Jmat.Real.erf

Percentage Accurate: 79.0% → 83.3%

Time: 14.4s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592\\ t_2 := e^{x \cdot x}\\ t_3 := \frac{t\_1}{t\_0 \cdot t\_2}\\ t_4 := \mathsf{fma}\left(t\_3, t\_3 + 1, 1\right)\\ t_5 := \frac{t\_1}{t\_2 \cdot t\_0}\\ t_6 := \left(1 + {t\_5}^{6}\right) + {t\_5}^{3}\\ \frac{\frac{1}{t\_6}}{t\_4} - \frac{\frac{{t\_5}^{9}}{t\_6}}{t\_4} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1
         (+
          (/
           (+
            (/
             (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741)
             t_0)
            -0.284496736)
           t_0)
          0.254829592))
        (t_2 (exp (* x x)))
        (t_3 (/ t_1 (* t_0 t_2)))
        (t_4 (fma t_3 (+ t_3 1.0) 1.0))
        (t_5 (/ t_1 (* t_2 t_0)))
        (t_6 (+ (+ 1.0 (pow t_5 6.0)) (pow t_5 3.0))))
   (- (/ (/ 1.0 t_6) t_4) (/ (/ (pow t_5 9.0) t_6) t_4))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = (((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592;
	double t_2 = exp((x * x));
	double t_3 = t_1 / (t_0 * t_2);
	double t_4 = fma(t_3, (t_3 + 1.0), 1.0);
	double t_5 = t_1 / (t_2 * t_0);
	double t_6 = (1.0 + pow(t_5, 6.0)) + pow(t_5, 3.0);
	return ((1.0 / t_6) / t_4) - ((pow(t_5, 9.0) / t_6) / t_4);
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = 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)
	t_2 = exp(Float64(x * x))
	t_3 = Float64(t_1 / Float64(t_0 * t_2))
	t_4 = fma(t_3, Float64(t_3 + 1.0), 1.0)
	t_5 = Float64(t_1 / Float64(t_2 * t_0))
	t_6 = Float64(Float64(1.0 + (t_5 ^ 6.0)) + (t_5 ^ 3.0))
	return Float64(Float64(Float64(1.0 / t_6) / t_4) - Float64(Float64((t_5 ^ 9.0) / t_6) / t_4))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(t$95$3 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 / N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(1.0 + N[Power[t$95$5, 6.0], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$5, 3.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / t$95$6), $MachinePrecision] / t$95$4), $MachinePrecision] - N[(N[(N[Power[t$95$5, 9.0], $MachinePrecision] / t$95$6), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.1%

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

5. Applied rewrites 79.2%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}\right)}}}{\mathsf{fma}\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}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \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}} + 1, 1\right)} \]

7. Applied rewrites 80.3%

   \[\leadsto \frac{\color{blue}{\frac{1}{\left(1 + {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{6}\right) + {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}} - \frac{{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{9}}{\left(1 + {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{6}\right) + {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}}}}{\mathsf{fma}\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}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \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}} + 1, 1\right)} \]

9. Applied rewrites 83.3%

   \[\leadsto \color{blue}{\frac{\frac{1}{\left(1 + {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{6}\right) + {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}}}{\mathsf{fma}\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}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \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}} + 1, 1\right)} - \frac{\frac{{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{9}}{\left(1 + {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{6}\right) + {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}}}{\mathsf{fma}\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}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \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}} + 1, 1\right)}} \]
10. Add Preprocessing

