Jmat.Real.erf

Percentage Accurate: 79.3% → 79.5%

Time: 7.9s

Alternatives: 15

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% 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: 79.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-x\right) \cdot x}\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_3 := \frac{\frac{\frac{\frac{1.061405429}{t\_2} - 1.453152027}{t\_2} - -1.421413741}{t\_2} + -0.284496736}{t\_2} + 0.254829592\\ t_4 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_5 := t\_0 \cdot \frac{t\_3}{t\_4}\\ t_6 := {t\_5}^{6}\\ \frac{\frac{\frac{1 - t\_6 \cdot t\_6}{1 + t\_6}}{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{t\_1} - 1.453152027}{t\_1} - -1.421413741}{t\_1} + -0.284496736}{t\_1} + 0.254829592}{t\_4} \cdot t\_0\right)}^{3}}}{\left(e^{\left(\log \left(\frac{t\_3}{t\_2}\right) - x \cdot x\right) \cdot 2} - t\_5\right) + 1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* (- x) x)))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2 (fma 0.3275911 (fabs x) 1.0))
        (t_3
         (+
          (/
           (+
            (/
             (- (/ (- (/ 1.061405429 t_2) 1.453152027) t_2) -1.421413741)
             t_2)
            -0.284496736)
           t_2)
          0.254829592))
        (t_4 (fma -0.3275911 (fabs x) -1.0))
        (t_5 (* t_0 (/ t_3 t_4)))
        (t_6 (pow t_5 6.0)))
   (/
    (/
     (/ (- 1.0 (* t_6 t_6)) (+ 1.0 t_6))
     (-
      1.0
      (pow
       (*
        (/
         (+
          (/
           (+
            (/
             (- (/ (- (/ 1.061405429 t_1) 1.453152027) t_1) -1.421413741)
             t_1)
            -0.284496736)
           t_1)
          0.254829592)
         t_4)
        t_0)
       3.0)))
    (+ (- (exp (* (- (log (/ t_3 t_2)) (* x x)) 2.0)) t_5) 1.0))))

C

double code(double x) {
	double t_0 = exp((-x * x));
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = fma(0.3275911, fabs(x), 1.0);
	double t_3 = (((((((1.061405429 / t_2) - 1.453152027) / t_2) - -1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592;
	double t_4 = fma(-0.3275911, fabs(x), -1.0);
	double t_5 = t_0 * (t_3 / t_4);
	double t_6 = pow(t_5, 6.0);
	return (((1.0 - (t_6 * t_6)) / (1.0 + t_6)) / (1.0 - pow(((((((((((1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592) / t_4) * t_0), 3.0))) / ((exp(((log((t_3 / t_2)) - (x * x)) * 2.0)) - t_5) + 1.0);
}

Julia

function code(x)
	t_0 = exp(Float64(Float64(-x) * x))
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = fma(0.3275911, abs(x), 1.0)
	t_3 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_2) - 1.453152027) / t_2) - -1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592)
	t_4 = fma(-0.3275911, abs(x), -1.0)
	t_5 = Float64(t_0 * Float64(t_3 / t_4))
	t_6 = t_5 ^ 6.0
	return Float64(Float64(Float64(Float64(1.0 - Float64(t_6 * t_6)) / Float64(1.0 + t_6)) / Float64(1.0 - (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592) / t_4) * t_0) ^ 3.0))) / Float64(Float64(exp(Float64(Float64(log(Float64(t_3 / t_2)) - Float64(x * x)) * 2.0)) - t_5) + 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$2), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$2), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$2), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$2), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$4 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 * N[(t$95$3 / t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$5, 6.0], $MachinePrecision]}, N[(N[(N[(N[(1.0 - N[(t$95$6 * t$95$6), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$6), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Exp[N[(N[(N[Log[N[(t$95$3 / t$95$2), $MachinePrecision]], $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] - t$95$5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

5. Applied rewrites 79.3%

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

7. Applied rewrites 79.4%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{6}}{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}}{\left(e^{\left(\log \left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - x \cdot x\right) \cdot 2} - e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right) + 1} \]

