Jmat.Real.erf

Percentage Accurate: 78.8% → 80.0%

Time: 11.2s

Alternatives: 15

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.8% 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: 80.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1}\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({t\_1}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{t\_1}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\\ t_3 := 1 + \left({t\_2}^{6} + 1 \cdot {t\_2}^{3}\right)\\ \frac{\frac{1}{t\_3} - \frac{{t\_2}^{9}}{t\_3}}{1 + \mathsf{fma}\left(t\_2, t\_2, 1 \cdot t\_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (* 0.3275911 (fabs x))) -1.0))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2
         (*
          (*
           t_0
           (+
            0.254829592
            (*
             t_0
             (+
              -0.284496736
              (fma
               (/
                (fma (pow t_1 -2.0) 1.061405429 1.421413741)
                (+ 1.0 (* -0.10731592879921 (* x x))))
               (- 1.0 (* (fabs x) 0.3275911))
               (/ (/ 1.453152027 (fma -0.3275911 (fabs x) -1.0)) t_1))))))
          (exp (* (- x) x))))
        (t_3 (+ 1.0 (+ (pow t_2 6.0) (* 1.0 (pow t_2 3.0))))))
   (/
    (- (/ 1.0 t_3) (/ (pow t_2 9.0) t_3))
    (+ 1.0 (fma t_2 t_2 (* 1.0 t_2))))))

C

double code(double x) {
	double t_0 = pow((1.0 + (0.3275911 * fabs(x))), -1.0);
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = (t_0 * (0.254829592 + (t_0 * (-0.284496736 + fma((fma(pow(t_1, -2.0), 1.061405429, 1.421413741) / (1.0 + (-0.10731592879921 * (x * x)))), (1.0 - (fabs(x) * 0.3275911)), ((1.453152027 / fma(-0.3275911, fabs(x), -1.0)) / t_1)))))) * exp((-x * x));
	double t_3 = 1.0 + (pow(t_2, 6.0) + (1.0 * pow(t_2, 3.0)));
	return ((1.0 / t_3) - (pow(t_2, 9.0) / t_3)) / (1.0 + fma(t_2, t_2, (1.0 * t_2)));
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x))) ^ -1.0
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + fma(Float64(fma((t_1 ^ -2.0), 1.061405429, 1.421413741) / Float64(1.0 + Float64(-0.10731592879921 * Float64(x * x)))), Float64(1.0 - Float64(abs(x) * 0.3275911)), Float64(Float64(1.453152027 / fma(-0.3275911, abs(x), -1.0)) / t_1)))))) * exp(Float64(Float64(-x) * x)))
	t_3 = Float64(1.0 + Float64((t_2 ^ 6.0) + Float64(1.0 * (t_2 ^ 3.0))))
	return Float64(Float64(Float64(1.0 / t_3) - Float64((t_2 ^ 9.0) / t_3)) / Float64(1.0 + fma(t_2, t_2, Float64(1.0 * t_2))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(N[(N[(N[Power[t$95$1, -2.0], $MachinePrecision] * 1.061405429 + 1.421413741), $MachinePrecision] / N[(1.0 + N[(-0.10731592879921 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] + N[(N[(1.453152027 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(N[Power[t$95$2, 6.0], $MachinePrecision] + N[(1.0 * N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / t$95$3), $MachinePrecision] - N[(N[Power[t$95$2, 9.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$2 * t$95$2 + N[(1.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

5. Applied rewrites 78.8%

   \[\leadsto \color{blue}{\frac{1 - {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \mathsf{fma}\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, \left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, 1 \cdot \left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)\right)}} \]

7. Applied rewrites 78.9%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left({\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}\right)}^{3}}{1 + \mathsf{fma}\left({\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}, {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}, 1 \cdot {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}\right)}}}{1 + \mathsf{fma}\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, \left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, 1 \cdot \left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

