Jmat.Real.erf

Percentage Accurate: 79.2% → 80.4%

Time: 8.0s

Alternatives: 18

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_2 := \frac{1.421413741}{t\_0}\\ t_3 := \mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)\\ t_4 := \frac{1}{t\_0}\\ t_5 := \frac{1.061405429}{t\_0} - 1.453152027\\ t_6 := e^{\left(-x\right) \cdot x}\\ t_7 := \left(t\_6 \cdot t\_4\right) \cdot \mathsf{fma}\left(t\_4, \left(t\_2 + \frac{t\_5}{t\_3 \cdot t\_3}\right) + -0.284496736, 0.254829592\right)\\ t_8 := {t\_7}^{3}\\ t_9 := \left({\left({t\_7}^{2}\right)}^{3} + t\_8\right) + 1\\ t_10 := \left(\mathsf{fma}\left(\frac{t\_5}{t\_1 \cdot t\_1} + \left(t\_2 + -0.284496736\right), t\_4, 0.254829592\right) \cdot t\_4\right) \cdot t\_6\\ \frac{\frac{1}{t\_9} - \frac{{t\_8}^{3}}{t\_9}}{1 + \left({t\_10}^{2} + 1 \cdot t\_10\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1 (fma -0.3275911 (fabs x) -1.0))
        (t_2 (/ 1.421413741 t_0))
        (t_3 (fma (fabs x) -0.3275911 -1.0))
        (t_4 (/ 1.0 t_0))
        (t_5 (- (/ 1.061405429 t_0) 1.453152027))
        (t_6 (exp (* (- x) x)))
        (t_7
         (*
          (* t_6 t_4)
          (fma t_4 (+ (+ t_2 (/ t_5 (* t_3 t_3))) -0.284496736) 0.254829592)))
        (t_8 (pow t_7 3.0))
        (t_9 (+ (+ (pow (pow t_7 2.0) 3.0) t_8) 1.0))
        (t_10
         (*
          (*
           (fma (+ (/ t_5 (* t_1 t_1)) (+ t_2 -0.284496736)) t_4 0.254829592)
           t_4)
          t_6)))
   (/
    (- (/ 1.0 t_9) (/ (pow t_8 3.0) t_9))
    (+ 1.0 (+ (pow t_10 2.0) (* 1.0 t_10))))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = fma(-0.3275911, fabs(x), -1.0);
	double t_2 = 1.421413741 / t_0;
	double t_3 = fma(fabs(x), -0.3275911, -1.0);
	double t_4 = 1.0 / t_0;
	double t_5 = (1.061405429 / t_0) - 1.453152027;
	double t_6 = exp((-x * x));
	double t_7 = (t_6 * t_4) * fma(t_4, ((t_2 + (t_5 / (t_3 * t_3))) + -0.284496736), 0.254829592);
	double t_8 = pow(t_7, 3.0);
	double t_9 = (pow(pow(t_7, 2.0), 3.0) + t_8) + 1.0;
	double t_10 = (fma(((t_5 / (t_1 * t_1)) + (t_2 + -0.284496736)), t_4, 0.254829592) * t_4) * t_6;
	return ((1.0 / t_9) - (pow(t_8, 3.0) / t_9)) / (1.0 + (pow(t_10, 2.0) + (1.0 * t_10)));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = fma(-0.3275911, abs(x), -1.0)
	t_2 = Float64(1.421413741 / t_0)
	t_3 = fma(abs(x), -0.3275911, -1.0)
	t_4 = Float64(1.0 / t_0)
	t_5 = Float64(Float64(1.061405429 / t_0) - 1.453152027)
	t_6 = exp(Float64(Float64(-x) * x))
	t_7 = Float64(Float64(t_6 * t_4) * fma(t_4, Float64(Float64(t_2 + Float64(t_5 / Float64(t_3 * t_3))) + -0.284496736), 0.254829592))
	t_8 = t_7 ^ 3.0
	t_9 = Float64(Float64(((t_7 ^ 2.0) ^ 3.0) + t_8) + 1.0)
	t_10 = Float64(Float64(fma(Float64(Float64(t_5 / Float64(t_1 * t_1)) + Float64(t_2 + -0.284496736)), t_4, 0.254829592) * t_4) * t_6)
	return Float64(Float64(Float64(1.0 / t_9) - Float64((t_8 ^ 3.0) / t_9)) / Float64(1.0 + Float64((t_10 ^ 2.0) + Float64(1.0 * t_10))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.421413741 / t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[x], $MachinePrecision] * -0.3275911 + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision]}, Block[{t$95$6 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$6 * t$95$4), $MachinePrecision] * N[(t$95$4 * N[(N[(t$95$2 + N[(t$95$5 / N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] + 0.254829592), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[Power[t$95$7, 3.0], $MachinePrecision]}, Block[{t$95$9 = N[(N[(N[Power[N[Power[t$95$7, 2.0], $MachinePrecision], 3.0], $MachinePrecision] + t$95$8), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$10 = N[(N[(N[(N[(N[(t$95$5 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + -0.284496736), $MachinePrecision]), $MachinePrecision] * t$95$4 + 0.254829592), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$6), $MachinePrecision]}, N[(N[(N[(1.0 / t$95$9), $MachinePrecision] - N[(N[Power[t$95$8, 3.0], $MachinePrecision] / t$95$9), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[t$95$10, 2.0], $MachinePrecision] + N[(1.0 * t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

