Jmat.Real.erf

Percentage Accurate: 79.2% → 85.5%

Time: 7.3s

Alternatives: 15

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|}\\ t_1 := \mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)\\ t_2 := 1 - -0.3275911 \cdot \left|x\_m\right|\\ t_3 := {t\_2}^{-3}\\ t_4 := \mathsf{fma}\left(t\_3, 1.421413741, \frac{t\_3}{t\_1} \cdot 1.453152027\right)\\ t_5 := \frac{0.284496736}{t\_1 \cdot t\_1} - -1.061405429 \cdot \frac{{t\_2}^{-4}}{t\_1}\\ t_6 := \frac{1 + {t\_5}^{3}}{1 + \left({t\_5}^{2} - 1 \cdot t\_5\right)} - \frac{-0.254829592}{t\_1}\\ \mathbf{if}\;x\_m \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{t\_6 \cdot t\_6 - t\_4 \cdot t\_4}{t\_6 + t\_4}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + \frac{\frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)}}{t\_1} - -1.421413741}{1 - 0.10731592879921 \cdot \left(x\_m \cdot x\_m\right)} \cdot \left(1 - \left|x\_m\right| \cdot 0.3275911\right)\right)\right)\right) \cdot e^{-\left|x\_m\right| \cdot \left|x\_m\right|}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m)))))
        (t_1 (fma -0.3275911 (fabs x_m) -1.0))
        (t_2 (- 1.0 (* -0.3275911 (fabs x_m))))
        (t_3 (pow t_2 -3.0))
        (t_4 (fma t_3 1.421413741 (* (/ t_3 t_1) 1.453152027)))
        (t_5
         (-
          (/ 0.284496736 (* t_1 t_1))
          (* -1.061405429 (/ (pow t_2 -4.0) t_1))))
        (t_6
         (-
          (/ (+ 1.0 (pow t_5 3.0)) (+ 1.0 (- (pow t_5 2.0) (* 1.0 t_5))))
          (/ -0.254829592 t_1))))
   (if (<= x_m 3.2e-9)
     (/ (- (* t_6 t_6) (* t_4 t_4)) (+ t_6 t_4))
     (-
      1.0
      (*
       (*
        t_0
        (+
         0.254829592
         (*
          t_0
          (+
           -0.284496736
           (*
            (/
             (-
              (/
               (- 1.453152027 (/ 1.061405429 (fma (fabs x_m) 0.3275911 1.0)))
               t_1)
              -1.421413741)
             (- 1.0 (* 0.10731592879921 (* x_m x_m))))
            (- 1.0 (* (fabs x_m) 0.3275911)))))))
       (exp (- (* (fabs x_m) (fabs x_m)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x_m)));
	double t_1 = fma(-0.3275911, fabs(x_m), -1.0);
	double t_2 = 1.0 - (-0.3275911 * fabs(x_m));
	double t_3 = pow(t_2, -3.0);
	double t_4 = fma(t_3, 1.421413741, ((t_3 / t_1) * 1.453152027));
	double t_5 = (0.284496736 / (t_1 * t_1)) - (-1.061405429 * (pow(t_2, -4.0) / t_1));
	double t_6 = ((1.0 + pow(t_5, 3.0)) / (1.0 + (pow(t_5, 2.0) - (1.0 * t_5)))) - (-0.254829592 / t_1);
	double tmp;
	if (x_m <= 3.2e-9) {
		tmp = ((t_6 * t_6) - (t_4 * t_4)) / (t_6 + t_4);
	} else {
		tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (((((1.453152027 - (1.061405429 / fma(fabs(x_m), 0.3275911, 1.0))) / t_1) - -1.421413741) / (1.0 - (0.10731592879921 * (x_m * x_m)))) * (1.0 - (fabs(x_m) * 0.3275911))))))) * exp(-(fabs(x_m) * fabs(x_m))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x_m))))
	t_1 = fma(-0.3275911, abs(x_m), -1.0)
	t_2 = Float64(1.0 - Float64(-0.3275911 * abs(x_m)))
	t_3 = t_2 ^ -3.0
	t_4 = fma(t_3, 1.421413741, Float64(Float64(t_3 / t_1) * 1.453152027))
	t_5 = Float64(Float64(0.284496736 / Float64(t_1 * t_1)) - Float64(-1.061405429 * Float64((t_2 ^ -4.0) / t_1)))
	t_6 = Float64(Float64(Float64(1.0 + (t_5 ^ 3.0)) / Float64(1.0 + Float64((t_5 ^ 2.0) - Float64(1.0 * t_5)))) - Float64(-0.254829592 / t_1))
	tmp = 0.0
	if (x_m <= 3.2e-9)
		tmp = Float64(Float64(Float64(t_6 * t_6) - Float64(t_4 * t_4)) / Float64(t_6 + t_4));
	else
		tmp = Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / fma(abs(x_m), 0.3275911, 1.0))) / t_1) - -1.421413741) / Float64(1.0 - Float64(0.10731592879921 * Float64(x_m * x_m)))) * Float64(1.0 - Float64(abs(x_m) * 0.3275911))))))) * exp(Float64(-Float64(abs(x_m) * abs(x_m))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, -3.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * 1.421413741 + N[(N[(t$95$3 / t$95$1), $MachinePrecision] * 1.453152027), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(0.284496736 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(-1.061405429 * N[(N[Power[t$95$2, -4.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(1.0 + N[Power[t$95$5, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[t$95$5, 2.0], $MachinePrecision] - N[(1.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.254829592 / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 3.2e-9], N[(N[(N[(t$95$6 * t$95$6), $MachinePrecision] - N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision] / N[(t$95$6 + t$95$4), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(1.0 - N[(0.10731592879921 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x$95$m], $MachinePrecision] * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.20000000000000012e-9