9. Applied rewrites 80.0%

   \[\leadsto \frac{\color{blue}{\frac{1}{1 + \left({\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{6} + 1 \cdot {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}\right)} - \frac{{\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{9}}{1 + \left({\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{6} + 1 \cdot {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}\right)}}}{1 + \mathsf{fma}\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, \left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, 1 \cdot \left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]
10. Add Preprocessing

Alternative 2: 78.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1}\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({t\_1}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{t\_1}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\\ t_3 := {t\_2}^{3}\\ \frac{\frac{1 - {t\_3}^{3}}{1 + \mathsf{fma}\left(t\_3, t\_3, 1 \cdot t\_3\right)}}{1 + \mathsf{fma}\left(t\_2, t\_2, 1 \cdot t\_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (* 0.3275911 (fabs x))) -1.0))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2
         (*
          (*
           t_0
           (+
            0.254829592
            (*
             t_0
             (+
              -0.284496736
              (fma
               (/
                (fma (pow t_1 -2.0) 1.061405429 1.421413741)
                (+ 1.0 (* -0.10731592879921 (* x x))))
               (- 1.0 (* (fabs x) 0.3275911))
               (/ (/ 1.453152027 (fma -0.3275911 (fabs x) -1.0)) t_1))))))
          (exp (* (- x) x))))
        (t_3 (pow t_2 3.0)))
   (/
    (/ (- 1.0 (pow t_3 3.0)) (+ 1.0 (fma t_3 t_3 (* 1.0 t_3))))
    (+ 1.0 (fma t_2 t_2 (* 1.0 t_2))))))

C

double code(double x) {
	double t_0 = pow((1.0 + (0.3275911 * fabs(x))), -1.0);
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = (t_0 * (0.254829592 + (t_0 * (-0.284496736 + fma((fma(pow(t_1, -2.0), 1.061405429, 1.421413741) / (1.0 + (-0.10731592879921 * (x * x)))), (1.0 - (fabs(x) * 0.3275911)), ((1.453152027 / fma(-0.3275911, fabs(x), -1.0)) / t_1)))))) * exp((-x * x));
	double t_3 = pow(t_2, 3.0);
	return ((1.0 - pow(t_3, 3.0)) / (1.0 + fma(t_3, t_3, (1.0 * t_3)))) / (1.0 + fma(t_2, t_2, (1.0 * t_2)));
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x))) ^ -1.0
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + fma(Float64(fma((t_1 ^ -2.0), 1.061405429, 1.421413741) / Float64(1.0 + Float64(-0.10731592879921 * Float64(x * x)))), Float64(1.0 - Float64(abs(x) * 0.3275911)), Float64(Float64(1.453152027 / fma(-0.3275911, abs(x), -1.0)) / t_1)))))) * exp(Float64(Float64(-x) * x)))
	t_3 = t_2 ^ 3.0
	return Float64(Float64(Float64(1.0 - (t_3 ^ 3.0)) / Float64(1.0 + fma(t_3, t_3, Float64(1.0 * t_3)))) / Float64(1.0 + fma(t_2, t_2, Float64(1.0 * t_2))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(N[(N[(N[Power[t$95$1, -2.0], $MachinePrecision] * 1.061405429 + 1.421413741), $MachinePrecision] / N[(1.0 + N[(-0.10731592879921 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] + N[(N[(1.453152027 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 3.0], $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$3 * t$95$3 + N[(1.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$2 * t$95$2 + N[(1.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

5. Applied rewrites 78.8%

   \[\leadsto \color{blue}{\frac{1 - {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \mathsf{fma}\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, \left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, 1 \cdot \left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)\right)}} \]

7. Applied rewrites 78.9%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left({\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}\right)}^{3}}{1 + \mathsf{fma}\left({\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}, {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}, 1 \cdot {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}\right)}}}{1 + \mathsf{fma}\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, \left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, 1 \cdot \left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]
8. Add Preprocessing