9. Applied rewrites 79.5%

   \[\leadsto \frac{\frac{\color{blue}{\frac{1 - {\left(e^{\left(-x\right) \cdot x} \cdot \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}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}^{6} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \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}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}^{6}}{1 + {\left(e^{\left(-x\right) \cdot x} \cdot \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}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}^{6}}}}{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(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}{\left(e^{\left(\log \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}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - x \cdot x\right) \cdot 2} - e^{\left(-x\right) \cdot x} \cdot \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}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right) + 1} \]
10. Add Preprocessing

Alternative 2: 79.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := e^{\left(-x\right) \cdot x}\\ t_2 := \frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592\\ t_3 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_4 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_5 := t\_1 \cdot \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_4} - 1.453152027}{t\_4} - -1.421413741}{t\_4} + -0.284496736}{t\_4} + 0.254829592}{t\_3}\\ t_6 := {t\_5}^{3}\\ \frac{\frac{{t\_6}^{3} + 1}{{t\_5}^{6} + \left(1 - t\_6 \cdot 1\right)}}{e^{\left(\log \left(\frac{t\_2}{t\_0}\right) - x \cdot x\right) \cdot 2} - \left(\frac{t\_2}{t\_3} \cdot t\_1 - 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1 (exp (* (- x) x)))
        (t_2
         (+
          (/
           (+
            (/
             (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741)
             t_0)
            -0.284496736)
           t_0)
          0.254829592))
        (t_3 (fma -0.3275911 (fabs x) -1.0))
        (t_4 (fma 0.3275911 (fabs x) 1.0))
        (t_5
         (*
          t_1
          (/
           (+
            (/
             (+
              (/
               (- (/ (- (/ 1.061405429 t_4) 1.453152027) t_4) -1.421413741)
               t_4)
              -0.284496736)
             t_4)
            0.254829592)
           t_3)))
        (t_6 (pow t_5 3.0)))
   (/
    (/ (+ (pow t_6 3.0) 1.0) (+ (pow t_5 6.0) (- 1.0 (* t_6 1.0))))
    (-
     (exp (* (- (log (/ t_2 t_0)) (* x x)) 2.0))
     (- (* (/ t_2 t_3) t_1) 1.0)))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 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;
	double t_3 = fma(-0.3275911, fabs(x), -1.0);
	double t_4 = fma(0.3275911, fabs(x), 1.0);
	double t_5 = t_1 * (((((((((1.061405429 / t_4) - 1.453152027) / t_4) - -1.421413741) / t_4) + -0.284496736) / t_4) + 0.254829592) / t_3);
	double t_6 = pow(t_5, 3.0);
	return ((pow(t_6, 3.0) + 1.0) / (pow(t_5, 6.0) + (1.0 - (t_6 * 1.0)))) / (exp(((log((t_2 / t_0)) - (x * x)) * 2.0)) - (((t_2 / t_3) * t_1) - 1.0));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = exp(Float64(Float64(-x) * x))
	t_2 = 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_3 = fma(-0.3275911, abs(x), -1.0)
	t_4 = fma(0.3275911, abs(x), 1.0)
	t_5 = Float64(t_1 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_4) - 1.453152027) / t_4) - -1.421413741) / t_4) + -0.284496736) / t_4) + 0.254829592) / t_3))
	t_6 = t_5 ^ 3.0
	return Float64(Float64(Float64((t_6 ^ 3.0) + 1.0) / Float64((t_5 ^ 6.0) + Float64(1.0 - Float64(t_6 * 1.0)))) / Float64(exp(Float64(Float64(log(Float64(t_2 / t_0)) - Float64(x * x)) * 2.0)) - Float64(Float64(Float64(t_2 / t_3) * 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[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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$3 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 * N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$4), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$4), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$4), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$4), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$5, 3.0], $MachinePrecision]}, N[(N[(N[(N[Power[t$95$6, 3.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Power[t$95$5, 6.0], $MachinePrecision] + N[(1.0 - N[(t$95$6 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[(N[(N[Log[N[(t$95$2 / t$95$0), $MachinePrecision]], $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(t$95$2 / t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

