Jmat.Real.erf

Percentage Accurate: 79.2% → 86.6%

Time: 10.8s

Alternatives: 11

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: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ t_1 := \frac{1.061405429}{t\_0}\\ t_2 := \mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)\\ t_3 := \mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right)\\ t_4 := t\_2 \cdot 1\\ t_5 := \frac{\frac{\frac{-1.421413741 - \frac{1.453152027 - t\_1}{t\_2}}{t\_2} - 0.284496736}{t\_0} - -0.254829592}{t\_4}\\ t_6 := 1 + {t\_5}^{2}\\ \mathbf{if}\;x\_m \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{t\_6} - \frac{{t\_5}^{4}}{t\_6}}{1 - \frac{\frac{\frac{\frac{1.453152027 - \frac{1.061405429}{t\_3}}{t\_2} - -1.421413741}{t\_3} - 0.284496736}{t\_3} - -0.254829592}{t\_4}}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|} \cdot \left(\frac{0.284496736}{t\_2} + \left(\frac{\frac{t\_1 - 1.453152027}{t\_0} - -1.421413741}{t\_0 \cdot t\_0} + 0.254829592\right)\right)\right) \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0))
        (t_1 (/ 1.061405429 t_0))
        (t_2 (fma -0.3275911 (fabs x_m) -1.0))
        (t_3 (fma 0.3275911 (fabs x_m) 1.0))
        (t_4 (* t_2 1.0))
        (t_5
         (/
          (-
           (/
            (-
             (/ (- -1.421413741 (/ (- 1.453152027 t_1) t_2)) t_2)
             0.284496736)
            t_0)
           -0.254829592)
          t_4))
        (t_6 (+ 1.0 (pow t_5 2.0))))
   (if (<= x_m 1.9e-10)
     (/
      (- (/ 1.0 t_6) (/ (pow t_5 4.0) t_6))
      (-
       1.0
       (/
        (-
         (/
          (-
           (/ (- (/ (- 1.453152027 (/ 1.061405429 t_3)) t_2) -1.421413741) t_3)
           0.284496736)
          t_3)
         -0.254829592)
        t_4)))
     (-
      1.0
      (*
       (*
        (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m))))
        (+
         (/ 0.284496736 t_2)
         (+
          (/ (- (/ (- t_1 1.453152027) t_0) -1.421413741) (* t_0 t_0))
          0.254829592)))
       (exp (* (- x_m) x_m)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double t_1 = 1.061405429 / t_0;
	double t_2 = fma(-0.3275911, fabs(x_m), -1.0);
	double t_3 = fma(0.3275911, fabs(x_m), 1.0);
	double t_4 = t_2 * 1.0;
	double t_5 = (((((-1.421413741 - ((1.453152027 - t_1) / t_2)) / t_2) - 0.284496736) / t_0) - -0.254829592) / t_4;
	double t_6 = 1.0 + pow(t_5, 2.0);
	double tmp;
	if (x_m <= 1.9e-10) {
		tmp = ((1.0 / t_6) - (pow(t_5, 4.0) / t_6)) / (1.0 - ((((((((1.453152027 - (1.061405429 / t_3)) / t_2) - -1.421413741) / t_3) - 0.284496736) / t_3) - -0.254829592) / t_4));
	} else {
		tmp = 1.0 - (((1.0 / (1.0 + (0.3275911 * fabs(x_m)))) * ((0.284496736 / t_2) + (((((t_1 - 1.453152027) / t_0) - -1.421413741) / (t_0 * t_0)) + 0.254829592))) * exp((-x_m * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	t_1 = Float64(1.061405429 / t_0)
	t_2 = fma(-0.3275911, abs(x_m), -1.0)
	t_3 = fma(0.3275911, abs(x_m), 1.0)
	t_4 = Float64(t_2 * 1.0)
	t_5 = Float64(Float64(Float64(Float64(Float64(Float64(-1.421413741 - Float64(Float64(1.453152027 - t_1) / t_2)) / t_2) - 0.284496736) / t_0) - -0.254829592) / t_4)
	t_6 = Float64(1.0 + (t_5 ^ 2.0))
	tmp = 0.0
	if (x_m <= 1.9e-10)
		tmp = Float64(Float64(Float64(1.0 / t_6) - Float64((t_5 ^ 4.0) / t_6)) / Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_3)) / t_2) - -1.421413741) / t_3) - 0.284496736) / t_3) - -0.254829592) / t_4)));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x_m)))) * Float64(Float64(0.284496736 / t_2) + Float64(Float64(Float64(Float64(Float64(t_1 - 1.453152027) / t_0) - -1.421413741) / Float64(t_0 * t_0)) + 0.254829592))) * exp(Float64(Float64(-x_m) * x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.061405429 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * 1.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(N[(N[(-1.421413741 - N[(N[(1.453152027 - t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - 0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.254829592), $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(1.0 + N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.9e-10], N[(N[(N[(1.0 / t$95$6), $MachinePrecision] - N[(N[Power[t$95$5, 4.0], $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$3), $MachinePrecision] - 0.284496736), $MachinePrecision] / t$95$3), $MachinePrecision] - -0.254829592), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.284496736 / t$95$2), $MachinePrecision] + N[(N[(N[(N[(N[(t$95$1 - 1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999e-10