Alternative 3: 78.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1}\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({t\_1}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{t\_1}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\\ \frac{\frac{1 - {t\_2}^{9}}{1 + \left({t\_2}^{6} + 1 \cdot {t\_2}^{3}\right)}}{1 + \mathsf{fma}\left(t\_2, t\_2, 1 \cdot t\_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (* 0.3275911 (fabs x))) -1.0))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2
         (*
          (*
           t_0
           (+
            0.254829592
            (*
             t_0
             (+
              -0.284496736
              (fma
               (/
                (fma (pow t_1 -2.0) 1.061405429 1.421413741)
                (+ 1.0 (* -0.10731592879921 (* x x))))
               (- 1.0 (* (fabs x) 0.3275911))
               (/ (/ 1.453152027 (fma -0.3275911 (fabs x) -1.0)) t_1))))))
          (exp (* (- x) x)))))
   (/
    (/ (- 1.0 (pow t_2 9.0)) (+ 1.0 (+ (pow t_2 6.0) (* 1.0 (pow t_2 3.0)))))
    (+ 1.0 (fma t_2 t_2 (* 1.0 t_2))))))

C

double code(double x) {
	double t_0 = pow((1.0 + (0.3275911 * fabs(x))), -1.0);
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = (t_0 * (0.254829592 + (t_0 * (-0.284496736 + fma((fma(pow(t_1, -2.0), 1.061405429, 1.421413741) / (1.0 + (-0.10731592879921 * (x * x)))), (1.0 - (fabs(x) * 0.3275911)), ((1.453152027 / fma(-0.3275911, fabs(x), -1.0)) / t_1)))))) * exp((-x * x));
	return ((1.0 - pow(t_2, 9.0)) / (1.0 + (pow(t_2, 6.0) + (1.0 * pow(t_2, 3.0))))) / (1.0 + fma(t_2, t_2, (1.0 * t_2)));
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x))) ^ -1.0
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + fma(Float64(fma((t_1 ^ -2.0), 1.061405429, 1.421413741) / Float64(1.0 + Float64(-0.10731592879921 * Float64(x * x)))), Float64(1.0 - Float64(abs(x) * 0.3275911)), Float64(Float64(1.453152027 / fma(-0.3275911, abs(x), -1.0)) / t_1)))))) * exp(Float64(Float64(-x) * x)))
	return Float64(Float64(Float64(1.0 - (t_2 ^ 9.0)) / Float64(1.0 + Float64((t_2 ^ 6.0) + Float64(1.0 * (t_2 ^ 3.0))))) / Float64(1.0 + fma(t_2, t_2, Float64(1.0 * t_2))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(N[(N[(N[Power[t$95$1, -2.0], $MachinePrecision] * 1.061405429 + 1.421413741), $MachinePrecision] / N[(1.0 + N[(-0.10731592879921 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] + N[(N[(1.453152027 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[t$95$2, 9.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[t$95$2, 6.0], $MachinePrecision] + N[(1.0 * N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$2 * t$95$2 + N[(1.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

5. Applied rewrites 78.8%

   \[\leadsto \color{blue}{\frac{1 - {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \mathsf{fma}\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, \left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, 1 \cdot \left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)\right)}} \]

7. Applied rewrites 78.9%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left({\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}\right)}^{3}}{1 + \mathsf{fma}\left({\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}, {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}, 1 \cdot {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}\right)}}}{1 + \mathsf{fma}\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, \left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, 1 \cdot \left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

9. Applied rewrites 78.9%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{9}}{1 + \left({\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{6} + 1 \cdot {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}\right)}}}{1 + \mathsf{fma}\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, \left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, 1 \cdot \left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]
10. Add Preprocessing

Alternative 4: 78.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1}\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({t\_1}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{t\_1}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\\ \frac{1 - {t\_2}^{3}}{1 + \left({t\_2}^{2} + 1 \cdot t\_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (* 0.3275911 (fabs x))) -1.0))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2
         (*
          (*
           t_0
           (+
            0.254829592
            (*
             t_0
             (+
              -0.284496736
              (fma
               (/
                (fma (pow t_1 -2.0) 1.061405429 1.421413741)
                (+ 1.0 (* -0.10731592879921 (* x x))))
               (- 1.0 (* (fabs x) 0.3275911))
               (/ (/ 1.453152027 (fma -0.3275911 (fabs x) -1.0)) t_1))))))
          (exp (* (- x) x)))))
   (/ (- 1.0 (pow t_2 3.0)) (+ 1.0 (+ (pow t_2 2.0) (* 1.0 t_2))))))

