Jmat.Real.erf

Percentage Accurate: 79.3% → 79.4%

Time: 7.4s

Alternatives: 10

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ \frac{1}{1 \cdot \frac{t\_1}{t\_1 - e^{\left(-x\right) \cdot x} \cdot \left(\left(\frac{0.284496736}{t\_0} + \frac{\frac{\frac{-1.061405429}{t\_0} - 1.453152027}{t\_1} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)}\right) - -0.254829592\right)}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.3275911 (fabs x) -1.0))
        (t_1 (fma (fabs x) 0.3275911 1.0)))
   (/
    1.0
    (*
     1.0
     (/
      t_1
      (-
       t_1
       (*
        (exp (* (- x) x))
        (-
         (+
          (/ 0.284496736 t_0)
          (/
           (- (/ (- (/ -1.061405429 t_0) 1.453152027) t_1) -1.421413741)
           (fma (* 0.10731592879921 x) x (fma 0.6551822 (fabs x) 1.0))))
         -0.254829592))))))))

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);
	return 1.0 / (1.0 * (t_1 / (t_1 - (exp((-x * x)) * (((0.284496736 / t_0) + (((((-1.061405429 / t_0) - 1.453152027) / t_1) - -1.421413741) / fma((0.10731592879921 * x), x, fma(0.6551822, fabs(x), 1.0)))) - -0.254829592)))));
}

Julia

function code(x)
	t_0 = fma(-0.3275911, abs(x), -1.0)
	t_1 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 / Float64(1.0 * Float64(t_1 / Float64(t_1 - Float64(exp(Float64(Float64(-x) * x)) * Float64(Float64(Float64(0.284496736 / t_0) + Float64(Float64(Float64(Float64(Float64(-1.061405429 / t_0) - 1.453152027) / t_1) - -1.421413741) / fma(Float64(0.10731592879921 * x), x, fma(0.6551822, abs(x), 1.0)))) - -0.254829592))))))
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]}, N[(1.0 / N[(1.0 * N[(t$95$1 / N[(t$95$1 - N[(N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(0.284496736 / t$95$0), $MachinePrecision] + N[(N[(N[(N[(N[(-1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(N[(0.10731592879921 * x), $MachinePrecision] * x + N[(0.6551822 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.4%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\left(\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921, x \cdot x, 0.6551822 \cdot \left|x\right|\right) + 1} + \frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
4. Step-by-step derivation

5. Applied rewrites 79.4%

   \[\leadsto 1 - \color{blue}{\frac{\left(\frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)}\right) + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \cdot e^{-x \cdot x}} \]

7. Applied rewrites 79.4%

   \[\leadsto \color{blue}{\frac{2 \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) - 2 \cdot \left(\left(\left(\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)} + \frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)}\right) - -0.254829592\right) \cdot e^{\left(-x\right) \cdot x}\right)}{2 \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

9. Applied rewrites 79.4%

   \[\leadsto \color{blue}{\frac{1}{1 \cdot \frac{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) - e^{\left(-x\right) \cdot x} \cdot \left(\left(\frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)}\right) - -0.254829592\right)}}} \]
10. Add Preprocessing

Alternative 2: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ 1 \cdot \frac{t\_1 - e^{\left(-x\right) \cdot x} \cdot \left(\left(\frac{0.284496736}{t\_0} + \frac{\frac{\frac{-1.061405429}{t\_0} - 1.453152027}{t\_1} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)}\right) - -0.254829592\right)}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.3275911 (fabs x) -1.0))
        (t_1 (fma (fabs x) 0.3275911 1.0)))
   (*
    1.0
    (/
     (-
      t_1
      (*
       (exp (* (- x) x))
       (-
        (+
         (/ 0.284496736 t_0)
         (/
          (- (/ (- (/ -1.061405429 t_0) 1.453152027) t_1) -1.421413741)
          (fma (* 0.10731592879921 x) x (fma 0.6551822 (fabs x) 1.0))))
        -0.254829592)))
     t_1))))

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);
	return 1.0 * ((t_1 - (exp((-x * x)) * (((0.284496736 / t_0) + (((((-1.061405429 / t_0) - 1.453152027) / t_1) - -1.421413741) / fma((0.10731592879921 * x), x, fma(0.6551822, fabs(x), 1.0)))) - -0.254829592))) / t_1);
}

Julia

function code(x)
	t_0 = fma(-0.3275911, abs(x), -1.0)
	t_1 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 * Float64(Float64(t_1 - Float64(exp(Float64(Float64(-x) * x)) * Float64(Float64(Float64(0.284496736 / t_0) + Float64(Float64(Float64(Float64(Float64(-1.061405429 / t_0) - 1.453152027) / t_1) - -1.421413741) / fma(Float64(0.10731592879921 * x), x, fma(0.6551822, abs(x), 1.0)))) - -0.254829592))) / t_1))
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]}, N[(1.0 * N[(N[(t$95$1 - N[(N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(0.284496736 / t$95$0), $MachinePrecision] + N[(N[(N[(N[(N[(-1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(N[(0.10731592879921 * x), $MachinePrecision] * x + N[(0.6551822 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.4%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\left(\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921, x \cdot x, 0.6551822 \cdot \left|x\right|\right) + 1} + \frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
4. Step-by-step derivation