5. Applied rewrites 79.3%

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

7. Applied rewrites 79.3%

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

9. Applied rewrites 79.5%

   \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{\left(-x\right) \cdot x} \cdot \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}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}^{3}\right)}^{3} + 1}{{\left(e^{\left(-x\right) \cdot x} \cdot \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}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}^{6} + \left(1 - {\left(e^{\left(-x\right) \cdot x} \cdot \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}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}^{3} \cdot 1\right)}}}{e^{\left(\log \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)}\right) - x \cdot x\right) \cdot 2} - \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(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x} - 1\right)} \]
10. Add Preprocessing

Alternative 3: 79.4% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* (- x) x)))
        (t_1 (fma 0.3275911 (fabs x) 1.0))
        (t_2
         (+
          (/
           (+
            (/
             (- (/ (- (/ 1.061405429 t_1) 1.453152027) t_1) -1.421413741)
             t_1)
            -0.284496736)
           t_1)
          0.254829592))
        (t_3 (fma -0.3275911 (fabs x) -1.0))
        (t_4 (* t_0 (/ t_2 t_3)))
        (t_5 (pow t_4 3.0))
        (t_6 (fma (fabs x) 0.3275911 1.0)))
   (/
    (/
     (- 1.0 (* t_5 t_5))
     (-
      1.0
      (pow
       (*
        (/
         (+
          (/
           (+
            (/
             (- (/ (- (/ 1.061405429 t_6) 1.453152027) t_6) -1.421413741)
             t_6)
            -0.284496736)
           t_6)
          0.254829592)
         t_3)
        t_0)
       3.0)))
    (+ (- (exp (* (- (log (/ t_2 t_1)) (* x x)) 2.0)) t_4) 1.0))))

C

double code(double x) {
	double t_0 = exp((-x * x));
	double t_1 = fma(0.3275911, fabs(x), 1.0);
	double t_2 = (((((((1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592;
	double t_3 = fma(-0.3275911, fabs(x), -1.0);
	double t_4 = t_0 * (t_2 / t_3);
	double t_5 = pow(t_4, 3.0);
	double t_6 = fma(fabs(x), 0.3275911, 1.0);
	return ((1.0 - (t_5 * t_5)) / (1.0 - pow(((((((((((1.061405429 / t_6) - 1.453152027) / t_6) - -1.421413741) / t_6) + -0.284496736) / t_6) + 0.254829592) / t_3) * t_0), 3.0))) / ((exp(((log((t_2 / t_1)) - (x * x)) * 2.0)) - t_4) + 1.0);
}

Julia

function code(x)
	t_0 = exp(Float64(Float64(-x) * x))
	t_1 = fma(0.3275911, abs(x), 1.0)
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592)
	t_3 = fma(-0.3275911, abs(x), -1.0)
	t_4 = Float64(t_0 * Float64(t_2 / t_3))
	t_5 = t_4 ^ 3.0
	t_6 = fma(abs(x), 0.3275911, 1.0)
	return Float64(Float64(Float64(1.0 - Float64(t_5 * t_5)) / Float64(1.0 - (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_6) - 1.453152027) / t_6) - -1.421413741) / t_6) + -0.284496736) / t_6) + 0.254829592) / t_3) * t_0) ^ 3.0))) / Float64(Float64(exp(Float64(Float64(log(Float64(t_2 / t_1)) - Float64(x * x)) * 2.0)) - t_4) + 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$3 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * N[(t$95$2 / t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[t$95$4, 3.0], $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(N[(N[(1.0 - N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$6), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$6), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$6), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$6), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Exp[N[(N[(N[Log[N[(t$95$2 / t$95$1), $MachinePrecision]], $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] - t$95$4), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

5. Applied rewrites 79.3%

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

7. Applied rewrites 79.4%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{6}}{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}}{\left(e^{\left(\log \left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - x \cdot x\right) \cdot 2} - e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right) + 1} \]

9. Applied rewrites 79.4%

   \[\leadsto \frac{\frac{1 - \color{blue}{{\left(e^{\left(-x\right) \cdot x} \cdot \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}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}^{3} \cdot {\left(e^{\left(-x\right) \cdot x} \cdot \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}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}^{3}}}{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(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}{\left(e^{\left(\log \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}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - x \cdot x\right) \cdot 2} - e^{\left(-x\right) \cdot x} \cdot \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}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right) + 1} \]
10. Add Preprocessing