Alternative 2: 80.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592\\ t_2 := e^{x \cdot x}\\ t_3 := \frac{t\_1}{t\_0 \cdot t\_2}\\ t_4 := \frac{t\_1}{t\_2 \cdot t\_0}\\ t_5 := \left(1 + {t\_4}^{6}\right) + {t\_4}^{3}\\ \frac{\frac{1}{t\_5} - \frac{{t\_4}^{9}}{t\_5}}{\mathsf{fma}\left(t\_3, t\_3 + 1, 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1
         (+
          (/
           (+
            (/
             (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741)
             t_0)
            -0.284496736)
           t_0)
          0.254829592))
        (t_2 (exp (* x x)))
        (t_3 (/ t_1 (* t_0 t_2)))
        (t_4 (/ t_1 (* t_2 t_0)))
        (t_5 (+ (+ 1.0 (pow t_4 6.0)) (pow t_4 3.0))))
   (/ (- (/ 1.0 t_5) (/ (pow t_4 9.0) t_5)) (fma t_3 (+ t_3 1.0) 1.0))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = (((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592;
	double t_2 = exp((x * x));
	double t_3 = t_1 / (t_0 * t_2);
	double t_4 = t_1 / (t_2 * t_0);
	double t_5 = (1.0 + pow(t_4, 6.0)) + pow(t_4, 3.0);
	return ((1.0 / t_5) - (pow(t_4, 9.0) / t_5)) / fma(t_3, (t_3 + 1.0), 1.0);
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = 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)
	t_2 = exp(Float64(x * x))
	t_3 = Float64(t_1 / Float64(t_0 * t_2))
	t_4 = Float64(t_1 / Float64(t_2 * t_0))
	t_5 = Float64(Float64(1.0 + (t_4 ^ 6.0)) + (t_4 ^ 3.0))
	return Float64(Float64(Float64(1.0 / t_5) - Float64((t_4 ^ 9.0) / t_5)) / fma(t_3, Float64(t_3 + 1.0), 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(1.0 + N[Power[t$95$4, 6.0], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$4, 3.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / t$95$5), $MachinePrecision] - N[(N[Power[t$95$4, 9.0], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * N[(t$95$3 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.1%

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

5. Applied rewrites 79.2%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}\right)}}}{\mathsf{fma}\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}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \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}} + 1, 1\right)} \]

7. Applied rewrites 80.3%

   \[\leadsto \frac{\color{blue}{\frac{1}{\left(1 + {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{6}\right) + {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}} - \frac{{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{9}}{\left(1 + {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{6}\right) + {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}}}}{\mathsf{fma}\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}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \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}} + 1, 1\right)} \]
8. Add Preprocessing

Alternative 3: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := e^{x \cdot x}\\ t_2 := \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{t\_1 \cdot t\_0}\\ t_3 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_4 := \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_3} - 1.453152027}{t\_3} - -1.421413741}{t\_3} + -0.284496736}{t\_3} + 0.254829592}{t\_1 \cdot t\_3}\\ t_5 := {t\_2}^{3}\\ \frac{\frac{1 - {t\_2}^{9}}{1 + \mathsf{fma}\left(t\_5, t\_5, 1 \cdot t\_5\right)}}{\left(t\_4 + 1\right) \cdot t\_4 + 1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1 (exp (* x x)))
        (t_2
         (/
          (+
           (/
            (+
             (/
              (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741)
              t_0)
             -0.284496736)
            t_0)
           0.254829592)
          (* t_1 t_0)))
        (t_3 (fma (fabs x) 0.3275911 1.0))
        (t_4
         (/
          (+
           (/
            (+
             (/
              (- (/ (- (/ 1.061405429 t_3) 1.453152027) t_3) -1.421413741)
              t_3)
             -0.284496736)
            t_3)
           0.254829592)
          (* t_1 t_3)))
        (t_5 (pow t_2 3.0)))
   (/
    (/ (- 1.0 (pow t_2 9.0)) (+ 1.0 (fma t_5 t_5 (* 1.0 t_5))))
    (+ (* (+ t_4 1.0) t_4) 1.0))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = exp((x * x));
	double t_2 = ((((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / (t_1 * t_0);
	double t_3 = fma(fabs(x), 0.3275911, 1.0);
	double t_4 = ((((((((1.061405429 / t_3) - 1.453152027) / t_3) - -1.421413741) / t_3) + -0.284496736) / t_3) + 0.254829592) / (t_1 * t_3);
	double t_5 = pow(t_2, 3.0);
	return ((1.0 - pow(t_2, 9.0)) / (1.0 + fma(t_5, t_5, (1.0 * t_5)))) / (((t_4 + 1.0) * t_4) + 1.0);
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = exp(Float64(x * x))
	t_2 = 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_1 * t_0))
	t_3 = fma(abs(x), 0.3275911, 1.0)
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_3) - 1.453152027) / t_3) - -1.421413741) / t_3) + -0.284496736) / t_3) + 0.254829592) / Float64(t_1 * t_3))
	t_5 = t_2 ^ 3.0
	return Float64(Float64(Float64(1.0 - (t_2 ^ 9.0)) / Float64(1.0 + fma(t_5, t_5, Float64(1.0 * t_5)))) / Float64(Float64(Float64(t_4 + 1.0) * t_4) + 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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$1 * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$3), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$3), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$3), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$3), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[t$95$2, 3.0], $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[t$95$2, 9.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$5 * t$95$5 + N[(1.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$4 + 1.0), $MachinePrecision] * t$95$4), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.1%

