Jmat.Real.erf

Percentage Accurate: 79.1% → 80.3%

Time: 9.8s

Alternatives: 12

Speedup: 1.3×

Specification

Math

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

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: 79.1% accurate, 1.0× speedup

Math

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

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: 80.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

5. Applied rewrites 79.2%

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

7. Applied rewrites 80.3%

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

Alternative 2: 79.2% accurate, 0.3× speedup

Math

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

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

5. Applied rewrites 79.2%

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

7. Applied rewrites 78.0%

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

9. Applied rewrites 78.0%

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

11. Applied rewrites 78.0%

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

13. Applied rewrites 79.2%

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

Alternative 3: 79.1% accurate, 0.4× speedup

Math

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

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
         (*
          (-
           (/
            (-
             (/
              (-
               (/
                (- (/ (- (/ -1.061405429 t_0) -1.453152027) t_0) 1.421413741)
                t_0)
               -0.284496736)
              t_1)
             -0.254829592)
            t_1))
          (exp (* (- x) x)))))
   (/ (- (* 1.0 1.0) (* t_2 t_2)) (+ 1.0 t_2))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

Alternative 4: 79.1% accurate, 0.5× speedup

Math

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

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_1) 1.421413741)
          t_1)))
   (/
    (fma
     (pow (/ (exp (* (- x) x)) t_1) 2.0)
     (pow (- (/ (- -0.284496736 t_2) t_1) -0.254829592) 2.0)
     -1.0)
    (-
     (/ (- (/ (- t_2 -0.284496736) t_0) -0.254829592) (* t_0 (exp (* x x))))
     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 = ((((-1.061405429 / t_1) - -1.453152027) / t_1) - 1.421413741) / t_1;
	return fma(pow((exp((-x * x)) / t_1), 2.0), pow((((-0.284496736 - t_2) / t_1) - -0.254829592), 2.0), -1.0) / (((((t_2 - -0.284496736) / t_0) - -0.254829592) / (t_0 * exp((x * x)))) - 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 = Float64(Float64(Float64(Float64(Float64(-1.061405429 / t_1) - -1.453152027) / t_1) - 1.421413741) / t_1)
	return Float64(fma((Float64(exp(Float64(Float64(-x) * x)) / t_1) ^ 2.0), (Float64(Float64(Float64(-0.284496736 - t_2) / t_1) - -0.254829592) ^ 2.0), -1.0) / Float64(Float64(Float64(Float64(Float64(t_2 - -0.284496736) / t_0) - -0.254829592) / Float64(t_0 * exp(Float64(x * x)))) - 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[(N[(N[(N[(N[(-1.061405429 / t$95$1), $MachinePrecision] - -1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision]}, N[(N[(N[Power[N[(N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(N[(N[(-0.284496736 - t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision] - -0.254829592), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(N[(N[(N[(t$95$2 - -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.254829592), $MachinePrecision] / N[(t$95$0 * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

5. Applied rewrites 79.2%

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

7. Applied rewrites 79.1%

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

Alternative 5: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

Alternative 6: 79.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

5. Applied rewrites 79.1%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. associate--r- N/A

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

   6. lower-+.f64 N/A

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

7. Applied rewrites 79.1%

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

Alternative 7: 79.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

Alternative 8: 79.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

5. Applied rewrites 79.1%

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

Alternative 9: 79.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

Alternative 10: 77.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

5. Applied rewrites 79.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 77.5%

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

Alternative 11: 77.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

5. Applied rewrites 79.1%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}, \frac{-0.284496736 - \frac{-1.421413741 - \frac{-1.453152027 - \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -0.254829592, 1\right) \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fabs.f64 77.5%

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

8. Applied rewrites 77.5%

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

10. Applied rewrites 77.5%

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

Alternative 12: 77.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Applied rewrites 79.1%

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

5. Applied rewrites 79.1%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}, \frac{-0.284496736 - \frac{-1.421413741 - \frac{-1.453152027 - \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -0.254829592, 1\right) \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fabs.f64 77.5%

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

8. Applied rewrites 77.5%

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

10. Applied rewrites 77.5%

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

Reproduce

herbie shell --seed 2025207 
(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)))))))