Alternative 4: 79.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_3 := \frac{\frac{\frac{\frac{1.061405429}{t\_1} - 1.453152027}{t\_1} - -1.421413741}{t\_1} + -0.284496736}{t\_1} + 0.254829592\\ t_4 := e^{\left(-x\right) \cdot x}\\ t_5 := \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{t\_2} \cdot t\_4\\ \frac{\frac{1 - {t\_5}^{6}}{1 - {t\_5}^{3}}}{\left(e^{\left(\log \left(\frac{t\_3}{t\_1}\right) - x \cdot x\right) \cdot 2} - t\_4 \cdot \frac{t\_3}{t\_2}\right) + 1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1 (fma 0.3275911 (fabs x) 1.0))
        (t_2 (fma -0.3275911 (fabs x) -1.0))
        (t_3
         (+
          (/
           (+
            (/
             (- (/ (- (/ 1.061405429 t_1) 1.453152027) t_1) -1.421413741)
             t_1)
            -0.284496736)
           t_1)
          0.254829592))
        (t_4 (exp (* (- x) x)))
        (t_5
         (*
          (/
           (+
            (/
             (+
              (/
               (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741)
               t_0)
              -0.284496736)
             t_0)
            0.254829592)
           t_2)
          t_4)))
   (/
    (/ (- 1.0 (pow t_5 6.0)) (- 1.0 (pow t_5 3.0)))
    (+
     (- (exp (* (- (log (/ t_3 t_1)) (* x x)) 2.0)) (* t_4 (/ t_3 t_2)))
     1.0))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = fma(0.3275911, fabs(x), 1.0);
	double t_2 = fma(-0.3275911, fabs(x), -1.0);
	double t_3 = (((((((1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592;
	double t_4 = exp((-x * x));
	double t_5 = (((((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_2) * t_4;
	return ((1.0 - pow(t_5, 6.0)) / (1.0 - pow(t_5, 3.0))) / ((exp(((log((t_3 / t_1)) - (x * x)) * 2.0)) - (t_4 * (t_3 / t_2))) + 1.0);
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = fma(0.3275911, abs(x), 1.0)
	t_2 = fma(-0.3275911, abs(x), -1.0)
	t_3 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592)
	t_4 = exp(Float64(Float64(-x) * x))
	t_5 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_2) * t_4)
	return Float64(Float64(Float64(1.0 - (t_5 ^ 6.0)) / Float64(1.0 - (t_5 ^ 3.0))) / Float64(Float64(exp(Float64(Float64(log(Float64(t_3 / t_1)) - Float64(x * x)) * 2.0)) - Float64(t_4 * Float64(t_3 / t_2))) + 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[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] + 0.254829592), $MachinePrecision]}, Block[{t$95$4 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$2), $MachinePrecision] * t$95$4), $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[t$95$5, 6.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[t$95$5, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Exp[N[(N[(N[Log[N[(t$95$3 / t$95$1), $MachinePrecision]], $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$4 * N[(t$95$3 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

5. Applied rewrites 79.3%

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

7. Applied rewrites 79.4%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{6}}{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}}{\left(e^{\left(\log \left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - x \cdot x\right) \cdot 2} - e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right) + 1} \]
8. Add Preprocessing

Alternative 5: 79.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-x\right) \cdot x}\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_3 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_4 := \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_1} - 1.453152027}{t\_1} - -1.421413741}{t\_1} + -0.284496736}{t\_1} + 0.254829592}{t\_3} \cdot t\_0\\ t_5 := \frac{\frac{\frac{\frac{1.061405429}{t\_2} - 1.453152027}{t\_2} - -1.421413741}{t\_2} + -0.284496736}{t\_2} + 0.254829592\\ \frac{\frac{1 - {t\_4}^{6}}{1 - {t\_4}^{3}}}{e^{\left(\log \left(\frac{t\_5}{t\_2}\right) - x \cdot x\right) \cdot 2} - \left(t\_0 \cdot \frac{t\_5}{t\_3} - 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* (- x) x)))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2 (fma 0.3275911 (fabs x) 1.0))
        (t_3 (fma -0.3275911 (fabs x) -1.0))
        (t_4
         (*
          (/
           (+
            (/
             (+
              (/
               (- (/ (- (/ 1.061405429 t_1) 1.453152027) t_1) -1.421413741)
               t_1)
              -0.284496736)
             t_1)
            0.254829592)
           t_3)
          t_0))
        (t_5
         (+
          (/
           (+
            (/
             (- (/ (- (/ 1.061405429 t_2) 1.453152027) t_2) -1.421413741)
             t_2)
            -0.284496736)
           t_2)
          0.254829592)))
   (/
    (/ (- 1.0 (pow t_4 6.0)) (- 1.0 (pow t_4 3.0)))
    (-
     (exp (* (- (log (/ t_5 t_2)) (* x x)) 2.0))
     (- (* t_0 (/ t_5 t_3)) 1.0)))))