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

7. Applied rewrites 79.3%

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

9. Applied rewrites 80.4%

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

Alternative 2: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := \frac{1.061405429}{t\_1} - 1.453152027\\ t_3 := \frac{1}{t\_1}\\ t_4 := e^{\left(-x\right) \cdot x}\\ t_5 := \mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)\\ t_6 := \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ t_7 := \frac{1.421413741}{t\_1} + -0.284496736\\ t_8 := \mathsf{fma}\left(t\_6, \frac{t\_2}{t\_5 \cdot t\_5} + t\_7, 0.254829592\right) \cdot \left(t\_6 \cdot t\_4\right)\\ t_9 := \left(\mathsf{fma}\left(\frac{t\_2}{t\_0 \cdot t\_0} + t\_7, t\_3, 0.254829592\right) \cdot t\_3\right) \cdot t\_4\\ t_10 := {t\_8}^{3}\\ \frac{\frac{1 - {t\_10}^{3}}{1 + \left({\left({t\_8}^{2}\right)}^{3} + t\_10\right)}}{1 + \left({t\_9}^{2} + 1 \cdot t\_9\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.3275911 (fabs x) -1.0))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2 (- (/ 1.061405429 t_1) 1.453152027))
        (t_3 (/ 1.0 t_1))
        (t_4 (exp (* (- x) x)))
        (t_5 (fma (fabs x) -0.3275911 -1.0))
        (t_6 (/ 1.0 (fma 0.3275911 (fabs x) 1.0)))
        (t_7 (+ (/ 1.421413741 t_1) -0.284496736))
        (t_8 (* (fma t_6 (+ (/ t_2 (* t_5 t_5)) t_7) 0.254829592) (* t_6 t_4)))
        (t_9 (* (* (fma (+ (/ t_2 (* t_0 t_0)) t_7) t_3 0.254829592) t_3) t_4))
        (t_10 (pow t_8 3.0)))
   (/
    (/ (- 1.0 (pow t_10 3.0)) (+ 1.0 (+ (pow (pow t_8 2.0) 3.0) t_10)))
    (+ 1.0 (+ (pow t_9 2.0) (* 1.0 t_9))))))

C

double code(double x) {
	double t_0 = fma(-0.3275911, fabs(x), -1.0);
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = (1.061405429 / t_1) - 1.453152027;
	double t_3 = 1.0 / t_1;
	double t_4 = exp((-x * x));
	double t_5 = fma(fabs(x), -0.3275911, -1.0);
	double t_6 = 1.0 / fma(0.3275911, fabs(x), 1.0);
	double t_7 = (1.421413741 / t_1) + -0.284496736;
	double t_8 = fma(t_6, ((t_2 / (t_5 * t_5)) + t_7), 0.254829592) * (t_6 * t_4);
	double t_9 = (fma(((t_2 / (t_0 * t_0)) + t_7), t_3, 0.254829592) * t_3) * t_4;
	double t_10 = pow(t_8, 3.0);
	return ((1.0 - pow(t_10, 3.0)) / (1.0 + (pow(pow(t_8, 2.0), 3.0) + t_10))) / (1.0 + (pow(t_9, 2.0) + (1.0 * t_9)));
}

