exp(x_m), -x_m), t_1, 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	t_1 = Float64(Float64(0.254829592 + Float64(Float64(0.284496736 - Float64(Float64(Float64(Float64(Float64(1.061405429 / fma(0.3275911, x_m, 1.0)) + -1.453152027) / fma(0.3275911, x_m, 1.0)) + 1.421413741) / fma(0.3275911, x_m, 1.0))) / fma(-0.3275911, x_m, -1.0))) / t_0)
	t_2 = exp(x_m) ^ x_m
	tmp = 0.0
	if (x_m <= 7.6e-6)
		tmp = Float64(Float64(fma(-0.00011824361065510943, x_m, 1.128386358070218) * x_m) + 1e-9);
	else
		tmp = Float64(Float64(1.0 - (Float64(t_1 / t_2) ^ 4.0)) / Float64(Float64((Float64(Float64(Float64(0.254829592 + Float64(Float64(0.284496736 - Float64(Float64(Float64(1.421413741 + Float64(Float64(1.061405429 / fma(x_m, 0.3275911, 1.0)) / fma(x_m, 0.3275911, 1.0))) + Float64(-1.453152027 / fma(x_m, 0.3275911, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(-0.3275911, x_m, -1.0))) / t_0) / t_2) ^ 2.0) + 1.0) * fma((exp(x_m) ^ Float64(-x_m)), t_1, 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.254829592 + N[(N[(0.284496736 - N[(N[(N[(N[(N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + 1.421413741), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Exp[x$95$m], $MachinePrecision], x$95$m], $MachinePrecision]}, If[LessEqual[x$95$m, 7.6e-6], N[(N[(N[(-0.00011824361065510943 * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m), $MachinePrecision] + 1e-9), $MachinePrecision], N[(N[(1.0 - N[Power[N[(t$95$1 / t$95$2), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[(N[(N[(0.254829592 + N[(N[(0.284496736 - N[(N[(N[(1.421413741 + N[(N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.453152027 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.6000000000000001e-6

   1. Initial program 71.0%

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

   4. Applied rewrites 39.3%

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

   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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right) + \frac{1}{1000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right) \cdot x} + \frac{1}{1000000000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x, x, \frac{1}{1000000000}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x + \frac{564193179035109}{500000000000000}}, x, \frac{1}{1000000000}\right) \]

      5. lower-fma.f64 67.2

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

   7. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.00011824361065510943, x, 1.128386358070218\right), x, 10^{-9}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.2%

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

   if 7.6000000000000001e-6 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.9%

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

   8. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. div-add N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