   1. Initial program 79.2%

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

   3. Applied rewrites 79.2%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\frac{8890523}{31250000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}\right)\right) - \left(\frac{31853699}{125000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \left(\frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}} + \frac{1453152027}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}\right)\right)} \]

   6. Applied rewrites 77.6%

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

   8. Applied rewrites 81.8%

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

   10. Applied rewrites 83.8%

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

   if 3.20000000000000012e-9 < x

   1. Initial program 79.2%

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

   3. Applied rewrites 79.2%

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

Alternative 2: 83.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)\\ t_1 := \frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|}\\ t_2 := 1 - -0.3275911 \cdot \left|x\_m\right|\\ t_3 := -1.061405429 \cdot \frac{{t\_2}^{-4}}{t\_0}\\ t_4 := \frac{0.284496736}{t\_0 \cdot t\_0} - t\_3\\ t_5 := {t\_2}^{-3}\\ t_6 := 1 \cdot t\_4\\ t_7 := {t\_4}^{2}\\ \mathbf{if}\;x\_m \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{1 + {\left(\frac{0.284496736}{t\_2 \cdot t\_2} - t\_3\right)}^{3}}{1 + \frac{{t\_7}^{3} - {t\_6}^{3}}{\mathsf{fma}\left(t\_7, t\_7, \mathsf{fma}\left(t\_6, t\_6, t\_7 \cdot t\_6\right)\right)}} - \frac{0.254829592}{t\_2}\right) - \mathsf{fma}\left(t\_5, 1.421413741, \frac{t\_5}{t\_0} \cdot 1.453152027\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(t\_1 \cdot \left(0.254829592 + t\_1 \cdot \left(-0.284496736 + \frac{\frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)}}{t\_0} - -1.421413741}{1 - 0.10731592879921 \cdot \left(x\_m \cdot x\_m\right)} \cdot \left(1 - \left|x\_m\right| \cdot 0.3275911\right)\right)\right)\right) \cdot e^{-\left|x\_m\right| \cdot \left|x\_m\right|}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma -0.3275911 (fabs x_m) -1.0))
        (t_1 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m)))))
        (t_2 (- 1.0 (* -0.3275911 (fabs x_m))))
        (t_3 (* -1.061405429 (/ (pow t_2 -4.0) t_0)))
        (t_4 (- (/ 0.284496736 (* t_0 t_0)) t_3))
        (t_5 (pow t_2 -3.0))
        (t_6 (* 1.0 t_4))
        (t_7 (pow t_4 2.0)))
   (if (<= x_m 3e-9)
     (-
      (-
       (/
        (+ 1.0 (pow (- (/ 0.284496736 (* t_2 t_2)) t_3) 3.0))
        (+
         1.0
         (/
          (- (pow t_7 3.0) (pow t_6 3.0))
          (fma t_7 t_7 (fma t_6 t_6 (* t_7 t_6))))))
       (/ 0.254829592 t_2))
      (fma t_5 1.421413741 (* (/ t_5 t_0) 1.453152027)))
     (-
      1.0
      (*
       (*
        t_1
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            (/
             (-
              (/
               (- 1.453152027 (/ 1.061405429 (fma (fabs x_m) 0.3275911 1.0)))
               t_0)
              -1.421413741)
             (- 1.0 (* 0.10731592879921 (* x_m x_m))))
            (- 1.0 (* (fabs x_m) 0.3275911)))))))
       (exp (- (* (fabs x_m) (fabs x_m)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(-0.3275911, fabs(x_m), -1.0);
	double t_1 = 1.0 / (1.0 + (0.3275911 * fabs(x_m)));
	double t_2 = 1.0 - (-0.3275911 * fabs(x_m));
	double t_3 = -1.061405429 * (pow(t_2, -4.0) / t_0);
	double t_4 = (0.284496736 / (t_0 * t_0)) - t_3;
	double t_5 = pow(t_2, -3.0);
	double t_6 = 1.0 * t_4;
	double t_7 = pow(t_4, 2.0);
	double tmp;
	if (x_m <= 3e-9) {
		tmp = (((1.0 + pow(((0.284496736 / (t_2 * t_2)) - t_3), 3.0)) / (1.0 + ((pow(t_7, 3.0) - pow(t_6, 3.0)) / fma(t_7, t_7, fma(t_6, t_6, (t_7 * t_6)))))) - (0.254829592 / t_2)) - fma(t_5, 1.421413741, ((t_5 / t_0) * 1.453152027));
	} else {
		tmp = 1.0 - ((t_1 * (0.254829592 + (t_1 * (-0.284496736 + (((((1.453152027 - (1.061405429 / fma(fabs(x_m), 0.3275911, 1.0))) / t_0) - -1.421413741) / (1.0 - (0.10731592879921 * (x_m * x_m)))) * (1.0 - (fabs(x_m) * 0.3275911))))))) * exp(-(fabs(x_m) * fabs(x_m))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(-0.3275911, abs(x_m), -1.0)
	t_1 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x_m))))
	t_2 = Float64(1.0 - Float64(-0.3275911 * abs(x_m)))
	t_3 = Float64(-1.061405429 * Float64((t_2 ^ -4.0) / t_0))
	t_4 = Float64(Float64(0.284496736 / Float64(t_0 * t_0)) - t_3)
	t_5 = t_2 ^ -3.0
	t_6 = Float64(1.0 * t_4)
	t_7 = t_4 ^ 2.0
	tmp = 0.0
	if (x_m <= 3e-9)
		tmp = Float64(Float64(Float64(Float64(1.0 + (Float64(Float64(0.284496736 / Float64(t_2 * t_2)) - t_3) ^ 3.0)) / Float64(1.0 + Float64(Float64((t_7 ^ 3.0) - (t_6 ^ 3.0)) / fma(t_7, t_7, fma(t_6, t_6, Float64(t_7 * t_6)))))) - Float64(0.254829592 / t_2)) - fma(t_5, 1.421413741, Float64(Float64(t_5 / t_0) * 1.453152027)));
	else
		tmp = Float64(1.0 - Float64(Float64(t_1 * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / fma(abs(x_m), 0.3275911, 1.0))) / t_0) - -1.421413741) / Float64(1.0 - Float64(0.10731592879921 * Float64(x_m * x_m)))) * Float64(1.0 - Float64(abs(x_m) * 0.3275911))))))) * exp(Float64(-Float64(abs(x_m) * abs(x_m))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-1.061405429 * N[(N[Power[t$95$2, -4.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.284496736 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Power[t$95$2, -3.0], $MachinePrecision]}, Block[{t$95$6 = N[(1.0 * t$95$4), $MachinePrecision]}, Block[{t$95$7 = N[Power[t$95$4, 2.0], $MachinePrecision]}, If[LessEqual[x$95$m, 3e-9], N[(N[(N[(N[(1.0 + N[Power[N[(N[(0.284496736 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Power[t$95$7, 3.0], $MachinePrecision] - N[Power[t$95$6, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$7 * t$95$7 + N[(t$95$6 * t$95$6 + N[(t$95$7 * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.254829592 / t$95$2), $MachinePrecision]), $MachinePrecision] - N[(t$95$5 * 1.421413741 + N[(N[(t$95$5 / t$95$0), $MachinePrecision] * 1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(t$95$1 * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(1.0 - N[(0.10731592879921 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x$95$m], $MachinePrecision] * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.99999999999999998e-9