Julia

function code(x)
	t_0 = fma(-0.3275911, abs(x), -1.0)
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = Float64(Float64(1.061405429 / t_1) - 1.453152027)
	t_3 = Float64(1.0 / t_1)
	t_4 = exp(Float64(Float64(-x) * x))
	t_5 = fma(abs(x), -0.3275911, -1.0)
	t_6 = Float64(1.0 / fma(0.3275911, abs(x), 1.0))
	t_7 = Float64(Float64(1.421413741 / t_1) + -0.284496736)
	t_8 = Float64(fma(t_6, Float64(Float64(t_2 / Float64(t_5 * t_5)) + t_7), 0.254829592) * Float64(t_6 * t_4))
	t_9 = Float64(Float64(fma(Float64(Float64(t_2 / Float64(t_0 * t_0)) + t_7), t_3, 0.254829592) * t_3) * t_4)
	t_10 = t_8 ^ 3.0
	return Float64(Float64(Float64(1.0 - (t_10 ^ 3.0)) / Float64(1.0 + Float64(((t_8 ^ 2.0) ^ 3.0) + t_10))) / Float64(1.0 + Float64((t_9 ^ 2.0) + Float64(1.0 * t_9))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.061405429 / t$95$1), $MachinePrecision] - 1.453152027), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[x], $MachinePrecision] * -0.3275911 + -1.0), $MachinePrecision]}, Block[{t$95$6 = N[(1.0 / N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(1.421413741 / t$95$1), $MachinePrecision] + -0.284496736), $MachinePrecision]}, Block[{t$95$8 = N[(N[(t$95$6 * N[(N[(t$95$2 / N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision] + t$95$7), $MachinePrecision] + 0.254829592), $MachinePrecision] * N[(t$95$6 * t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[(N[(N[(N[(t$95$2 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$7), $MachinePrecision] * t$95$3 + 0.254829592), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$4), $MachinePrecision]}, Block[{t$95$10 = N[Power[t$95$8, 3.0], $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[t$95$10, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[Power[t$95$8, 2.0], $MachinePrecision], 3.0], $MachinePrecision] + t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[t$95$9, 2.0], $MachinePrecision] + N[(1.0 * t$95$9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

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

7. Applied rewrites 79.3%

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

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{1 - {\left({\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}, \color{blue}{\left(\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right) + \frac{-8890523}{31250000}}, \frac{31853699}{125000000}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)}^{3}\right)}^{3}}{1 + \left({\left({\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}, \left(\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right) + \frac{-8890523}{31250000}, \frac{31853699}{125000000}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)}^{2}\right)}^{3} + {\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}, \left(\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right) + \frac{-8890523}{31250000}, \frac{31853699}{125000000}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)}^{3}\right)}}{1 + \left({\left(\left(\mathsf{fma}\left(\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \left(\frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, \frac{31853699}{125000000}\right) \cdot \frac{1}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{2} + 1 \cdot \left(\left(\mathsf{fma}\left(\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \left(\frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, \frac{31853699}{125000000}\right) \cdot \frac{1}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}\right) \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

9. Applied rewrites 79.3%

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

    1. lift-+.f64 N/A

       \[\leadsto \frac{\frac{1 - {\left({\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}, \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) \cdot \mathsf{fma}\left(\left|x\right|, \frac{-3275911}{10000000}, -1\right)} + \left(\frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}\right), \frac{31853699}{125000000}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)}^{3}\right)}^{3}}{1 + \left({\left({\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}, \color{blue}{\left(\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right) + \frac{-8890523}{31250000}}, \frac{31853699}{125000000}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)}^{2}\right)}^{3} + {\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}, \left(\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right) + \frac{-8890523}{31250000}, \frac{31853699}{125000000}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)}^{3}\right)}}{1 + \left({\left(\left(\mathsf{fma}\left(\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \left(\frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, \frac{31853699}{125000000}\right) \cdot \frac{1}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{2} + 1 \cdot \left(\left(\mathsf{fma}\left(\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \left(\frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, \frac{31853699}{125000000}\right) \cdot \frac{1}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}\right) \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