   10. Applied rewrites 99.9%

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

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ t_1 := \frac{0.254829592 + \frac{0.284496736 - \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)}}{t\_0}\\ t_2 := {\left(e^{x\_m}\right)}^{x\_m}\\ \mathbf{if}\;x\_m \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.00011824361065510943, x\_m, 1.128386358070218\right) \cdot x\_m + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\frac{t\_1}{t\_2}\right)}^{4}}{\left({\left(\frac{\frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)} + 0.254829592}{t\_2 \cdot t\_0}\right)}^{2} + 1\right) \cdot \mathsf{fma}\left({\left(e^{x\_m}\right)}^{\left(-x\_m\right)}, t\_1, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0))
        (t_1
         (/
          (+
           0.254829592
           (/
            (-
             0.284496736
             (/
              (+
               (/
                (+ (/ 1.061405429 (fma 0.3275911 x_m 1.0)) -1.453152027)
                (fma 0.3275911 x_m 1.0))
               1.421413741)
              (fma 0.3275911 x_m 1.0)))
            (fma -0.3275911 x_m -1.0)))
          t_0))
        (t_2 (pow (exp x_m) x_m)))
   (if (<= x_m 7.6e-6)
     (+ (* (fma -0.00011824361065510943 x_m 1.128386358070218) x_m) 1e-9)
     (/
      (- 1.0 (pow (/ t_1 t_2) 4.0))
      (*
       (+
        (pow
         (/
          (+
           (/
            (-
             0.284496736
             (/
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
                (fma x_m 0.3275911 1.0)))
              (fma x_m 0.3275911 1.0)))
            (fma -0.3275911 x_m -1.0))
           0.254829592)
          (* t_2 t_0))
         2.0)
        1.0)
       (fma (pow (exp x_m) (- x_m)) t_1 1.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double t_1 = (0.254829592 + ((0.284496736 - (((((1.061405429 / fma(0.3275911, x_m, 1.0)) + -1.453152027) / fma(0.3275911, x_m, 1.0)) + 1.421413741) / fma(0.3275911, x_m, 1.0))) / fma(-0.3275911, x_m, -1.0))) / t_0;
	double t_2 = pow(exp(x_m), x_m);
	double tmp;
	if (x_m <= 7.6e-6) {
		tmp = (fma(-0.00011824361065510943, x_m, 1.128386358070218) * x_m) + 1e-9;
	} else {
		tmp = (1.0 - pow((t_1 / t_2), 4.0)) / ((pow(((((0.284496736 - ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(-0.3275911, x_m, -1.0)) + 0.254829592) / (t_2 * t_0)), 2.0) + 1.0) * fma(pow(exp(x_m), -x_m), t_1, 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	t_1 = Float64(Float64(0.254829592 + Float64(Float64(0.284496736 - Float64(Float64(Float64(Float64(Float64(1.061405429 / fma(0.3275911, x_m, 1.0)) + -1.453152027) / fma(0.3275911, x_m, 1.0)) + 1.421413741) / fma(0.3275911, x_m, 1.0))) / fma(-0.3275911, x_m, -1.0))) / t_0)
	t_2 = exp(x_m) ^ x_m
	tmp = 0.0
	if (x_m <= 7.6e-6)
		tmp = Float64(Float64(fma(-0.00011824361065510943, x_m, 1.128386358070218) * x_m) + 1e-9);
	else
		tmp = Float64(Float64(1.0 - (Float64(t_1 / t_2) ^ 4.0)) / Float64(Float64((Float64(Float64(Float64(Float64(0.284496736 - Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(-0.3275911, x_m, -1.0)) + 0.254829592) / Float64(t_2 * t_0)) ^ 2.0) + 1.0) * fma((exp(x_m) ^ Float64(-x_m)), t_1, 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.254829592 + N[(N[(0.284496736 - N[(N[(N[(N[(N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + 1.421413741), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Exp[x$95$m], $MachinePrecision], x$95$m], $MachinePrecision]}, If[LessEqual[x$95$m, 7.6e-6], N[(N[(N[(-0.00011824361065510943 * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m), $MachinePrecision] + 1e-9), $MachinePrecision], N[(N[(1.0 - N[Power[N[(t$95$1 / t$95$2), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[(N[(N[(N[(0.284496736 - N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.6000000000000001e-6

   1. Initial program 71.0%

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

   4. Applied rewrites 39.3%

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

   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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right) + \frac{1}{1000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right) \cdot x} + \frac{1}{1000000000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x, x, \frac{1}{1000000000}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x + \frac{564193179035109}{500000000000000}}, x, \frac{1}{1000000000}\right) \]

      5. lower-fma.f64 67.2

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

   7. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.00011824361065510943, x, 1.128386358070218\right), x, 10^{-9}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.2%

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

   if 7.6000000000000001e-6 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.9%

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

   8. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   10. Applied rewrites 99.9%