C

double code(double x) {
	double t_0 = pow((1.0 + (0.3275911 * fabs(x))), -1.0);
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = (t_0 * (0.254829592 + (t_0 * (-0.284496736 + fma((fma(pow(t_1, -2.0), 1.061405429, 1.421413741) / (1.0 + (-0.10731592879921 * (x * x)))), (1.0 - (fabs(x) * 0.3275911)), ((1.453152027 / fma(-0.3275911, fabs(x), -1.0)) / t_1)))))) * exp((-x * x));
	return (1.0 - pow(t_2, 3.0)) / (1.0 + (pow(t_2, 2.0) + (1.0 * t_2)));
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x))) ^ -1.0
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + fma(Float64(fma((t_1 ^ -2.0), 1.061405429, 1.421413741) / Float64(1.0 + Float64(-0.10731592879921 * Float64(x * x)))), Float64(1.0 - Float64(abs(x) * 0.3275911)), Float64(Float64(1.453152027 / fma(-0.3275911, abs(x), -1.0)) / t_1)))))) * exp(Float64(Float64(-x) * x)))
	return Float64(Float64(1.0 - (t_2 ^ 3.0)) / Float64(1.0 + Float64((t_2 ^ 2.0) + Float64(1.0 * t_2))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(N[(N[(N[Power[t$95$1, -2.0], $MachinePrecision] * 1.061405429 + 1.421413741), $MachinePrecision] / N[(1.0 + N[(-0.10731592879921 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] + N[(N[(1.453152027 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[t$95$2, 2.0], $MachinePrecision] + N[(1.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

5. Applied rewrites 78.8%

   \[\leadsto \color{blue}{\frac{1 - {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \mathsf{fma}\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, \left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}, 1 \cdot \left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)\right)}} \]

7. Applied rewrites 78.8%

   \[\leadsto \color{blue}{\frac{1 - {\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{3}}{1 + \left({\left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{2} + 1 \cdot \left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)\right)}} \]
8. Add Preprocessing

Alternative 5: 78.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1}\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({t\_1}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{t\_1}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\\ \frac{1 - t\_2 \cdot t\_2}{1 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (* 0.3275911 (fabs x))) -1.0))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2
         (*
          (*
           t_0
           (+
            0.254829592
            (*
             t_0
             (+
              -0.284496736
              (fma
               (/
                (fma (pow t_1 -2.0) 1.061405429 1.421413741)
                (+ 1.0 (* -0.10731592879921 (* x x))))
               (- 1.0 (* (fabs x) 0.3275911))
               (/ (/ 1.453152027 (fma -0.3275911 (fabs x) -1.0)) t_1))))))
          (exp (* (- x) x)))))
   (/ (- 1.0 (* t_2 t_2)) (+ 1.0 t_2))))

C

double code(double x) {
	double t_0 = pow((1.0 + (0.3275911 * fabs(x))), -1.0);
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = (t_0 * (0.254829592 + (t_0 * (-0.284496736 + fma((fma(pow(t_1, -2.0), 1.061405429, 1.421413741) / (1.0 + (-0.10731592879921 * (x * x)))), (1.0 - (fabs(x) * 0.3275911)), ((1.453152027 / fma(-0.3275911, fabs(x), -1.0)) / t_1)))))) * exp((-x * x));
	return (1.0 - (t_2 * t_2)) / (1.0 + t_2);
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x))) ^ -1.0
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + fma(Float64(fma((t_1 ^ -2.0), 1.061405429, 1.421413741) / Float64(1.0 + Float64(-0.10731592879921 * Float64(x * x)))), Float64(1.0 - Float64(abs(x) * 0.3275911)), Float64(Float64(1.453152027 / fma(-0.3275911, abs(x), -1.0)) / t_1)))))) * exp(Float64(Float64(-x) * x)))
	return Float64(Float64(1.0 - Float64(t_2 * t_2)) / Float64(1.0 + t_2))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(N[(N[(N[Power[t$95$1, -2.0], $MachinePrecision] * 1.061405429 + 1.421413741), $MachinePrecision] / N[(1.0 + N[(-0.10731592879921 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] + N[(N[(1.453152027 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