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

5. Applied rewrites 79.2%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}\right)}}}{\mathsf{fma}\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}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \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}} + 1, 1\right)} \]

7. Applied rewrites 79.2%

   \[\leadsto \frac{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}\right)}}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1\right) \cdot \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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1}} \]
8. Add Preprocessing

Alternative 4: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{e^{x \cdot x} \cdot t\_0}\\ \frac{1 - {t\_1}^{9}}{\left(\left(1 + {t\_1}^{6}\right) + {t\_1}^{3}\right) \cdot \mathsf{fma}\left(t\_1 + 1, t\_1, 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1
         (/
          (+
           (/
            (+
             (/
              (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741)
              t_0)
             -0.284496736)
            t_0)
           0.254829592)
          (* (exp (* x x)) t_0))))
   (/
    (- 1.0 (pow t_1 9.0))
    (* (+ (+ 1.0 (pow t_1 6.0)) (pow t_1 3.0)) (fma (+ t_1 1.0) t_1 1.0)))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = ((((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / (exp((x * x)) * t_0);
	return (1.0 - pow(t_1, 9.0)) / (((1.0 + pow(t_1, 6.0)) + pow(t_1, 3.0)) * fma((t_1 + 1.0), t_1, 1.0));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = 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(exp(Float64(x * x)) * t_0))
	return Float64(Float64(1.0 - (t_1 ^ 9.0)) / Float64(Float64(Float64(1.0 + (t_1 ^ 6.0)) + (t_1 ^ 3.0)) * fma(Float64(t_1 + 1.0), t_1, 1.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = 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[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - N[Power[t$95$1, 9.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + N[Power[t$95$1, 6.0], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 + 1.0), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.1%

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

5. Applied rewrites 79.2%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{9}}{1 + \mathsf{fma}\left({\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}, 1 \cdot {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{3}\right)}}}{\mathsf{fma}\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}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}, \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}} + 1, 1\right)} \]

7. Applied rewrites 79.2%

   \[\leadsto \color{blue}{\frac{1 - {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{9}}{\left(\left(1 + {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{6}\right) + {\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}^{3}\right) \cdot \mathsf{fma}\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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1, \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}{e^{x \cdot x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right)}} \]
8. Add Preprocessing

Alternative 5: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{\left(x \cdot x\right) \cdot 0.10731592879921 - 1}{t\_0 - 1}\\ t_2 := \frac{1}{1 + t\_0}\\ 1 - \left(t\_2 \cdot \left(0.254829592 + t\_2 \cdot \mathsf{fma}\left(\frac{1}{t\_1}, \frac{\frac{1.061405429}{t\_1} - 1.453152027}{t\_1} - -1.421413741, -0.284496736\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.3275911 (fabs x)))
        (t_1 (/ (- (* (* x x) 0.10731592879921) 1.0) (- t_0 1.0)))
        (t_2 (/ 1.0 (+ 1.0 t_0))))
   (-
    1.0
    (*
     (*
      t_2
      (+
       0.254829592
       (*
        t_2
        (fma
         (/ 1.0 t_1)
         (- (/ (- (/ 1.061405429 t_1) 1.453152027) t_1) -1.421413741)
         -0.284496736))))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = 0.3275911 * fabs(x);
	double t_1 = (((x * x) * 0.10731592879921) - 1.0) / (t_0 - 1.0);
	double t_2 = 1.0 / (1.0 + t_0);
	return 1.0 - ((t_2 * (0.254829592 + (t_2 * fma((1.0 / t_1), ((((1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741), -0.284496736)))) * exp(-(fabs(x) * fabs(x))));
}