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

Alternative 3: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ \mathbf{if}\;x\_m \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.00011824361065510943, x\_m, 1.128386358070218\right) \cdot x\_m + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\frac{\frac{\frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)} + 0.254829592}{t\_0}}{{\left(e^{x\_m}\right)}^{x\_m}}\right)}^{2}}{\mathsf{fma}\left({\left(e^{x\_m}\right)}^{\left(-x\_m\right)}, \frac{\frac{0.284496736 - \frac{\frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + \left(\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + 1.421413741\right)}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)} + 0.254829592}{t\_0}, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (if (<= x_m 7.6e-6)
     (+ (* (fma -0.00011824361065510943 x_m 1.128386358070218) x_m) 1e-9)
     (/
      (-
       1.0
       (pow
        (/
         (/
          (+
           (/
            (-
             0.284496736
             (/
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
                (fma x_m 0.3275911 1.0)))
              (fma x_m 0.3275911 1.0)))
            (fma -0.3275911 x_m -1.0))
           0.254829592)
          t_0)
         (pow (exp x_m) x_m))
        2.0))
      (fma
       (pow (exp x_m) (- x_m))
       (/
        (+
         (/
          (-
           0.284496736
           (/
            (+
             (/ -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))
              1.421413741))
            (fma x_m 0.3275911 1.0)))
          (fma -0.3275911 x_m -1.0))
         0.254829592)
        t_0)
       1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double tmp;
	if (x_m <= 7.6e-6) {
		tmp = (fma(-0.00011824361065510943, x_m, 1.128386358070218) * x_m) + 1e-9;
	} else {
		tmp = (1.0 - pow((((((0.284496736 - ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(-0.3275911, x_m, -1.0)) + 0.254829592) / t_0) / pow(exp(x_m), x_m)), 2.0)) / fma(pow(exp(x_m), -x_m), ((((0.284496736 - (((-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)) + 1.421413741)) / fma(x_m, 0.3275911, 1.0))) / fma(-0.3275911, x_m, -1.0)) + 0.254829592) / t_0), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	tmp = 0.0
	if (x_m <= 7.6e-6)
		tmp = Float64(Float64(fma(-0.00011824361065510943, x_m, 1.128386358070218) * x_m) + 1e-9);
	else
		tmp = Float64(Float64(1.0 - (Float64(Float64(Float64(Float64(Float64(0.284496736 - Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(-0.3275911, x_m, -1.0)) + 0.254829592) / t_0) / (exp(x_m) ^ x_m)) ^ 2.0)) / fma((exp(x_m) ^ Float64(-x_m)), Float64(Float64(Float64(Float64(0.284496736 - Float64(Float64(Float64(-1.453152027 / fma(0.3275911, x_m, 1.0)) + Float64(Float64(Float64(1.061405429 / fma(0.3275911, x_m, 1.0)) / fma(0.3275911, x_m, 1.0)) + 1.421413741)) / fma(x_m, 0.3275911, 1.0))) / fma(-0.3275911, x_m, -1.0)) + 0.254829592) / t_0), 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 7.6e-6], N[(N[(N[(-0.00011824361065510943 * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m), $MachinePrecision] + 1e-9), $MachinePrecision], N[(N[(1.0 - N[Power[N[(N[(N[(N[(N[(0.284496736 - N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision] / N[Power[N[Exp[x$95$m], $MachinePrecision], x$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision] * N[(N[(N[(N[(0.284496736 - N[(N[(N[(-1.453152027 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + 1.421413741), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.6000000000000001e-6

   1. Initial program 71.0%

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

   4. Applied rewrites 39.3%

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

   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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right) + \frac{1}{1000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right) \cdot x} + \frac{1}{1000000000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x, x, \frac{1}{1000000000}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x + \frac{564193179035109}{500000000000000}}, x, \frac{1}{1000000000}\right) \]

      5. lower-fma.f64 67.2

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

   7. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.00011824361065510943, x, 1.128386358070218\right), x, 10^{-9}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.2%

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

   if 7.6000000000000001e-6 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. div-add N/A