5. Applied rewrites 78.8%

   \[\leadsto \color{blue}{\frac{1 - \left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right) \cdot \left(\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\right)}{1 + \left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 + -0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}}} \]
6. Add Preprocessing

Alternative 6: 78.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t\_1 \cdot \left(0.254829592 + t\_1 \cdot \left(-0.284496736 + \mathsf{fma}\left(\frac{\mathsf{fma}\left({t\_0}^{-2}, 1.061405429, 1.421413741\right)}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{t\_0}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))))
   (-
    1.0
    (*
     (*
      t_1
      (+
       0.254829592
       (*
        t_1
        (+
         -0.284496736
         (fma
          (/
           (fma (pow t_0 -2.0) 1.061405429 1.421413741)
           (- 1.0 (* 0.10731592879921 (* x x))))
          (- 1.0 (* (fabs x) 0.3275911))
          (/ (/ 1.453152027 (fma -0.3275911 (fabs x) -1.0)) t_0))))))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	return 1.0 - ((t_1 * (0.254829592 + (t_1 * (-0.284496736 + fma((fma(pow(t_0, -2.0), 1.061405429, 1.421413741) / (1.0 - (0.10731592879921 * (x * x)))), (1.0 - (fabs(x) * 0.3275911)), ((1.453152027 / fma(-0.3275911, fabs(x), -1.0)) / t_0)))))) * exp(-(fabs(x) * fabs(x))));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	return Float64(1.0 - Float64(Float64(t_1 * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + fma(Float64(fma((t_0 ^ -2.0), 1.061405429, 1.421413741) / Float64(1.0 - Float64(0.10731592879921 * Float64(x * x)))), Float64(1.0 - Float64(abs(x) * 0.3275911)), Float64(Float64(1.453152027 / fma(-0.3275911, abs(x), -1.0)) / t_0)))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$1 * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(N[(N[(N[Power[t$95$0, -2.0], $MachinePrecision] * 1.061405429 + 1.421413741), $MachinePrecision] / N[(1.0 - N[(0.10731592879921 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] + N[(N[(1.453152027 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $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 78.8%

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

3. Applied rewrites 78.8%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-2}, 1.061405429, 1.421413741\right)}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
4. Add Preprocessing

Alternative 7: 78.8% accurate, 1.2× speedup

Math

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

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 78.8%

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

3. Applied rewrites 78.8%

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

Alternative 8: 78.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \mathsf{fma}\left(\frac{1.061405429}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, -1.453152027\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + 0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)} \end{array} \end{array} \]

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
            (fma
             (/ 1.061405429 (- 1.0 (* 0.10731592879921 (* x x))))
             (- 1.0 (* (fabs x) 0.3275911))
             -1.453152027))))))))
     (/
      1.0
      (+
       1.0
       (*
        (* x x)
        (+ 1.0 (* (* x x) (+ 0.5 (* 0.16666666666666666 (* x 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 * fma((1.061405429 / (1.0 - (0.10731592879921 * (x * x)))), (1.0 - (fabs(x) * 0.3275911)), -1.453152027)))))))) * (1.0 / (1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + (0.16666666666666666 * (x * 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 * fma(Float64(1.061405429 / Float64(1.0 - Float64(0.10731592879921 * Float64(x * x)))), Float64(1.0 - Float64(abs(x) * 0.3275911)), -1.453152027)))))))) * Float64(1.0 / Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(0.16666666666666666 * Float64(x * 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[(N[(1.061405429 / N[(1.0 - N[(0.10731592879921 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-*l/ N/A