   1. Initial program 79.2%

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

   3. Applied rewrites 79.2%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\frac{8890523}{31250000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}\right)\right) - \left(\frac{31853699}{125000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \left(\frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}} + \frac{1453152027}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}\right)\right)} \]

   6. Applied rewrites 77.6%

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

   8. Applied rewrites 81.8%

      \[\leadsto \left(\frac{1 + {\left(\frac{0.284496736}{\left(1 - -0.3275911 \cdot \left|x\right|\right) \cdot \left(1 - -0.3275911 \cdot \left|x\right|\right)} - -1.061405429 \cdot \frac{{\left(1 - -0.3275911 \cdot \left|x\right|\right)}^{-4}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}^{3}}{1 + \left(\left(\frac{0.284496736}{\left(1 - -0.3275911 \cdot \left|x\right|\right) \cdot \left(1 - -0.3275911 \cdot \left|x\right|\right)} - -1.061405429 \cdot \frac{{\left(1 - -0.3275911 \cdot \left|x\right|\right)}^{-4}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right) \cdot \left(\frac{0.284496736}{\left(1 - -0.3275911 \cdot \left|x\right|\right) \cdot \left(1 - -0.3275911 \cdot \left|x\right|\right)} - -1.061405429 \cdot \frac{{\left(1 - -0.3275911 \cdot \left|x\right|\right)}^{-4}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right) - 1 \cdot \left(\frac{0.284496736}{\left(1 - -0.3275911 \cdot \left|x\right|\right) \cdot \left(1 - -0.3275911 \cdot \left|x\right|\right)} - -1.061405429 \cdot \frac{{\left(1 - -0.3275911 \cdot \left|x\right|\right)}^{-4}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)\right)} - \frac{0.254829592}{1 - -0.3275911 \cdot \left|x\right|}\right) - \mathsf{fma}\left({\color{blue}{\left(1 - -0.3275911 \cdot \left|x\right|\right)}}^{-3}, 1.421413741, \frac{{\left(1 - -0.3275911 \cdot \left|x\right|\right)}^{-3}}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot 1.453152027\right) \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\frac{1 + {\left(\frac{\frac{8890523}{31250000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)} - \frac{-1061405429}{1000000000} \cdot \frac{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{-4}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)}^{3}}{1 + \left(\left(\frac{\frac{8890523}{31250000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)} - \frac{-1061405429}{1000000000} \cdot \frac{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{-4}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right) \cdot \left(\frac{\frac{8890523}{31250000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)} - \frac{-1061405429}{1000000000} \cdot \frac{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{-4}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right) - 1 \cdot \left(\frac{\frac{8890523}{31250000}}{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)} - \frac{-1061405429}{1000000000} \cdot \frac{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{-4}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}\right)\right)} - \frac{\frac{31853699}{125000000}}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|}\right) - \mathsf{fma}\left({\left(1 - \frac{-3275911}{10000000} \cdot \color{blue}{\left|x\right|}\right)}^{-3}, \frac{1421413741}{1000000000}, \frac{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{-3}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot \frac{1453152027}{1000000000}\right) \]

