Jmat.Real.erf

Percentage Accurate: 79.3% → 79.3%

Time: 7.5s

Alternatives: 13

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% 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: 79.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. flip-+ N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. lower-unsound--.f64 N/A

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

   8. lower-unsound-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-unsound-*.f64 N/A

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

   12. lower-unsound--.f64 N/A

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

   13. lower-*.f64 79.3

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

5. Applied rewrites 79.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. flip-+ N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. lower-unsound--.f64 N/A

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

   8. lower-unsound-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-unsound-*.f64 N/A

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

   12. lower-unsound--.f64 N/A

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

   13. lower-*.f64 79.3

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

7. Applied rewrites 79.3%

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

Alternative 2: 79.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. flip-+ N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. lower-unsound--.f64 N/A

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

   8. lower-unsound-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-unsound-*.f64 N/A

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

   12. lower-unsound--.f64 N/A

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

   13. lower-*.f64 79.3

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

5. Applied rewrites 79.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. flip-+ N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. lower-unsound--.f64 N/A

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

   8. lower-unsound-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-unsound-*.f64 N/A

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

   12. lower-unsound--.f64 N/A

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

   13. lower-*.f64 79.3

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

7. Applied rewrites 79.3%

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

9. Applied rewrites 79.3%

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

Alternative 3: 79.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. mult-flip N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-neg-in N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. lift-fma.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

5. Applied rewrites 79.3%

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

Alternative 4: 79.3% accurate, 1.2× speedup

Math

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

FPCore

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

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. mult-flip N/A

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

   5. distribute-neg-frac2 N/A

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

   6. div-flip N/A

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

   7. lift-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. div-flip N/A

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

   13. lift-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. div-flip N/A

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

   17. lift-+.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. +-commutative N/A

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

   21. lift-fma.f64 N/A

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

5. Applied rewrites 79.3%

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

Alternative 5: 79.3% accurate, 1.2× speedup

Math

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

FPCore

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

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. mult-flip N/A

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

   7. distribute-neg-frac2 N/A

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

   8. div-flip N/A

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

   9. lift-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. div-flip N/A

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

   15. lift-/.f64 N/A

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

   16. lower-fma.f64 N/A

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

5. Applied rewrites 79.3%

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

Alternative 6: 79.3% accurate, 1.3× speedup

Math

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

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)
        (fma -0.3275911 (fabs x) -1.0))
       -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) / fma(-0.3275911, fabs(x), -1.0)) - -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) / fma(-0.3275911, abs(x), -1.0)) - -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] / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $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%

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

Alternative 7: 79.3% accurate, 1.3× speedup

Math

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

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)
       (fma -0.3275911 (fabs x) -1.0))
      -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) / fma(-0.3275911, fabs(x), -1.0)) - -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) / fma(-0.3275911, abs(x), -1.0)) - -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] / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $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%

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

Alternative 8: 77.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.7%

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

8. Applied rewrites 77.7%

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

Alternative 9: 77.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.7%

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

8. Applied rewrites 77.7%

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

Alternative 10: 77.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.7%

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

8. Applied rewrites 77.7%

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

Alternative 11: 77.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.7%

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

8. Applied rewrites 77.7%

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

10. Applied rewrites 77.7%

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

Alternative 12: 77.7% accurate, 1.5× speedup

Math

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

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)
       (fma -0.3275911 (fabs x) -1.0))
      -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) / fma(-0.3275911, fabs(x), -1.0)) - -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(-1.061405429 / t_0) - -1.453152027) / t_0) - 1.421413741) / t_0) - -0.284496736) / fma(-0.3275911, abs(x), -1.0)) - -0.254829592) / Float64(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[(-1.061405429 / t$95$0), $MachinePrecision] - -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.284496736), $MachinePrecision] / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(t$95$0 * 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%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.7%

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

Alternative 13: 55.9% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) 0.3275911)))
   (-
    1.0
    (*
     (* (/ 1.0 (/ (- (* t_0 t_0) (* 1.0 1.0)) (- t_0 1.0))) 0.254829592)
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = fabs(x) * 0.3275911;
	return 1.0 - (((1.0 / (((t_0 * t_0) - (1.0 * 1.0)) / (t_0 - 1.0))) * 0.254829592) * 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 = abs(x) * 0.3275911d0
    code = 1.0d0 - (((1.0d0 / (((t_0 * t_0) - (1.0d0 * 1.0d0)) / (t_0 - 1.0d0))) * 0.254829592d0) * exp(-(abs(x) * abs(x))))
end function

Java

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

Python

def code(x):
	t_0 = math.fabs(x) * 0.3275911
	return 1.0 - (((1.0 / (((t_0 * t_0) - (1.0 * 1.0)) / (t_0 - 1.0))) * 0.254829592) * math.exp(-(math.fabs(x) * math.fabs(x))))

Julia

function code(x)
	t_0 = Float64(abs(x) * 0.3275911)
	return Float64(1.0 - Float64(Float64(Float64(1.0 / Float64(Float64(Float64(t_0 * t_0) - Float64(1.0 * 1.0)) / Float64(t_0 - 1.0))) * 0.254829592) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * 0.3275911;
	tmp = 1.0 - (((1.0 / (((t_0 * t_0) - (1.0 * 1.0)) / (t_0 - 1.0))) * 0.254829592) * exp(-(abs(x) * abs(x))));
end

Wolfram

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

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. flip-+ N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. lower-unsound--.f64 N/A

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

   8. lower-unsound-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-unsound-*.f64 N/A

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

   12. lower-unsound--.f64 N/A

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

   13. lower-*.f64 79.3

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

5. Applied rewrites 79.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. flip-+ N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. lower-unsound--.f64 N/A

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

   8. lower-unsound-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-unsound-*.f64 N/A

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

   12. lower-unsound--.f64 N/A

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

   13. lower-*.f64 79.3

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

7. Applied rewrites 79.3%

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

9. Applied rewrites 79.3%

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

10. Taylor expanded in x around inf

    \[\leadsto 1 - \left(\frac{1}{\frac{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right) \cdot \left(\left|x\right| \cdot \frac{3275911}{10000000}\right) - 1 \cdot 1}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} \cdot \color{blue}{\frac{31853699}{125000000}}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
11. Step-by-step derivation

12. Applied rewrites 55.9%

    \[\leadsto 1 - \left(\frac{1}{\frac{\left(\left|x\right| \cdot 0.3275911\right) \cdot \left(\left|x\right| \cdot 0.3275911\right) - 1 \cdot 1}{\left|x\right| \cdot 0.3275911 - 1}} \cdot \color{blue}{0.254829592}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(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)))))))