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

   9. metadata-eval N/A

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

   10. flip-+ N/A

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

   11. associate-/r/ N/A

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

   12. lower-fma.f64 N/A

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

3. Applied rewrites 78.8%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. exp-neg N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. sqr-abs-rev N/A

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

   9. pow2 N/A

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

   10. lower-exp.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 78.8

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

5. Applied rewrites 78.8%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot \frac{3275911}{10000000}, \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}} \]
7. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot \frac{3275911}{10000000}, \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot \frac{3275911}{10000000}, \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot \frac{1}{1 + {x}^{2} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}} \]

   3. pow2 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot \frac{3275911}{10000000}, \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(x \cdot x\right) \cdot \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot \frac{3275911}{10000000}, \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(x \cdot x\right) \cdot \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot \frac{3275911}{10000000}, \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(x \cdot x\right) \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot \frac{3275911}{10000000}, \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(x \cdot x\right) \cdot \left(1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)} \]

   7. pow2 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot \frac{3275911}{10000000}, \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot {x}^{2}\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot \frac{3275911}{10000000}, \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot {x}^{2}\right)\right)} \]

   9. lower-+.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot \frac{3275911}{10000000}, \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{6} \cdot {x}^{2}}\right)\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot \frac{3275911}{10000000}, \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \color{blue}{{x}^{2}}\right)\right)} \]

   11. pow2 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 - \frac{10731592879921}{100000000000000} \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot \frac{3275911}{10000000}, \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)} \]

   12. lift-*.f64 78.6

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{1.061405429}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, -1.453152027\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + 0.16666666666666666 \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)} \]

8. Applied rewrites 78.6%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{1.061405429}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, -1.453152027\right)\right)\right)\right)\right) \cdot \frac{1}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + 0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}} \]
9. Add Preprocessing

Alternative 9: 78.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \mathsf{fma}\left(\frac{1.061405429}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, -1.453152027\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(x \cdot x\right) \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)} \end{array} \end{array} \]

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
            (fma
             (/ 1.061405429 (- 1.0 (* 0.10731592879921 (* x x))))
             (- 1.0 (* (fabs x) 0.3275911))
             -1.453152027))))))))
     (/ 1.0 (+ 1.0 (* (* x x) (+ 1.0 (* 0.5 (* x 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 * fma((1.061405429 / (1.0 - (0.10731592879921 * (x * x)))), (1.0 - (fabs(x) * 0.3275911)), -1.453152027)))))))) * (1.0 / (1.0 + ((x * x) * (1.0 + (0.5 * (x * 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 * fma(Float64(1.061405429 / Float64(1.0 - Float64(0.10731592879921 * Float64(x * x)))), Float64(1.0 - Float64(abs(x) * 0.3275911)), -1.453152027)))))))) * Float64(1.0 / Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(0.5 * Float64(x * 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[(N[(1.061405429 / N[(1.0 - N[(0.10731592879921 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-*l/ N/A

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

   9. metadata-eval N/A

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

   10. flip-+ N/A

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

   11. associate-/r/ N/A

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

   12. lower-fma.f64 N/A

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

3. Applied rewrites 78.8%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. exp-neg N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. sqr-abs-rev N/A

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

   9. pow2 N/A

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

   10. lower-exp.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 78.8

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

5. Applied rewrites 78.8%

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

6. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 78.5

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{1.061405429}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, -1.453152027\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(x \cdot x\right) \cdot \left(1 + 0.5 \cdot \left(x \cdot \color{blue}{x}\right)\right)} \]

8. Applied rewrites 78.5%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \mathsf{fma}\left(\frac{1.061405429}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, -1.453152027\right)\right)\right)\right)\right) \cdot \frac{1}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}} \]
9. Add Preprocessing