   10. Applied rewrites 44.3%

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

   if 2.99999999999999998e-9 < x

   1. Initial program 79.2%

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

   3. Applied rewrites 79.2%

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

Alternative 3: 83.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 - -0.3275911 \cdot \left|x\_m\right|\\ t_1 := {t\_0}^{-3}\\ t_2 := \mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)\\ t_3 := \frac{0.284496736}{t\_2 \cdot t\_2}\\ t_4 := \frac{1.061405429}{{t\_0}^{4} \cdot t\_2}\\ t_5 := t\_4 + t\_3\\ t_6 := \frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|}\\ \mathbf{if}\;x\_m \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{1 + {t\_5}^{3}}{\left(1 + {t\_5}^{2}\right) - \left(t\_3 + t\_4\right)} - \frac{-0.254829592}{t\_2}\right) - \mathsf{fma}\left(t\_1, 1.421413741, \frac{t\_1}{t\_2} \cdot 1.453152027\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(t\_6 \cdot \left(0.254829592 + t\_6 \cdot \left(-0.284496736 + \frac{\frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)}}{t\_2} - -1.421413741}{1 - 0.10731592879921 \cdot \left(x\_m \cdot x\_m\right)} \cdot \left(1 - \left|x\_m\right| \cdot 0.3275911\right)\right)\right)\right) \cdot e^{-\left|x\_m\right| \cdot \left|x\_m\right|}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* -0.3275911 (fabs x_m))))
        (t_1 (pow t_0 -3.0))
        (t_2 (fma -0.3275911 (fabs x_m) -1.0))
        (t_3 (/ 0.284496736 (* t_2 t_2)))
        (t_4 (/ 1.061405429 (* (pow t_0 4.0) t_2)))
        (t_5 (+ t_4 t_3))
        (t_6 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m))))))
   (if (<= x_m 3e-9)
     (-
      (-
       (/ (+ 1.0 (pow t_5 3.0)) (- (+ 1.0 (pow t_5 2.0)) (+ t_3 t_4)))
       (/ -0.254829592 t_2))
      (fma t_1 1.421413741 (* (/ t_1 t_2) 1.453152027)))
     (-
      1.0
      (*
       (*
        t_6
        (+
         0.254829592
         (*
          t_6
          (+
           -0.284496736
           (*
            (/
             (-
              (/
               (- 1.453152027 (/ 1.061405429 (fma (fabs x_m) 0.3275911 1.0)))
               t_2)
              -1.421413741)
             (- 1.0 (* 0.10731592879921 (* x_m x_m))))
            (- 1.0 (* (fabs x_m) 0.3275911)))))))
       (exp (- (* (fabs x_m) (fabs x_m)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 - (-0.3275911 * fabs(x_m));
	double t_1 = pow(t_0, -3.0);
	double t_2 = fma(-0.3275911, fabs(x_m), -1.0);
	double t_3 = 0.284496736 / (t_2 * t_2);
	double t_4 = 1.061405429 / (pow(t_0, 4.0) * t_2);
	double t_5 = t_4 + t_3;
	double t_6 = 1.0 / (1.0 + (0.3275911 * fabs(x_m)));
	double tmp;
	if (x_m <= 3e-9) {
		tmp = (((1.0 + pow(t_5, 3.0)) / ((1.0 + pow(t_5, 2.0)) - (t_3 + t_4))) - (-0.254829592 / t_2)) - fma(t_1, 1.421413741, ((t_1 / t_2) * 1.453152027));
	} else {
		tmp = 1.0 - ((t_6 * (0.254829592 + (t_6 * (-0.284496736 + (((((1.453152027 - (1.061405429 / fma(fabs(x_m), 0.3275911, 1.0))) / t_2) - -1.421413741) / (1.0 - (0.10731592879921 * (x_m * x_m)))) * (1.0 - (fabs(x_m) * 0.3275911))))))) * exp(-(fabs(x_m) * fabs(x_m))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 - Float64(-0.3275911 * abs(x_m)))
	t_1 = t_0 ^ -3.0
	t_2 = fma(-0.3275911, abs(x_m), -1.0)
	t_3 = Float64(0.284496736 / Float64(t_2 * t_2))
	t_4 = Float64(1.061405429 / Float64((t_0 ^ 4.0) * t_2))
	t_5 = Float64(t_4 + t_3)
	t_6 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x_m))))
	tmp = 0.0
	if (x_m <= 3e-9)
		tmp = Float64(Float64(Float64(Float64(1.0 + (t_5 ^ 3.0)) / Float64(Float64(1.0 + (t_5 ^ 2.0)) - Float64(t_3 + t_4))) - Float64(-0.254829592 / t_2)) - fma(t_1, 1.421413741, Float64(Float64(t_1 / t_2) * 1.453152027)));
	else
		tmp = Float64(1.0 - Float64(Float64(t_6 * Float64(0.254829592 + Float64(t_6 * Float64(-0.284496736 + Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / fma(abs(x_m), 0.3275911, 1.0))) / t_2) - -1.421413741) / Float64(1.0 - Float64(0.10731592879921 * Float64(x_m * x_m)))) * Float64(1.0 - Float64(abs(x_m) * 0.3275911))))))) * exp(Float64(-Float64(abs(x_m) * abs(x_m))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 - N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, -3.0], $MachinePrecision]}, Block[{t$95$2 = N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.284496736 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.061405429 / N[(N[Power[t$95$0, 4.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 + t$95$3), $MachinePrecision]}, Block[{t$95$6 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 3e-9], N[(N[(N[(N[(1.0 + N[Power[t$95$5, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision] - N[(t$95$3 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.254829592 / t$95$2), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * 1.421413741 + N[(N[(t$95$1 / t$95$2), $MachinePrecision] * 1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(t$95$6 * N[(0.254829592 + N[(t$95$6 * N[(-0.284496736 + N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(1.0 - N[(0.10731592879921 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x$95$m], $MachinePrecision] * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.99999999999999998e-9