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

      6. associate-+l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 99.9

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 99.9

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

      17. lift-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

Alternative 4: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)}\\ \mathbf{if}\;x\_m \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.00011824361065510943, x\_m, 1.128386358070218\right) \cdot x\_m + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\frac{t\_0}{{\left(e^{x\_m}\right)}^{x\_m}}\right)}^{2}}{\mathsf{fma}\left({\left(e^{x\_m}\right)}^{\left(-x\_m\right)}, t\_0, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (/
          (+
           (/
            (-
             0.284496736
             (/
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
                (fma x_m 0.3275911 1.0)))
              (fma x_m 0.3275911 1.0)))
            (fma -0.3275911 x_m -1.0))
           0.254829592)
          (fma (fabs x_m) 0.3275911 1.0))))
   (if (<= x_m 7.6e-6)
     (+ (* (fma -0.00011824361065510943 x_m 1.128386358070218) x_m) 1e-9)
     (/
      (- 1.0 (pow (/ t_0 (pow (exp x_m) x_m)) 2.0))
      (fma (pow (exp x_m) (- x_m)) t_0 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (((0.284496736 - ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(-0.3275911, x_m, -1.0)) + 0.254829592) / fma(fabs(x_m), 0.3275911, 1.0);
	double tmp;
	if (x_m <= 7.6e-6) {
		tmp = (fma(-0.00011824361065510943, x_m, 1.128386358070218) * x_m) + 1e-9;
	} else {
		tmp = (1.0 - pow((t_0 / pow(exp(x_m), x_m)), 2.0)) / fma(pow(exp(x_m), -x_m), t_0, 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(Float64(Float64(0.284496736 - Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(-0.3275911, x_m, -1.0)) + 0.254829592) / fma(abs(x_m), 0.3275911, 1.0))
	tmp = 0.0
	if (x_m <= 7.6e-6)
		tmp = Float64(Float64(fma(-0.00011824361065510943, x_m, 1.128386358070218) * x_m) + 1e-9);
	else
		tmp = Float64(Float64(1.0 - (Float64(t_0 / (exp(x_m) ^ x_m)) ^ 2.0)) / fma((exp(x_m) ^ Float64(-x_m)), t_0, 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(0.284496736 - N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 7.6e-6], N[(N[(N[(-0.00011824361065510943 * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m), $MachinePrecision] + 1e-9), $MachinePrecision], N[(N[(1.0 - N[Power[N[(t$95$0 / N[Power[N[Exp[x$95$m], $MachinePrecision], x$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.6000000000000001e-6

   1. Initial program 71.0%

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

   4. Applied rewrites 39.3%

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

   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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right) + \frac{1}{1000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right) \cdot x} + \frac{1}{1000000000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x, x, \frac{1}{1000000000}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x + \frac{564193179035109}{500000000000000}}, x, \frac{1}{1000000000}\right) \]

      5. lower-fma.f64 67.2

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

   7. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.00011824361065510943, x, 1.128386358070218\right), x, 10^{-9}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.2%

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

   if 7.6000000000000001e-6 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.9%

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

Alternative 5: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(1 + 0.3275911 \cdot \left|x\_m\right|\right)}^{-1}\\ \mathbf{if}\;\left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\_m\right) \cdot x\_m} \leq 0.999:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (* 0.3275911 (fabs x_m))) -1.0)))
   (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)))
        0.999)
     1.0
     1e-9)))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = pow((1.0 + (0.3275911 * fabs(x_m))), -1.0);
	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))) <= 0.999) {
		tmp = 1.0;
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (0.3275911d0 * abs(x_m))) ** (-1.0d0)
    if (((t_0 * (0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (t_0 * 1.061405429d0))))))))) * exp((-x_m * x_m))) <= 0.999d0) then
        tmp = 1.0d0
    else
        tmp = 1d-9
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.pow((1.0 + (0.3275911 * Math.abs(x_m))), -1.0);
	double tmp;
	if (((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * Math.exp((-x_m * x_m))) <= 0.999) {
		tmp = 1.0;
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.pow((1.0 + (0.3275911 * math.fabs(x_m))), -1.0)
	tmp = 0
	if ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * math.exp((-x_m * x_m))) <= 0.999:
		tmp = 1.0
	else:
		tmp = 1e-9
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x_m))) ^ -1.0
	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))) <= 0.999)
		tmp = 1.0;
	else
		tmp = 1e-9;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = (1.0 + (0.3275911 * abs(x_m))) ^ -1.0;
	tmp = 0.0;
	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))) <= 0.999)
		tmp = 1.0;
	else
		tmp = 1e-9;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $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], 0.999], 1.0, 1e-9]]