Alternative 10: 54.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := -0.3275911 \cdot \left|x\right|\\ \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{\frac{t\_1 \cdot t\_1 - 1}{t\_1 - -1}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) - 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)) (t_1 (* -0.3275911 (fabs x))))
   (if (<= x 1.25)
     (fma
      (/
       (+
        (/
         (+
          (/ (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741) t_0)
          -0.284496736)
         t_0)
        0.254829592)
       (/ (- (* t_1 t_1) 1.0) (- t_1 -1.0)))
      (+
       1.0
       (*
        (* x x)
        (- (* (* x x) (+ 0.5 (* -0.16666666666666666 (* x x)))) 1.0)))
      1.0)
     1.0)))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = -0.3275911 * fabs(x);
	double tmp;
	if (x <= 1.25) {
		tmp = fma((((((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / (((t_1 * t_1) - 1.0) / (t_1 - -1.0))), (1.0 + ((x * x) * (((x * x) * (0.5 + (-0.16666666666666666 * (x * x)))) - 1.0))), 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = Float64(-0.3275911 * abs(x))
	tmp = 0.0
	if (x <= 1.25)
		tmp = 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) / Float64(Float64(Float64(t_1 * t_1) - 1.0) / Float64(t_1 - -1.0))), Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(0.5 + Float64(-0.16666666666666666 * Float64(x * x)))) - 1.0))), 1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(-0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.25], 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[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - 1.0), $MachinePrecision] / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 71.8%

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

   3. Applied rewrites 71.8%

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

   4. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, \color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right)}, 1\right) \]
   5. Step-by-step derivation

      1. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      4. sqr-abs-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right)}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + {x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right)}, 1\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right)} - 1\right), 1\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right)} - 1\right), 1\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - \color{blue}{1}\right), 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      16. lift-*.f64 40.3

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

   6. Applied rewrites 40.3%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, \color{blue}{1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) - 1\right)}, 1\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\color{blue}{\frac{-3275911}{10000000} \cdot \left|x\right| + -1}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      2. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{-3275911}{10000000} \cdot \color{blue}{\left|x\right|} + -1}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\color{blue}{\frac{\left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) - -1 \cdot -1}{\frac{-3275911}{10000000} \cdot \left|x\right| - -1}}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\color{blue}{\frac{\left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) - -1 \cdot -1}{\frac{-3275911}{10000000} \cdot \left|x\right| - -1}}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) - \color{blue}{1}}{\frac{-3275911}{10000000} \cdot \left|x\right| - -1}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\color{blue}{\left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) - 1}}{\frac{-3275911}{10000000} \cdot \left|x\right| - -1}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\color{blue}{\left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right|\right)} - 1}{\frac{-3275911}{10000000} \cdot \left|x\right| - -1}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      8. lift-fabs.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\left(\frac{-3275911}{10000000} \cdot \color{blue}{\left|x\right|}\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) - 1}{\frac{-3275911}{10000000} \cdot \left|x\right| - -1}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\color{blue}{\left(\frac{-3275911}{10000000} \cdot \left|x\right|\right)} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) - 1}{\frac{-3275911}{10000000} \cdot \left|x\right| - -1}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      10. lift-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \color{blue}{\left|x\right|}\right) - 1}{\frac{-3275911}{10000000} \cdot \left|x\right| - -1}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \color{blue}{\left(\frac{-3275911}{10000000} \cdot \left|x\right|\right)} - 1}{\frac{-3275911}{10000000} \cdot \left|x\right| - -1}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) - 1}{\color{blue}{\frac{-3275911}{10000000} \cdot \left|x\right| - -1}}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      13. lift-fabs.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) - 1}{\frac{-3275911}{10000000} \cdot \color{blue}{\left|x\right|} - -1}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      14. lower-*.f64 39.8

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\frac{\left(-0.3275911 \cdot \left|x\right|\right) \cdot \left(-0.3275911 \cdot \left|x\right|\right) - 1}{\color{blue}{-0.3275911 \cdot \left|x\right|} - -1}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