11. Applied rewrites 79.3%

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

    1. lift-+.f64 N/A

       \[\leadsto \frac{\frac{1 - {\left({\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}, \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) \cdot \mathsf{fma}\left(\left|x\right|, \frac{-3275911}{10000000}, -1\right)} + \left(\frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}\right), \frac{31853699}{125000000}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)}^{3}\right)}^{3}}{1 + \left({\left({\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}, \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) \cdot \mathsf{fma}\left(\left|x\right|, \frac{-3275911}{10000000}, -1\right)} + \left(\frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}\right), \frac{31853699}{125000000}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)}^{2}\right)}^{3} + {\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}, \color{blue}{\left(\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right) + \frac{-8890523}{31250000}}, \frac{31853699}{125000000}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} \cdot e^{\left(-x\right) \cdot x}\right)\right)}^{3}\right)}}{1 + \left({\left(\left(\mathsf{fma}\left(\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \left(\frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, \frac{31853699}{125000000}\right) \cdot \frac{1}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}\right) \cdot e^{\left(-x\right) \cdot x}\right)}^{2} + 1 \cdot \left(\left(\mathsf{fma}\left(\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right) \cdot \mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \left(\frac{\frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, \frac{31853699}{125000000}\right) \cdot \frac{1}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}\right) \cdot e^{\left(-x\right) \cdot x}\right)\right)} \]

13. Applied rewrites 79.3%

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

Alternative 3: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \frac{1}{t\_0}\\ t_2 := e^{\left(-x\right) \cdot x}\\ t_3 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_4 := t\_3 \cdot t\_3\\ t_5 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_6 := \frac{1}{t\_5}\\ t_7 := \mathsf{fma}\left(t\_6, \left(\frac{\frac{1.061405429}{t\_5} - 1.453152027}{t\_4} + \frac{1.421413741}{t\_5}\right) + -0.284496736, 0.254829592\right) \cdot \left(t\_6 \cdot t\_2\right)\\ t_8 := {t\_7}^{3}\\ t_9 := \left(\mathsf{fma}\left(\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_4} + \left(\frac{1.421413741}{t\_0} + -0.284496736\right), t\_1, 0.254829592\right) \cdot t\_1\right) \cdot t\_2\\ \frac{\frac{1 - {t\_8}^{3}}{1 + \left({\left({t\_7}^{2}\right)}^{3} + t\_8\right)}}{1 + \left({t\_9}^{2} + 1 \cdot t\_9\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1 (/ 1.0 t_0))
        (t_2 (exp (* (- x) x)))
        (t_3 (fma -0.3275911 (fabs x) -1.0))
        (t_4 (* t_3 t_3))
        (t_5 (fma 0.3275911 (fabs x) 1.0))
        (t_6 (/ 1.0 t_5))
        (t_7
         (*
          (fma
           t_6
           (+
            (+ (/ (- (/ 1.061405429 t_5) 1.453152027) t_4) (/ 1.421413741 t_5))
            -0.284496736)
           0.254829592)
          (* t_6 t_2)))
        (t_8 (pow t_7 3.0))
        (t_9
         (*
          (*
           (fma
            (+
             (/ (- (/ 1.061405429 t_0) 1.453152027) t_4)
             (+ (/ 1.421413741 t_0) -0.284496736))
            t_1
            0.254829592)
           t_1)
          t_2)))
   (/
    (/ (- 1.0 (pow t_8 3.0)) (+ 1.0 (+ (pow (pow t_7 2.0) 3.0) t_8)))
    (+ 1.0 (+ (pow t_9 2.0) (* 1.0 t_9))))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = 1.0 / t_0;
	double t_2 = exp((-x * x));
	double t_3 = fma(-0.3275911, fabs(x), -1.0);
	double t_4 = t_3 * t_3;
	double t_5 = fma(0.3275911, fabs(x), 1.0);
	double t_6 = 1.0 / t_5;
	double t_7 = fma(t_6, (((((1.061405429 / t_5) - 1.453152027) / t_4) + (1.421413741 / t_5)) + -0.284496736), 0.254829592) * (t_6 * t_2);
	double t_8 = pow(t_7, 3.0);
	double t_9 = (fma(((((1.061405429 / t_0) - 1.453152027) / t_4) + ((1.421413741 / t_0) + -0.284496736)), t_1, 0.254829592) * t_1) * t_2;
	return ((1.0 - pow(t_8, 3.0)) / (1.0 + (pow(pow(t_7, 2.0), 3.0) + t_8))) / (1.0 + (pow(t_9, 2.0) + (1.0 * t_9)));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = Float64(1.0 / t_0)
	t_2 = exp(Float64(Float64(-x) * x))
	t_3 = fma(-0.3275911, abs(x), -1.0)
	t_4 = Float64(t_3 * t_3)
	t_5 = fma(0.3275911, abs(x), 1.0)
	t_6 = Float64(1.0 / t_5)
	t_7 = Float64(fma(t_6, Float64(Float64(Float64(Float64(Float64(1.061405429 / t_5) - 1.453152027) / t_4) + Float64(1.421413741 / t_5)) + -0.284496736), 0.254829592) * Float64(t_6 * t_2))
	t_8 = t_7 ^ 3.0
	t_9 = Float64(Float64(fma(Float64(Float64(Float64(Float64(1.061405429 / t_0) - 1.453152027) / t_4) + Float64(Float64(1.421413741 / t_0) + -0.284496736)), t_1, 0.254829592) * t_1) * t_2)
	return Float64(Float64(Float64(1.0 - (t_8 ^ 3.0)) / Float64(1.0 + Float64(((t_7 ^ 2.0) ^ 3.0) + t_8))) / Float64(1.0 + Float64((t_9 ^ 2.0) + Float64(1.0 * t_9))))
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 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$6 = N[(1.0 / t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$6 * N[(N[(N[(N[(N[(1.061405429 / t$95$5), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$4), $MachinePrecision] + N[(1.421413741 / t$95$5), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] + 0.254829592), $MachinePrecision] * N[(t$95$6 * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[Power[t$95$7, 3.0], $MachinePrecision]}, Block[{t$95$9 = N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$4), $MachinePrecision] + N[(N[(1.421413741 / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision]), $MachinePrecision] * t$95$1 + 0.254829592), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[t$95$8, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[Power[t$95$7, 2.0], $MachinePrecision], 3.0], $MachinePrecision] + t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[t$95$9, 2.0], $MachinePrecision] + N[(1.0 * t$95$9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