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))))) < 0.998999999999999999

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.3%

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

   if 0.998999999999999999 < (*.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 57.8%

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 95.8%

      \[\leadsto \color{blue}{10^{-9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(1.421413741 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-1.453152027 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x} \leq 0.999:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.00011824361065510943, x\_m, 1.128386358070218\right) \cdot x\_m + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.05e-5)
   (+ (* (fma -0.00011824361065510943 x_m 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 tmp;
	if (x_m <= 1.05e-5) {
		tmp = (fma(-0.00011824361065510943, x_m, 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)
	tmp = 0.0
	if (x_m <= 1.05e-5)
		tmp = Float64(Float64(fma(-0.00011824361065510943, x_m, 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_] := If[LessEqual[x$95$m, 1.05e-5], N[(N[(N[(-0.00011824361065510943 * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m), $MachinePrecision] + 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 < 1.04999999999999994e-5

   1. Initial program 71.0%

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

   4. Applied rewrites 39.3%

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

   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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right) + \frac{1}{1000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right) \cdot x} + \frac{1}{1000000000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x, x, \frac{1}{1000000000}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x + \frac{564193179035109}{500000000000000}}, x, \frac{1}{1000000000}\right) \]

      5. lower-fma.f64 67.2

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

   7. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.00011824361065510943, x, 1.128386358070218\right), x, 10^{-9}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.2%

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

   if 1.04999999999999994e-5 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

      \[\leadsto 1 - \color{blue}{\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(x, 0.3275911, 1\right)}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   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 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.00011824361065510943, x, 1.128386358070218\right) \cdot x + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{\left(-x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 12.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88)
   (+ (* (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(-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 = Float64(Float64(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[(N[(-0.00011824361065510943 * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m), $MachinePrecision] + 1e-9), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.1%

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

   4. Applied rewrites 39.4%

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

   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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right) + \frac{1}{1000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right) \cdot x} + \frac{1}{1000000000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x, x, \frac{1}{1000000000}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x + \frac{564193179035109}{500000000000000}}, x, \frac{1}{1000000000}\right) \]

      5. lower-fma.f64 67.2

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

   7. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.00011824361065510943, x, 1.128386358070218\right), x, 10^{-9}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.2%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. 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: 99.4% 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 71.1%

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

   4. Applied rewrites 39.4%

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

   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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right) + \frac{1}{1000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x\right) \cdot x} + \frac{1}{1000000000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{564193179035109}{500000000000000} + \frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x, x, \frac{1}{1000000000}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-7094616632211949400058292842768868679}{59999999940000000020000000000000000000000} \cdot x + \frac{564193179035109}{500000000000000}}, x, \frac{1}{1000000000}\right) \]

      5. lower-fma.f64 67.2

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

   7. Applied rewrites 67.2%

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

   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 100.0%

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

   5. 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, 17.4× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = 1e-9 - (-1.128386358070218 * x_m);
	} 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 <= 0.88d0) then
        tmp = 1d-9 - ((-1.128386358070218d0) * x_m)
    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 <= 0.88) {
		tmp = 1e-9 - (-1.128386358070218 * x_m);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.88:
		tmp = 1e-9 - (-1.128386358070218 * x_m)
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.88)
		tmp = Float64(1e-9 - Float64(-1.128386358070218 * x_m));
	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 <= 0.88)
		tmp = 1e-9 - (-1.128386358070218 * x_m);
	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, 0.88], N[(1e-9 - N[(-1.128386358070218 * x$95$m), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.1%

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 67.2

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

   7. Applied rewrites 67.2%

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

   9. Applied rewrites 67.2%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. 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: 99.3% accurate, 20.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88) (fma 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(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(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[(1.128386358070218 * x$95$m + 1e-9), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.1%

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 67.2

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

   7. Applied rewrites 67.2%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. 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: 55.1% accurate, 262.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

4. Applied rewrites 77.2%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 52.4%

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

Reproduce

herbie shell --seed 2024352 
(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)))))))