   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%

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 85.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 85.6%

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

   14. Taylor expanded in x around 0

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

   16. Applied rewrites 85.6%

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

   17. Taylor expanded in x around 0

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

   19. Applied rewrites 85.0%

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

   if 1.8999999999999999e-10 < 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 78.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. sqr-abs-rev N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 78.0

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

   5. Applied rewrites 78.0%

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

Alternative 2: 86.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double t_1 = fma(0.3275911, fabs(x_m), 1.0);
	double t_2 = fma(-0.3275911, fabs(x_m), -1.0);
	double t_3 = t_2 * exp((x_m * x_m));
	double t_4 = (((((-1.421413741 - ((1.453152027 - (1.061405429 / t_0)) / t_2)) / t_2) - 0.284496736) / t_0) - -0.254829592) / t_3;
	double t_5 = 1.0 + pow(t_4, 2.0);
	return ((1.0 / t_5) - (pow(t_4, 4.0) / t_5)) / (1.0 - ((((((((1.453152027 - (1.061405429 / t_1)) / t_2) - -1.421413741) / t_1) - 0.284496736) / t_1) - -0.254829592) / t_3));
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	t_1 = fma(0.3275911, abs(x_m), 1.0)
	t_2 = fma(-0.3275911, abs(x_m), -1.0)
	t_3 = Float64(t_2 * exp(Float64(x_m * x_m)))
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(-1.421413741 - Float64(Float64(1.453152027 - Float64(1.061405429 / t_0)) / t_2)) / t_2) - 0.284496736) / t_0) - -0.254829592) / t_3)
	t_5 = Float64(1.0 + (t_4 ^ 2.0))
	return Float64(Float64(Float64(1.0 / t_5) - Float64((t_4 ^ 4.0) / t_5)) / Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_1)) / t_2) - -1.421413741) / t_1) - 0.284496736) / t_1) - -0.254829592) / t_3)))
end

Wolfram

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

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

5. Applied rewrites 79.2%

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

7. Applied rewrites 86.6%

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

Alternative 3: 79.2% accurate, 0.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0))
        (t_1 (fma -0.3275911 (fabs x_m) -1.0))
        (t_2
         (/
          (-
           (/
            (-
             (/
              (- -1.421413741 (/ (- 1.453152027 (/ 1.061405429 t_0)) t_1))
              t_1)
             0.284496736)
            t_0)
           -0.254829592)
          (* t_1 (exp (* x_m x_m))))))
   (/ (- 1.0 (pow t_2 4.0)) (* (+ 1.0 (pow t_2 2.0)) (- 1.0 t_2)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double t_1 = fma(-0.3275911, fabs(x_m), -1.0);
	double t_2 = (((((-1.421413741 - ((1.453152027 - (1.061405429 / t_0)) / t_1)) / t_1) - 0.284496736) / t_0) - -0.254829592) / (t_1 * exp((x_m * x_m)));
	return (1.0 - pow(t_2, 4.0)) / ((1.0 + pow(t_2, 2.0)) * (1.0 - t_2));
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	t_1 = fma(-0.3275911, abs(x_m), -1.0)
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(-1.421413741 - Float64(Float64(1.453152027 - Float64(1.061405429 / t_0)) / t_1)) / t_1) - 0.284496736) / t_0) - -0.254829592) / Float64(t_1 * exp(Float64(x_m * x_m))))
	return Float64(Float64(1.0 - (t_2 ^ 4.0)) / Float64(Float64(1.0 + (t_2 ^ 2.0)) * Float64(1.0 - t_2)))
end