Julia

function code(x)
	t_0 = Float64(0.3275911 * abs(x))
	t_1 = Float64(Float64(Float64(Float64(x * x) * 0.10731592879921) - 1.0) / Float64(t_0 - 1.0))
	t_2 = Float64(1.0 / Float64(1.0 + t_0))
	return Float64(1.0 - Float64(Float64(t_2 * Float64(0.254829592 + Float64(t_2 * fma(Float64(1.0 / t_1), Float64(Float64(Float64(Float64(1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741), -0.284496736)))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] * 0.10731592879921), $MachinePrecision] - 1.0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$2 * N[(0.254829592 + N[(t$95$2 * N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] + -0.284496736), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-fabs.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-fabs.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-fabs.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lift-fabs.f64 N/A

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

5. Applied rewrites 79.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-fabs.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-fabs.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-fabs.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lift-fabs.f64 N/A

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

7. Applied rewrites 79.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-fabs.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-fabs.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-fabs.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lift-fabs.f64 N/A

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

9. Applied rewrites 79.0%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. swap-sqr N/A

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

    5. metadata-eval N/A

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

    6. lift-fabs.f64 N/A

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

    7. lift-fabs.f64 N/A

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

    8. sqr-abs-rev N/A

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

    9. pow2 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{1}{\frac{\frac{10731592879921}{100000000000000} \cdot \color{blue}{{x}^{2}} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}}, \frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{1453152027}{1000000000}}{\frac{\left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{-1421413741}{1000000000}, \frac{-8890523}{31250000}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

    10. *-commutative N/A

        \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{1}{\frac{\color{blue}{{x}^{2} \cdot \frac{10731592879921}{100000000000000}} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}}, \frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{1453152027}{1000000000}}{\frac{\left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{-1421413741}{1000000000}, \frac{-8890523}{31250000}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

    11. lower-*.f64 N/A

        \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{1}{\frac{\color{blue}{{x}^{2} \cdot \frac{10731592879921}{100000000000000}} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}}, \frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{1453152027}{1000000000}}{\frac{\left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{-1421413741}{1000000000}, \frac{-8890523}{31250000}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

    12. pow2 N/A

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

    13. lift-*.f64 79.0

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

11. Applied rewrites 79.0%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. swap-sqr N/A

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

    5. metadata-eval N/A

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

    6. lift-fabs.f64 N/A

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

    7. lift-fabs.f64 N/A

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

    8. sqr-abs-rev N/A

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

    9. pow2 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{1}{\frac{\left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}}, \frac{\frac{\frac{1061405429}{1000000000}}{\frac{\frac{10731592879921}{100000000000000} \cdot \color{blue}{{x}^{2}} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{1453152027}{1000000000}}{\frac{\left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{-1421413741}{1000000000}, \frac{-8890523}{31250000}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

    10. *-commutative N/A

        \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{1}{\frac{\left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}}, \frac{\frac{\frac{1061405429}{1000000000}}{\frac{\color{blue}{{x}^{2} \cdot \frac{10731592879921}{100000000000000}} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{1453152027}{1000000000}}{\frac{\left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{-1421413741}{1000000000}, \frac{-8890523}{31250000}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

    11. lower-*.f64 N/A

        \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{1}{\frac{\left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}}, \frac{\frac{\frac{1061405429}{1000000000}}{\frac{\color{blue}{{x}^{2} \cdot \frac{10731592879921}{100000000000000}} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{1453152027}{1000000000}}{\frac{\left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{-1421413741}{1000000000}, \frac{-8890523}{31250000}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

    12. pow2 N/A

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

    13. lift-*.f64 79.0

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

13. Applied rewrites 79.0%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. swap-sqr N/A

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

    5. metadata-eval N/A

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

    6. lift-fabs.f64 N/A

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

    7. lift-fabs.f64 N/A

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

    8. sqr-abs-rev N/A

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

    9. pow2 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{1}{\frac{\left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}}, \frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{1453152027}{1000000000}}{\frac{\frac{10731592879921}{100000000000000} \cdot \color{blue}{{x}^{2}} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{-1421413741}{1000000000}, \frac{-8890523}{31250000}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

    10. *-commutative N/A

        \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{1}{\frac{\left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}}, \frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{1453152027}{1000000000}}{\frac{\color{blue}{{x}^{2} \cdot \frac{10731592879921}{100000000000000}} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{-1421413741}{1000000000}, \frac{-8890523}{31250000}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