   1. Initial program 79.2%

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

   3. Applied rewrites 79.2%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\frac{8890523}{31250000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}\right)\right) - \left(\frac{31853699}{125000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \left(\frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}} + \frac{1453152027}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}\right)\right)} \]

   6. Applied rewrites 77.6%

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(\frac{1}{\left(1 + {\left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{4} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)} + \frac{8890523}{31250000} \cdot \frac{1}{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)}^{2}\right) - \left(\frac{8890523}{31250000} \cdot \frac{1}{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{2}} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{4} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}\right)} + \frac{{\left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{4} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)} + \frac{8890523}{31250000} \cdot \frac{1}{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)}^{3}}{\left(1 + {\left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{4} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)} + \frac{8890523}{31250000} \cdot \frac{1}{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)}^{2}\right) - \left(\frac{8890523}{31250000} \cdot \frac{1}{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{2}} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{4} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}\right)}\right) - \frac{31853699}{125000000} \cdot \frac{1}{1 - \frac{-3275911}{10000000} \cdot \left|x\right|}\right) - \mathsf{fma}\left(\color{blue}{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{-3}}, \frac{1421413741}{1000000000}, \frac{{\left(1 - \frac{-3275911}{10000000} \cdot \left|x\right|\right)}^{-3}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} \cdot \frac{1453152027}{1000000000}\right) \]

