Jmat.Real.erf

Percentage Accurate: 79.2% → 86.6%

Time: 8.6s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) -0.3275911 -1.0))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2 (fma -0.3275911 (fabs x) -1.0))
        (t_3 (exp (* (- x) x)))
        (t_4
         (*
          (-
           (/
            (-
             0.284496736
             (/
              (- -1.421413741 (/ (- (/ 1.061405429 t_1) 1.453152027) t_1))
              t_2))
            t_2)
           -0.254829592)
          (/ t_3 t_1)))
        (t_5 (+ 1.0 (pow t_4 2.0)))
        (t_6 (fma 0.3275911 (fabs x) 1.0)))
   (/
    (- (/ 1.0 t_5) (/ (pow t_4 4.0) t_5))
    (+
     1.0
     (*
      (/ t_3 t_6)
      (-
       (/
        (-
         0.284496736
         (/ (- (/ (- 1.453152027 (/ 1.061405429 t_6)) t_0) -1.421413741) t_6))
        t_0)
       -0.254829592))))))

C

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

Julia

function code(x)
	t_0 = fma(abs(x), -0.3275911, -1.0)
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = fma(-0.3275911, abs(x), -1.0)
	t_3 = exp(Float64(Float64(-x) * x))
	t_4 = Float64(Float64(Float64(Float64(0.284496736 - Float64(Float64(-1.421413741 - Float64(Float64(Float64(1.061405429 / t_1) - 1.453152027) / t_1)) / t_2)) / t_2) - -0.254829592) * Float64(t_3 / t_1))
	t_5 = Float64(1.0 + (t_4 ^ 2.0))
	t_6 = fma(0.3275911, abs(x), 1.0)
	return Float64(Float64(Float64(1.0 / t_5) - Float64((t_4 ^ 4.0) / t_5)) / Float64(1.0 + Float64(Float64(t_3 / t_6) * Float64(Float64(Float64(0.284496736 - Float64(Float64(Float64(Float64(1.453152027 - Float64(1.061405429 / t_6)) / t_0) - -1.421413741) / t_6)) / t_0) - -0.254829592))))
end

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 78.1%

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

7. Applied rewrites 79.3%

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

9. Applied rewrites 86.6%

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

Alternative 2: 79.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 78.1%

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

7. Applied rewrites 79.3%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. metadata-eval N/A

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

   4. add-flip N/A

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

   5. distribute-neg-in N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip-rev N/A

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

   8. lift-/.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-fma.f64 N/A

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

   16. metadata-eval 79.3

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

9. Applied rewrites 79.3%

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

    1. lift--.f64 N/A

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

    2. sub-negate-rev N/A

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

    3. metadata-eval N/A

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

    4. add-flip N/A

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

    5. distribute-neg-in N/A

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

    6. lift-/.f64 N/A

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

    7. mult-flip-rev N/A

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

    8. lift-/.f64 N/A

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

    9. *-commutative N/A

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

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

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

    11. metadata-eval N/A

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

    12. lower-fma.f64 N/A

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

    13. lift-fma.f64 N/A

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

    14. *-commutative N/A

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

    15. lift-fma.f64 N/A

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

    16. metadata-eval 79.3

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

11. Applied rewrites 79.3%

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

Alternative 3: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$1 * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(N[(1.453152027 / t$95$0), $MachinePrecision] + N[(N[(-1.061405429 / N[(N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.421413741), $MachinePrecision]), $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 79.2%

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

3. Applied rewrites 79.2%

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

Alternative 4: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, -0.6881592437815868, 1\right), \frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, 1.421413741\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

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
          (fma
           (fma
            (/ 1.061405429 (fma (fabs x) 0.3275911 1.0))
            -0.6881592437815868
            1.0)
           (/ 1.453152027 (fma -0.3275911 (fabs x) -1.0))
           1.421413741))))))
     (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 * fma(fma((1.061405429 / fma(fabs(x), 0.3275911, 1.0)), -0.6881592437815868, 1.0), (1.453152027 / fma(-0.3275911, fabs(x), -1.0)), 1.421413741)))))) * exp(-(fabs(x) * 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 * fma(fma(Float64(1.061405429 / fma(abs(x), 0.3275911, 1.0)), -0.6881592437815868, 1.0), Float64(1.453152027 / fma(-0.3275911, abs(x), -1.0)), 1.421413741)))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(N[(N[(1.061405429 / N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] * -0.6881592437815868 + 1.0), $MachinePrecision] * N[(1.453152027 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.421413741), $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 79.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. sum-to-mult N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

3. Applied rewrites 79.2%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, -0.6881592437815868, 1\right), \frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, 1.421413741\right)}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
4. Add Preprocessing

Alternative 5: 79.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-/.f64 N/A

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

   4. div-flip N/A

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

   5. associate-/r/ N/A

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

   6. lift-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 79.2%

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

Alternative 6: 79.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

Alternative 7: 79.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

Alternative 8: 77.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.7%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. lift-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lower-fma.f64 N/A

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

8. Applied rewrites 77.7%

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

Alternative 9: 77.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.7%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. lift-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 77.7

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

   13. lift-+.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. +-commutative N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 77.7

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

8. Applied rewrites 77.7%

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

Alternative 10: 77.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.7%

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

Reproduce

herbie shell --seed 2025156 
(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)))))))