   8. Applied rewrites 39.8%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\color{blue}{\frac{\left(-0.3275911 \cdot \left|x\right|\right) \cdot \left(-0.3275911 \cdot \left|x\right|\right) - 1}{-0.3275911 \cdot \left|x\right| - -1}}}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

   if 1.25 < x

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 55.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) - 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (if (<= x 1.25)
     (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))
      (+
       1.0
       (*
        (* x x)
        (- (* (* x x) (+ 0.5 (* -0.16666666666666666 (* x x)))) 1.0)))
      1.0)
     1.0)))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double tmp;
	if (x <= 1.25) {
		tmp = 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)), (1.0 + ((x * x) * (((x * x) * (0.5 + (-0.16666666666666666 * (x * x)))) - 1.0))), 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	tmp = 0.0
	if (x <= 1.25)
		tmp = 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)), Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(0.5 + Float64(-0.16666666666666666 * Float64(x * x)))) - 1.0))), 1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, If[LessEqual[x, 1.25], 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[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 71.8%

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

   3. Applied rewrites 71.8%

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

   4. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, \color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right)}, 1\right) \]
   5. Step-by-step derivation

      1. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      4. sqr-abs-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right)}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + {x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right)}, 1\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right)} - 1\right), 1\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right)} - 1\right), 1\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - \color{blue}{1}\right), 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right), 1\right) \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

      16. lift-*.f64 40.3

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) - 1\right), 1\right) \]

   6. Applied rewrites 40.3%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, \color{blue}{1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) - 1\right)}, 1\right) \]

   if 1.25 < x

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 54.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(0.5 \cdot \left(x \cdot x\right) - 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (if (<= x 1.15)
     (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))
      (+ 1.0 (* (* x x) (- (* 0.5 (* x x)) 1.0)))
      1.0)
     1.0)))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double tmp;
	if (x <= 1.15) {
		tmp = 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)), (1.0 + ((x * x) * ((0.5 * (x * x)) - 1.0))), 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	tmp = 0.0
	if (x <= 1.15)
		tmp = 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)), Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.5 * Float64(x * x)) - 1.0))), 1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, If[LessEqual[x, 1.15], 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[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 71.8%

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

   3. Applied rewrites 71.8%

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

   4. Taylor expanded in x around 0

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

      1. distribute-lft-neg-out N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. sqr-abs-rev N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 39.3

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, 1 + \left(x \cdot x\right) \cdot \left(0.5 \cdot \left(x \cdot x\right) - 1\right), 1\right) \]

   6. Applied rewrites 39.3%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 \cdot \left(x \cdot x\right) - 1\right)}, 1\right) \]

   if 1.1499999999999999 < x

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 55.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double tmp;
	if (x <= 1.0) {
		tmp = 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)), (1.0 + (-1.0 * (x * x))), 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	tmp = 0.0
	if (x <= 1.0)
		tmp = 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)), Float64(1.0 + Float64(-1.0 * Float64(x * x))), 1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 71.8%

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

   3. Applied rewrites 71.8%

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

   4. Taylor expanded in x around 0

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

      1. distribute-lft-neg-out N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. sqr-abs-rev N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 40.6

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

   6. Applied rewrites 40.6%

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

   if 1 < x

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 77.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ \mathbf{if}\;x \leq 0.84:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (if (<= x 0.84)
     (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))
      1.0
      1.0)
     1.0)))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double tmp;
	if (x <= 0.84) {
		tmp = 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)), 1.0, 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	tmp = 0.0
	if (x <= 0.84)
		tmp = 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)), 1.0, 1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, If[LessEqual[x, 0.84], 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] * 1.0 + 1.0), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < 0.839999999999999969

   1. Initial program 71.8%

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

   3. Applied rewrites 71.8%

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

   4. Taylor expanded in x around 0

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

      1. distribute-lft-neg-out 70.5

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

      2. pow2 70.5

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

      3. pow2 70.5

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

      4. sqr-abs-rev 70.5

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

   6. Applied rewrites 70.5%

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

   if 0.839999999999999969 < x

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 54.8% accurate, 262.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 78.8%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 54.8%

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

Reproduce

herbie shell --seed 2025093 
(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)))))))