   11. Applied rewrites 81.8%

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

   if 2.99999999999999998e-9 < x

   1. Initial program 79.2%

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

   3. Applied rewrites 79.2%

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

Alternative 4: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|}\\ t_1 := \mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)\\ t_2 := 1 - -0.3275911 \cdot \left|x\_m\right|\\ t_3 := \frac{0.284496736}{t\_2 \cdot t\_2} - -1.061405429 \cdot \frac{{t\_2}^{-4}}{t\_1}\\ t_4 := {t\_2}^{-3}\\ \mathbf{if}\;x\_m \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{1 - t\_3 \cdot t\_3}{1 - t\_3} - \frac{0.254829592}{t\_2}\right) - \mathsf{fma}\left(t\_4, 1.421413741, \frac{t\_4}{t\_1} \cdot 1.453152027\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + \frac{\frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)}}{t\_1} - -1.421413741}{1 - 0.10731592879921 \cdot \left(x\_m \cdot x\_m\right)} \cdot \left(1 - \left|x\_m\right| \cdot 0.3275911\right)\right)\right)\right) \cdot e^{-\left|x\_m\right| \cdot \left|x\_m\right|}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m)))))
        (t_1 (fma -0.3275911 (fabs x_m) -1.0))
        (t_2 (- 1.0 (* -0.3275911 (fabs x_m))))
        (t_3
         (-
          (/ 0.284496736 (* t_2 t_2))
          (* -1.061405429 (/ (pow t_2 -4.0) t_1))))
        (t_4 (pow t_2 -3.0)))
   (if (<= x_m 2.5e-9)
     (-
      (- (/ (- 1.0 (* t_3 t_3)) (- 1.0 t_3)) (/ 0.254829592 t_2))
      (fma t_4 1.421413741 (* (/ t_4 t_1) 1.453152027)))
     (-
      1.0
      (*
       (*
        t_0
        (+
         0.254829592
         (*
          t_0
          (+
           -0.284496736
           (*
            (/
             (-
              (/
               (- 1.453152027 (/ 1.061405429 (fma (fabs x_m) 0.3275911 1.0)))
               t_1)
              -1.421413741)
             (- 1.0 (* 0.10731592879921 (* x_m x_m))))
            (- 1.0 (* (fabs x_m) 0.3275911)))))))
       (exp (- (* (fabs x_m) (fabs x_m)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x_m)));
	double t_1 = fma(-0.3275911, fabs(x_m), -1.0);
	double t_2 = 1.0 - (-0.3275911 * fabs(x_m));
	double t_3 = (0.284496736 / (t_2 * t_2)) - (-1.061405429 * (pow(t_2, -4.0) / t_1));
	double t_4 = pow(t_2, -3.0);
	double tmp;
	if (x_m <= 2.5e-9) {
		tmp = (((1.0 - (t_3 * t_3)) / (1.0 - t_3)) - (0.254829592 / t_2)) - fma(t_4, 1.421413741, ((t_4 / t_1) * 1.453152027));
	} else {
		tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (((((1.453152027 - (1.061405429 / fma(fabs(x_m), 0.3275911, 1.0))) / t_1) - -1.421413741) / (1.0 - (0.10731592879921 * (x_m * x_m)))) * (1.0 - (fabs(x_m) * 0.3275911))))))) * exp(-(fabs(x_m) * fabs(x_m))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x_m))))
	t_1 = fma(-0.3275911, abs(x_m), -1.0)
	t_2 = Float64(1.0 - Float64(-0.3275911 * abs(x_m)))
	t_3 = Float64(Float64(0.284496736 / Float64(t_2 * t_2)) - Float64(-1.061405429 * Float64((t_2 ^ -4.0) / t_1)))
	t_4 = t_2 ^ -3.0
	tmp = 0.0
	if (x_m <= 2.5e-9)
		tmp = Float64(Float64(Float64(Float64(1.0 - Float64(t_3 * t_3)) / Float64(1.0 - t_3)) - Float64(0.254829592 / t_2)) - fma(t_4, 1.421413741, Float64(Float64(t_4 / t_1) * 1.453152027)));
	else
		tmp = Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / fma(abs(x_m), 0.3275911, 1.0))) / t_1) - -1.421413741) / Float64(1.0 - Float64(0.10731592879921 * Float64(x_m * x_m)))) * Float64(1.0 - Float64(abs(x_m) * 0.3275911))))))) * exp(Float64(-Float64(abs(x_m) * abs(x_m))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.284496736 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(-1.061405429 * N[(N[Power[t$95$2, -4.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$2, -3.0], $MachinePrecision]}, If[LessEqual[x$95$m, 2.5e-9], N[(N[(N[(N[(1.0 - N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$3), $MachinePrecision]), $MachinePrecision] - N[(0.254829592 / t$95$2), $MachinePrecision]), $MachinePrecision] - N[(t$95$4 * 1.421413741 + N[(N[(t$95$4 / t$95$1), $MachinePrecision] * 1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(1.0 - N[(0.10731592879921 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x$95$m], $MachinePrecision] * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5000000000000001e-9

   1. Initial program 79.2%

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

   3. Applied rewrites 79.2%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\frac{8890523}{31250000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}\right)\right) - \left(\frac{31853699}{125000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \left(\frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}} + \frac{1453152027}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}\right)\right)} \]

   6. Applied rewrites 77.6%

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

   8. Applied rewrites 81.8%

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

   if 2.5000000000000001e-9 < x

   1. Initial program 79.2%

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

   3. Applied rewrites 79.2%

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

Alternative 5: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|}\\ 1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + \frac{\frac{1.453152027 - \frac{1.061405429}{\mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)} - -1.421413741}{1 - 0.10731592879921 \cdot \left(x\_m \cdot x\_m\right)} \cdot \left(1 - \left|x\_m\right| \cdot 0.3275911\right)\right)\right)\right) \cdot e^{-\left|x\_m\right| \cdot \left|x\_m\right|} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m))))))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (+
         -0.284496736
         (*
          (/
           (-
            (/
             (- 1.453152027 (/ 1.061405429 (fma (fabs x_m) 0.3275911 1.0)))
             (fma -0.3275911 (fabs x_m) -1.0))
            -1.421413741)
           (- 1.0 (* 0.10731592879921 (* x_m x_m))))
          (- 1.0 (* (fabs x_m) 0.3275911)))))))
     (exp (- (* (fabs x_m) (fabs x_m))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x_m)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (((((1.453152027 - (1.061405429 / fma(fabs(x_m), 0.3275911, 1.0))) / fma(-0.3275911, fabs(x_m), -1.0)) - -1.421413741) / (1.0 - (0.10731592879921 * (x_m * x_m)))) * (1.0 - (fabs(x_m) * 0.3275911))))))) * exp(-(fabs(x_m) * fabs(x_m))));
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x_m))))
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / fma(abs(x_m), 0.3275911, 1.0))) / fma(-0.3275911, abs(x_m), -1.0)) - -1.421413741) / Float64(1.0 - Float64(0.10731592879921 * Float64(x_m * x_m)))) * Float64(1.0 - Float64(abs(x_m) * 0.3275911))))))) * exp(Float64(-Float64(abs(x_m) * abs(x_m))))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(1.0 - N[(0.10731592879921 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x$95$m], $MachinePrecision] * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