    11. lower-*.f64 N/A

        \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{1}{\frac{\left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}}, \frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{1453152027}{1000000000}}{\frac{\color{blue}{{x}^{2} \cdot \frac{10731592879921}{100000000000000}} - 1}{\frac{3275911}{10000000} \cdot \left|x\right| - 1}} - \frac{-1421413741}{1000000000}, \frac{-8890523}{31250000}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

    12. pow2 N/A

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

    13. lift-*.f64 79.0

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

15. Applied rewrites 79.0%

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

Alternative 6: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ 1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \mathsf{fma}\left(\frac{\frac{\frac{1.061405429}{t\_1} - 1.453152027}{t\_1} - -1.421413741}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, -0.284496736\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))))
        (t_1 (fma (fabs x) 0.3275911 1.0)))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (fma
         (/
          (- (/ (- (/ 1.061405429 t_1) 1.453152027) t_1) -1.421413741)
          (- 1.0 (* 0.10731592879921 (* x x))))
         (- 1.0 (* (fabs x) 0.3275911))
         -0.284496736))))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * fma((((((1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / (1.0 - (0.10731592879921 * (x * x)))), (1.0 - (fabs(x) * 0.3275911)), -0.284496736)))) * exp(-(fabs(x) * fabs(x))));
}

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	t_1 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * fma(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / Float64(1.0 - Float64(0.10731592879921 * Float64(x * x)))), Float64(1.0 - Float64(abs(x) * 0.3275911)), -0.284496736)))) * exp(Float64(-Float64(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]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(N[(N[(N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(1.0 - N[(0.10731592879921 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

Alternative 7: 79.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.0 / N[(N[(N[(x * x), $MachinePrecision] * 0.10731592879921), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 1.061405429), $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 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

   \[\leadsto \color{blue}{1 - \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
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. lift-fma.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-fabs.f64 N/A

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

   7. +-commutative N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-fabs.f64 N/A

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

   15. lower-fma.f64 79.0

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

5. Applied rewrites 79.0%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. flip-+ N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. pow2 N/A

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

   8. lift-fabs.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. metadata-eval N/A

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\left(\frac{1}{\color{blue}{\frac{10731592879921}{100000000000000}} \cdot {\left(\left|x\right|\right)}^{2} - 1} \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right| - 1\right)\right) \cdot \frac{1061405429}{1000000000} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}} \]

   11. lift-fabs.f64 N/A

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\left(\frac{1}{\frac{10731592879921}{100000000000000} \cdot {\color{blue}{\left(\left|x\right|\right)}}^{2} - 1} \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right| - 1\right)\right) \cdot \frac{1061405429}{1000000000} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}} \]

   12. lower-*.f64 N/A

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\color{blue}{\left(\frac{1}{\frac{10731592879921}{100000000000000} \cdot {\left(\left|x\right|\right)}^{2} - 1} \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right| - 1\right)\right)} \cdot \frac{1061405429}{1000000000} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right) \cdot e^{x \cdot x}} \]

7. Applied rewrites 79.0%

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

Alternative 8: 79.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$1 * N[(0.254829592 + N[(t$95$1 * N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] + -0.284496736), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

Alternative 9: 79.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

   \[\leadsto \color{blue}{1 - \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
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. lift-fma.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-fabs.f64 N/A

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

   7. +-commutative N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-fabs.f64 N/A

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

   15. lower-fma.f64 79.0

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

5. Applied rewrites 79.0%

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

Alternative 10: 79.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 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 x)))))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 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 * x))));
}

Julia

function code(x)
	t_0 = fma(abs(x), 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 * x)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $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 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

Alternative 11: 78.5% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 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 (fma x x 1.0))))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 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 * fma(x, x, 1.0)));
}

Julia

function code(x)
	t_0 = fma(abs(x), 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 * fma(x, x, 1.0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $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[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. lower-fma.f64 78.5

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

6. Applied rewrites 78.5%

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

Alternative 12: 77.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = fma(fabs(x), 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) / fma(0.3275911, fabs(x), 1.0));
}

Julia

function code(x)
	t_0 = fma(abs(x), 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) / fma(0.3275911, abs(x), 1.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $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[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lift-fabs.f64 N/A

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

   3. lower-fma.f64 77.6

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

6. Applied rewrites 77.6%

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

Alternative 13: 55.4% accurate, 99.9× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

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
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 79.0%

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

3. Applied rewrites 79.0%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 55.4%

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

Reproduce

herbie shell --seed 2025124 
(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)))))))