C

double code(double x) {
	double t_0 = exp((-x * x));
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = fma(0.3275911, fabs(x), 1.0);
	double t_3 = fma(-0.3275911, fabs(x), -1.0);
	double t_4 = (((((((((1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592) / t_3) * t_0;
	double t_5 = (((((((1.061405429 / t_2) - 1.453152027) / t_2) - -1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592;
	return ((1.0 - pow(t_4, 6.0)) / (1.0 - pow(t_4, 3.0))) / (exp(((log((t_5 / t_2)) - (x * x)) * 2.0)) - ((t_0 * (t_5 / t_3)) - 1.0));
}

Julia

function code(x)
	t_0 = exp(Float64(Float64(-x) * x))
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = fma(0.3275911, abs(x), 1.0)
	t_3 = fma(-0.3275911, abs(x), -1.0)
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_1) - 1.453152027) / t_1) - -1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592) / t_3) * t_0)
	t_5 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_2) - 1.453152027) / t_2) - -1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592)
	return Float64(Float64(Float64(1.0 - (t_4 ^ 6.0)) / Float64(1.0 - (t_4 ^ 3.0))) / Float64(exp(Float64(Float64(log(Float64(t_5 / t_2)) - Float64(x * x)) * 2.0)) - Float64(Float64(t_0 * Float64(t_5 / t_3)) - 1.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$2), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$2), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$2), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$2), $MachinePrecision] + 0.254829592), $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[t$95$4, 6.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[t$95$4, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[(N[(N[Log[N[(t$95$5 / t$95$2), $MachinePrecision]], $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] - N[(N[(t$95$0 * N[(t$95$5 / t$95$3), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

5. Applied rewrites 79.3%

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

7. Applied rewrites 79.4%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{6}}{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}}{\left(e^{\left(\log \left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - x \cdot x\right) \cdot 2} - e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right) + 1} \]

9. Applied rewrites 79.4%

   \[\leadsto \frac{\frac{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{6}}{1 - {\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}}{\color{blue}{e^{\left(\log \left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - x \cdot x\right) \cdot 2} - \left(e^{\left(-x\right) \cdot x} \cdot \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - 1\right)}} \]
10. Add Preprocessing

Alternative 6: 79.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = Float64(0.3275911 * abs(x))
	t_2 = fma(-0.3275911, abs(x), -1.0)
	return Float64(1.0 - Float64(fma(Float64(1.0 / fma(0.3275911, abs(x), 1.0)), 0.254829592, Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / Float64(Float64(Float64(t_1 * t_1) - 1.0) / Float64(t_1 - 1.0))) + -0.284496736) / Float64(t_2 * t_2))) * 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[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[(1.0 / N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.254829592 + N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - 1.0), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

   1. lift-fma.f64 N/A

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

   2. flip-+ N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-fabs.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-fabs.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-fabs.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-fabs.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lift-fabs.f64 N/A

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

   19. lift-*.f64 79.3

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

5. Applied rewrites 79.3%

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

   1. lift-fma.f64 N/A

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

   2. flip-+ N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-fabs.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-fabs.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-fabs.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-fabs.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lift-fabs.f64 N/A

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

   19. lift-*.f64 79.3

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

7. Applied rewrites 79.3%

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

   1. lift-fma.f64 N/A

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

   2. flip-+ N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-fabs.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-fabs.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-fabs.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-fabs.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lift-fabs.f64 N/A