Alternative 6: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ 1 - \left(\frac{\frac{\frac{\frac{1.453152027 - \frac{1.061405429}{t\_0}}{\mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)} - -1.421413741}{t\_0} - 0.284496736}{t\_0} - -0.254829592}{1 - 0.10731592879921 \cdot \left(x\_m \cdot x\_m\right)} \cdot \left(1 - \left|x\_m\right| \cdot 0.3275911\right)\right) \cdot e^{-\left|x\_m\right| \cdot \left|x\_m\right|} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (-
    1.0
    (*
     (*
      (/
       (-
        (/
         (-
          (/
           (-
            (/
             (- 1.453152027 (/ 1.061405429 t_0))
             (fma -0.3275911 (fabs x_m) -1.0))
            -1.421413741)
           t_0)
          0.284496736)
         t_0)
        -0.254829592)
       (- 1.0 (* 0.10731592879921 (* x_m x_m))))
      (- 1.0 (* (fabs x_m) 0.3275911)))
     (exp (- (* (fabs x_m) (fabs x_m))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	return 1.0 - ((((((((((1.453152027 - (1.061405429 / t_0)) / fma(-0.3275911, fabs(x_m), -1.0)) - -1.421413741) / t_0) - 0.284496736) / t_0) - -0.254829592) / (1.0 - (0.10731592879921 * (x_m * x_m)))) * (1.0 - (fabs(x_m) * 0.3275911))) * exp(-(fabs(x_m) * fabs(x_m))));
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_0)) / fma(-0.3275911, abs(x_m), -1.0)) - -1.421413741) / t_0) - 0.284496736) / t_0) - -0.254829592) / Float64(1.0 - Float64(0.10731592879921 * Float64(x_m * x_m)))) * Float64(1.0 - Float64(abs(x_m) * 0.3275911))) * exp(Float64(-Float64(abs(x_m) * abs(x_m))))))
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]}, N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] - 0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(1.0 - N[(0.10731592879921 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x$95$m], $MachinePrecision] * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

Alternative 7: 79.2% accurate, 1.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (-
    1.0
    (/
     (-
      (/
       (-
        (/
         (-
          (/
           (/ (- (* 1.453152027 t_0) 1.061405429) t_0)
           (fma -0.3275911 (fabs x_m) -1.0))
          -1.421413741)
         t_0)
        0.284496736)
       t_0)
      -0.254829592)
     (* (exp (* x_m x_m)) t_0)))))

C

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

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 * t_0) - 1.061405429) / t_0) / fma(-0.3275911, abs(x_m), -1.0)) - -1.421413741) / t_0) - 0.284496736) / t_0) - -0.254829592) / Float64(exp(Float64(x_m * x_m)) * t_0)))
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]}, N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.453152027 * t$95$0), $MachinePrecision] - 1.061405429), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] - 0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(N[Exp[N[(x$95$m * x$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. sub-to-fraction N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 79.2

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

5. Applied rewrites 79.2%

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

Alternative 8: 79.2% accurate, 1.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (-
    1.0
    (/
     (-
      (/
       (fma
        (/ 1.0 t_0)
        (-
         (/
          (- 1.453152027 (/ 1.061405429 t_0))
          (fma -0.3275911 (fabs x_m) -1.0))
         -1.421413741)
        -0.284496736)
       t_0)
      -0.254829592)
     (* (exp (* x_m x_m)) t_0)))))

C

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

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(fma(Float64(1.0 / t_0), Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_0)) / fma(-0.3275911, abs(x_m), -1.0)) - -1.421413741), -0.284496736) / t_0) - -0.254829592) / Float64(exp(Float64(x_m * x_m)) * t_0)))
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]}, N[(1.0 - N[(N[(N[(N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - -1.421413741), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(N[Exp[N[(x$95$m * x$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. metadata-eval N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. lift-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-fma.f64 79.2

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

5. Applied rewrites 79.2%

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

Alternative 9: 79.2% accurate, 1.3× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (-
    1.0
    (/
     (-
      (/
       (-
        (/
         (-
          (/
           (- 1.453152027 (/ 1.061405429 t_0))
           (fma -0.3275911 (fabs x_m) -1.0))
          -1.421413741)
         t_0)
        0.284496736)
       t_0)
      -0.254829592)
     (* (exp (* x_m x_m)) t_0)))))

C

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

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_0)) / fma(-0.3275911, abs(x_m), -1.0)) - -1.421413741) / t_0) - 0.284496736) / t_0) - -0.254829592) / Float64(exp(Float64(x_m * x_m)) * t_0)))
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]}, N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] - 0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(N[Exp[N[(x$95$m * x$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