Wolfram

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

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

5. Applied rewrites 79.2%

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

7. Applied rewrites 79.2%

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

Alternative 4: 79.2% accurate, 1.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (-
    1.0
    (*
     (*
      (/
       (-
        (/
         (-
          (/ (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_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 - (((((((((((1.061405429 / t_0) - 1.453152027) / t_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(Float64(1.061405429 / t_0) - 1.453152027) / t_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[(N[(1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$0), $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%

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

Alternative 5: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right)\\ 1 - \frac{\frac{\mathsf{fma}\left(\frac{1}{t\_1}, \frac{1.453152027 - \frac{1.061405429}{t\_1}}{\mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)} - -1.421413741, -0.284496736\right)}{t\_0} - -0.254829592}{t\_0 \cdot e^{x\_m \cdot x\_m}} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0))
        (t_1 (fma 0.3275911 (fabs x_m) 1.0)))
   (-
    1.0
    (/
     (-
      (/
       (fma
        (/ 1.0 t_1)
        (-
         (/
          (- 1.453152027 (/ 1.061405429 t_1))
          (fma -0.3275911 (fabs x_m) -1.0))
         -1.421413741)
        -0.284496736)
       t_0)
      -0.254829592)
     (* t_0 (exp (* x_m x_m)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double t_1 = fma(0.3275911, fabs(x_m), 1.0);
	return 1.0 - (((fma((1.0 / t_1), (((1.453152027 - (1.061405429 / t_1)) / fma(-0.3275911, fabs(x_m), -1.0)) - -1.421413741), -0.284496736) / t_0) - -0.254829592) / (t_0 * exp((x_m * x_m))));
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	t_1 = fma(0.3275911, abs(x_m), 1.0)
	return Float64(1.0 - Float64(Float64(Float64(fma(Float64(1.0 / t_1), Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_1)) / fma(-0.3275911, abs(x_m), -1.0)) - -1.421413741), -0.284496736) / t_0) - -0.254829592) / Float64(t_0 * exp(Float64(x_m * 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]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[(N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(N[(N[(1.453152027 - N[(1.061405429 / t$95$1), $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[(t$95$0 * N[Exp[N[(x$95$m * 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%

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

5. Applied rewrites 79.2%

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

Alternative 6: 79.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-/.f64 N/A

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

   4. div-flip N/A

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

   5. associate-/r/ N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lower-fma.f64 N/A

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

5. Applied rewrites 79.2%

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

Alternative 7: 79.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	return 1.0 - (((((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) - 0.284496736) / t_0) - -0.254829592) / (t_0 * exp((x_m * 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(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) - 0.284496736) / t_0) - -0.254829592) / Float64(t_0 * exp(Float64(x_m * 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[(1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] - 0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(t$95$0 * N[Exp[N[(x$95$m * 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%

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

Alternative 8: 77.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fabs.f64 77.6

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

8. Applied rewrites 77.6%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lower--.f64 N/A

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

10. Applied rewrites 77.6%

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

Alternative 9: 77.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fabs.f64 77.6

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

8. Applied rewrites 77.6%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-/.f64 N/A

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

   4. div-flip N/A

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

   5. metadata-eval N/A

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

   6. lift-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-fma.f64 N/A

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

   9. mult-flip N/A

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

   10. times-frac N/A

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

   11. lower-fma.f64 N/A

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

10. Applied rewrites 77.6%

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

Alternative 10: 77.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fabs.f64 77.6

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

8. Applied rewrites 77.6%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval 77.6

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

10. Applied rewrites 77.6%

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

Alternative 11: 77.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fabs.f64 77.6

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

8. Applied rewrites 77.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 77.6

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

10. Applied rewrites 77.6%

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

Reproduce

herbie shell --seed 2025149 
(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)))))))