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

7. Applied rewrites 79.3%

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

Alternative 4: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_3 := \frac{1}{t\_2}\\ t_4 := e^{\left(-x\right) \cdot x}\\ t_5 := \mathsf{fma}\left(t\_3, \left(\frac{\frac{1.061405429}{t\_2} - 1.453152027}{t\_1} + \frac{1.421413741}{t\_2}\right) + -0.284496736, 0.254829592\right) \cdot \left(t\_3 \cdot t\_4\right)\\ t_6 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_7 := \frac{1}{t\_6}\\ t_8 := \left(\mathsf{fma}\left(\frac{\frac{1.061405429}{t\_6} - 1.453152027}{t\_1} + \left(\frac{1.421413741}{t\_6} + -0.284496736\right), t\_7, 0.254829592\right) \cdot t\_7\right) \cdot t\_4\\ \frac{\frac{1 - {\left({t\_5}^{2}\right)}^{3}}{1 + {t\_5}^{3}}}{1 + \left({t\_8}^{2} + 1 \cdot t\_8\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.3275911 (fabs x) -1.0))
        (t_1 (* t_0 t_0))
        (t_2 (fma 0.3275911 (fabs x) 1.0))
        (t_3 (/ 1.0 t_2))
        (t_4 (exp (* (- x) x)))
        (t_5
         (*
          (fma
           t_3
           (+
            (+ (/ (- (/ 1.061405429 t_2) 1.453152027) t_1) (/ 1.421413741 t_2))
            -0.284496736)
           0.254829592)
          (* t_3 t_4)))
        (t_6 (fma (fabs x) 0.3275911 1.0))
        (t_7 (/ 1.0 t_6))
        (t_8
         (*
          (*
           (fma
            (+
             (/ (- (/ 1.061405429 t_6) 1.453152027) t_1)
             (+ (/ 1.421413741 t_6) -0.284496736))
            t_7
            0.254829592)
           t_7)
          t_4)))
   (/
    (/ (- 1.0 (pow (pow t_5 2.0) 3.0)) (+ 1.0 (pow t_5 3.0)))
    (+ 1.0 (+ (pow t_8 2.0) (* 1.0 t_8))))))

C

double code(double x) {
	double t_0 = fma(-0.3275911, fabs(x), -1.0);
	double t_1 = t_0 * t_0;
	double t_2 = fma(0.3275911, fabs(x), 1.0);
	double t_3 = 1.0 / t_2;
	double t_4 = exp((-x * x));
	double t_5 = fma(t_3, (((((1.061405429 / t_2) - 1.453152027) / t_1) + (1.421413741 / t_2)) + -0.284496736), 0.254829592) * (t_3 * t_4);
	double t_6 = fma(fabs(x), 0.3275911, 1.0);
	double t_7 = 1.0 / t_6;
	double t_8 = (fma(((((1.061405429 / t_6) - 1.453152027) / t_1) + ((1.421413741 / t_6) + -0.284496736)), t_7, 0.254829592) * t_7) * t_4;
	return ((1.0 - pow(pow(t_5, 2.0), 3.0)) / (1.0 + pow(t_5, 3.0))) / (1.0 + (pow(t_8, 2.0) + (1.0 * t_8)));
}