Alternative 10: 78.6% accurate, 1.3× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (-
    1.0
    (/
     (-
      (/
       (-
        (/
         (-
          (/
           (- 1.453152027 (/ 1.061405429 t_0))
           (fma -0.3275911 (fabs x_m) -1.0))
          -1.421413741)
         t_0)
        0.284496736)
       t_0)
      -0.254829592)
     (+
      1.0
      (fma
       (fabs x_m)
       0.3275911
       (* (* x_m x_m) (- 1.0 (* -0.3275911 (fabs x_m))))))))))

C

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

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_0)) / fma(-0.3275911, abs(x_m), -1.0)) - -1.421413741) / t_0) - 0.284496736) / t_0) - -0.254829592) / Float64(1.0 + fma(abs(x_m), 0.3275911, Float64(Float64(x_m * x_m) * Float64(1.0 - Float64(-0.3275911 * abs(x_m))))))))
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]}, N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] - 0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(1.0 - N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

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

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

   10. metadata-eval N/A

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

   11. lift-fabs.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-fabs.f64 N/A

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

   14. lower-*.f64 78.6

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

6. Applied rewrites 78.6%

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

Alternative 11: 78.6% accurate, 1.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (-
    1.0
    (/
     (-
      (/
       (fma
        (/ 1.0 t_0)
        (-
         (/
          (- 1.453152027 (/ 1.061405429 t_0))
          (fma -0.3275911 (fabs x_m) -1.0))
         -1.421413741)
        -0.284496736)
       t_0)
      -0.254829592)
     (* (+ 1.0 (* x_m x_m)) t_0)))))

C

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

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(fma(Float64(1.0 / t_0), Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_0)) / fma(-0.3275911, abs(x_m), -1.0)) - -1.421413741), -0.284496736) / t_0) - -0.254829592) / Float64(Float64(1.0 + Float64(x_m * x_m)) * t_0)))
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]}, N[(1.0 - N[(N[(N[(N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - -1.421413741), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(N[(1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. metadata-eval N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. lift-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-fma.f64 79.2

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 78.6

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

8. Applied rewrites 78.6%

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

Alternative 12: 78.6% accurate, 1.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (-
    1.0
    (/
     (-
      (/
       (-
        (/
         (-
          (/
           (- 1.453152027 (/ 1.061405429 t_0))
           (fma -0.3275911 (fabs x_m) -1.0))
          -1.421413741)
         t_0)
        0.284496736)
       t_0)
      -0.254829592)
     (* (+ 1.0 (* x_m x_m)) t_0)))))

C

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

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_0)) / fma(-0.3275911, abs(x_m), -1.0)) - -1.421413741) / t_0) - 0.284496736) / t_0) - -0.254829592) / Float64(Float64(1.0 + Float64(x_m * x_m)) * t_0)))
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]}, N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] - 0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(N[(1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 78.6

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

6. Applied rewrites 78.6%

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

Alternative 13: 77.6% accurate, 1.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (-
    1.0
    (/
     (-
      (/
       (fma
        (/ 1.0 t_0)
        (-
         (/
          (- 1.453152027 (/ 1.061405429 t_0))
          (fma -0.3275911 (fabs x_m) -1.0))
         -1.421413741)
        -0.284496736)
       t_0)
      -0.254829592)
     (* 1.0 t_0)))))

C

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

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(fma(Float64(1.0 / t_0), Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_0)) / fma(-0.3275911, abs(x_m), -1.0)) - -1.421413741), -0.284496736) / t_0) - -0.254829592) / Float64(1.0 * t_0)))
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]}, N[(1.0 - N[(N[(N[(N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - -1.421413741), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(1.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. metadata-eval N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. lift-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-fma.f64 79.2

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 78.6

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

8. Applied rewrites 78.6%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 77.6%

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

Alternative 14: 55.7% accurate, 2.3× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (-
  1.0
  (*
   (exp (- (* x_m x_m)))
   (/
    (+ 0.254829592 (/ 0.284496736 (fma -0.3275911 (fabs x_m) -1.0)))
    (- 1.0 (* -0.3275911 (fabs x_m)))))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 - N[(N[Exp[(-N[(x$95$m * x$95$m), $MachinePrecision])], $MachinePrecision] * N[(N[(0.254829592 + N[(0.284496736 / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 55.7%

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

Alternative 15: 54.6% accurate, 3.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (-
  1.0
  (*
   1.0
   (/
    (+ 0.254829592 (/ 0.284496736 (fma -0.3275911 (fabs x_m) -1.0)))
    (- 1.0 (* -0.3275911 (fabs x_m)))))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 - N[(1.0 * N[(N[(0.254829592 + N[(0.284496736 / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 55.7%

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

7. Taylor expanded in x around 0

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

   1. lift-neg.f64 N/A

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

   2. sqr-abs-rev N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-fabs.f64 54.6

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

9. Applied rewrites 54.6%

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

Reproduce

herbie shell --seed 2025142 
(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)))))))