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

   19. lift-*.f64 79.3

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

9. Applied rewrites 79.3%

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

    1. lift-fma.f64 N/A

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

    2. flip-+ N/A

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

    3. lower-/.f64 N/A

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

    4. *-commutative N/A

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

    5. lift-fabs.f64 N/A

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

    6. *-commutative N/A

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

    7. lift-fabs.f64 N/A

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

    8. metadata-eval N/A

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

    9. lower--.f64 N/A

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

    10. lower-*.f64 N/A

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

    11. lift-fabs.f64 N/A

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

    12. lift-*.f64 N/A

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

    13. lift-fabs.f64 N/A

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

    14. lift-*.f64 N/A

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

    15. *-commutative N/A

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

    16. lift-fabs.f64 N/A

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

    17. lower--.f64 N/A

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

    18. lift-fabs.f64 N/A

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

    19. lift-*.f64 79.3

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

11. Applied rewrites 79.3%

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

    1. lift-/.f64 N/A

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

    2. lift--.f64 N/A

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

    3. div-sub N/A

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

13. Applied rewrites 79.3%

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

    1. lift-/.f64 N/A

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

    2. lift--.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-fabs.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-fabs.f64 N/A

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

    8. metadata-eval N/A

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

    9. lift--.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. lift-fabs.f64 N/A

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

    12. flip-+ N/A

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

    13. lift-fabs.f64 N/A

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

    14. lift-fma.f64 79.3

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

15. Applied rewrites 79.3%

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

Alternative 7: 79.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (-
    1.0
    (*
     (*
      (/
       (+
        (/
         (+
          (/ (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741) t_0)
          -0.284496736)
         t_0)
        0.254829592)
       (- 1.0 (* 0.10731592879921 (* x x))))
      (- 1.0 (* (fabs x) 0.3275911)))
     (exp (- (* (fabs x) (fabs 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) / (1.0 - (0.10731592879921 * (x * x)))) * (1.0 - (fabs(x) * 0.3275911))) * exp(-(fabs(x) * fabs(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(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / Float64(1.0 - Float64(0.10731592879921 * Float64(x * x)))) * Float64(1.0 - Float64(abs(x) * 0.3275911))) * 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]}, N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(1.0 - N[(0.10731592879921 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

Alternative 8: 79.3% accurate, 1.3× speedup

Math

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

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(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_0) * exp(Float64(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[(1.061405429 / 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] / t$95$0), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

Alternative 9: 79.3% accurate, 1.3× speedup

Math

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

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)
      (exp (* x x)))
     t_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) / exp((x * x))) / t_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(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / exp(Float64(x * x))) / t_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[(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[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

Alternative 10: 79.3% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (fma
    (/
     (+
      (/
       (+
        (/ (- (/ (- (/ 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))
    (exp (* (- x) x))
    1.0)))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	return fma((((((((((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)), exp((-x * x)), 1.0);
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return fma(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)), exp(Float64(Float64(-x) * x)), 1.0)
end

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

Alternative 11: 79.3% 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 79.3%

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

3. Applied rewrites 79.3%

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

Alternative 12: 78.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (-
    1.0
    (/
     (+
      (/
       (+
        (/
         (-
          (/
           (- (* (/ 1.0 (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 (fma x x 1.0))))))

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 * 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(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 * 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[(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[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

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 78.7

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

6. Applied rewrites 78.7%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-fabs.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-fabs.f64 N/A

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

   13. lift-fma.f64 78.7

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

8. Applied rewrites 78.7%

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

Alternative 13: 78.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 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(0.3275911, fabs(x), 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(0.3275911, abs(x), 1.0)
	return Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_0) / fma(x, x, 1.0)))
end

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

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 78.7

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

6. Applied rewrites 78.7%

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

8. Applied rewrites 78.7%

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

Alternative 14: 78.7% 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 79.3%

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

3. Applied rewrites 79.3%

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

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 78.7

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

6. Applied rewrites 78.7%

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

Alternative 15: 77.6% accurate, 1.5× speedup

Math

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

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)
     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) * 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(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_0) * 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[(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] / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.6%

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

Reproduce

herbie shell --seed 2025139 
(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)))))))