Julia

function code(x)
	t_0 = fma(-0.3275911, abs(x), -1.0)
	t_1 = Float64(t_0 * t_0)
	t_2 = fma(0.3275911, abs(x), 1.0)
	t_3 = Float64(1.0 / t_2)
	t_4 = exp(Float64(Float64(-x) * x))
	t_5 = Float64(fma(t_3, Float64(Float64(Float64(Float64(Float64(1.061405429 / t_2) - 1.453152027) / t_1) + Float64(1.421413741 / t_2)) + -0.284496736), 0.254829592) * Float64(t_3 * t_4))
	t_6 = fma(abs(x), 0.3275911, 1.0)
	t_7 = Float64(1.0 / t_6)
	t_8 = Float64(Float64(fma(Float64(Float64(Float64(Float64(1.061405429 / t_6) - 1.453152027) / t_1) + Float64(Float64(1.421413741 / t_6) + -0.284496736)), t_7, 0.254829592) * t_7) * t_4)
	return Float64(Float64(Float64(1.0 - ((t_5 ^ 2.0) ^ 3.0)) / Float64(1.0 + (t_5 ^ 3.0))) / Float64(1.0 + Float64((t_8 ^ 2.0) + Float64(1.0 * t_8))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$3 * N[(N[(N[(N[(N[(1.061405429 / t$95$2), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(1.421413741 / t$95$2), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] + 0.254829592), $MachinePrecision] * N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$7 = N[(1.0 / t$95$6), $MachinePrecision]}, Block[{t$95$8 = N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$6), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(1.421413741 / t$95$6), $MachinePrecision] + -0.284496736), $MachinePrecision]), $MachinePrecision] * t$95$7 + 0.254829592), $MachinePrecision] * t$95$7), $MachinePrecision] * t$95$4), $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[N[Power[t$95$5, 2.0], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Power[t$95$5, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[t$95$8, 2.0], $MachinePrecision] + N[(1.0 * t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

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

7. Applied rewrites 79.3%

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

Alternative 5: 79.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

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

Alternative 6: 79.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

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

Alternative 7: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

Alternative 8: 79.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

Alternative 9: 79.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

6. Applied rewrites 79.2%

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

Alternative 10: 79.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

5. Applied rewrites 79.2%

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

Alternative 11: 79.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

Alternative 12: 79.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

Alternative 13: 79.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

Alternative 14: 78.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-fabs.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-fma.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 78.6

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

6. Applied rewrites 78.6%

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

Alternative 15: 78.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. lower-fma.f64 78.6

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

6. Applied rewrites 78.6%

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

Alternative 16: 77.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-fabs.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-fma.f64 N/A

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

   11. lift-fabs.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 78.6

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

6. Applied rewrites 78.6%

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

7. Taylor expanded in x around 0

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 77.6

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

9. Applied rewrites 77.6%

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

Alternative 17: 55.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

6. Applied rewrites 55.7%

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

Alternative 18: 54.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{0.254829592 - \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  1.0
  (/
   (- 0.254829592 (/ 0.284496736 (fma 0.3275911 (fabs x) 1.0)))
   (fma (fabs x) 0.3275911 1.0))))

C

double code(double x) {
	return 1.0 - ((0.254829592 - (0.284496736 / fma(0.3275911, fabs(x), 1.0))) / fma(fabs(x), 0.3275911, 1.0));
}

Julia

function code(x)
	return Float64(1.0 - Float64(Float64(0.254829592 - Float64(0.284496736 / fma(0.3275911, abs(x), 1.0))) / fma(abs(x), 0.3275911, 1.0)))
end

Wolfram

code[x_] := N[(1.0 - N[(N[(0.254829592 - N[(0.284496736 / N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

3. Applied rewrites 79.2%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

6. Applied rewrites 55.7%

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

7. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-fabs.f64 N/A

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

   7. lower-fma.f64 54.6

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

9. Applied rewrites 54.6%

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

Reproduce

herbie shell --seed 2025142 
(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)))))))