5. Applied rewrites 79.4%

   \[\leadsto 1 - \color{blue}{\frac{\left(\frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)}\right) + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \cdot e^{-x \cdot x}} \]

7. Applied rewrites 79.4%

   \[\leadsto \color{blue}{\frac{2 \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) - 2 \cdot \left(\left(\left(\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)} + \frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)}\right) - -0.254829592\right) \cdot e^{\left(-x\right) \cdot x}\right)}{2 \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

9. Applied rewrites 79.4%

   \[\leadsto \color{blue}{1 \cdot \frac{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) - e^{\left(-x\right) \cdot x} \cdot \left(\left(\frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + \frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)}\right) - -0.254829592\right)}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \]
10. Add Preprocessing

Alternative 3: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)\\ 1 - \frac{\left(\left(\frac{\frac{\frac{-1.061405429}{t\_1} - 1.453152027}{t\_0} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)} + \frac{0.284496736}{t\_1}\right) - -0.254829592\right) \cdot e^{\left(-x\right) \cdot x}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1 (fma (fabs x) -0.3275911 -1.0)))
   (-
    1.0
    (/
     (*
      (-
       (+
        (/
         (- (/ (- (/ -1.061405429 t_1) 1.453152027) t_0) -1.421413741)
         (fma (* 0.10731592879921 x) x (fma 0.6551822 (fabs x) 1.0)))
        (/ 0.284496736 t_1))
       -0.254829592)
      (exp (* (- x) x)))
     t_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);
	return 1.0 - (((((((((-1.061405429 / t_1) - 1.453152027) / t_0) - -1.421413741) / fma((0.10731592879921 * x), x, fma(0.6551822, fabs(x), 1.0))) + (0.284496736 / t_1)) - -0.254829592) * exp((-x * x))) / t_0);
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = fma(abs(x), -0.3275911, -1.0)
	return Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1.061405429 / t_1) - 1.453152027) / t_0) - -1.421413741) / fma(Float64(0.10731592879921 * x), x, fma(0.6551822, abs(x), 1.0))) + Float64(0.284496736 / t_1)) - -0.254829592) * exp(Float64(Float64(-x) * x))) / t_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]}, N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(N[(-1.061405429 / t$95$1), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(N[(0.10731592879921 * x), $MachinePrecision] * x + N[(0.6551822 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.284496736 / t$95$1), $MachinePrecision]), $MachinePrecision] - -0.254829592), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.4%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\left(\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921, x \cdot x, 0.6551822 \cdot \left|x\right|\right) + 1} + \frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
4. Step-by-step derivation

5. Applied rewrites 79.4%

   \[\leadsto 1 - \color{blue}{\frac{\left(\frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)}\right) + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \cdot e^{-x \cdot x}} \]

7. Applied rewrites 79.4%

   \[\leadsto 1 - \color{blue}{\frac{\left(\left(\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)} + \frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)}\right) - -0.254829592\right) \cdot e^{\left(-x\right) \cdot x}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]
8. Add Preprocessing

Alternative 4: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ 1 - \frac{\frac{\left(\frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + \frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)}\right) + 0.254829592}{t\_0}}{e^{x \cdot x}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (-
    1.0
    (/
     (/
      (+
       (+
        (/ 0.284496736 (fma -0.3275911 (fabs x) -1.0))
        (/
         (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741)
         (fma (* 0.10731592879921 x) x (fma 0.6551822 (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 - (((((0.284496736 / fma(-0.3275911, fabs(x), -1.0)) + (((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / fma((0.10731592879921 * x), x, fma(0.6551822, 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(0.284496736 / fma(-0.3275911, abs(x), -1.0)) + Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / fma(Float64(0.10731592879921 * x), x, fma(0.6551822, abs(x), 1.0)))) + 0.254829592) / 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[(0.284496736 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(N[(0.10731592879921 * x), $MachinePrecision] * x + N[(0.6551822 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision] / N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.4%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\left(\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921, x \cdot x, 0.6551822 \cdot \left|x\right|\right) + 1} + \frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

5. Applied rewrites 79.4%

   \[\leadsto 1 - \color{blue}{\frac{\frac{\left(\frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)}\right) + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{e^{x \cdot x}}} \]
6. Add Preprocessing

Alternative 5: 79.3% accurate, 1.3× speedup

Math

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

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 79.3%

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

3. Applied rewrites 79.3%

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

Alternative 6: 79.3% accurate, 1.3× speedup

Math

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

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 79.3%

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

3. Applied rewrites 79.3%

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

Alternative 7: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ 1 - \frac{\left(\frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + \frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)}\right) + 0.254829592}{t\_0} \cdot 1 \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (-
    1.0
    (*
     (/
      (+
       (+
        (/ 0.284496736 (fma -0.3275911 (fabs x) -1.0))
        (/
         (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741)
         (fma (* 0.10731592879921 x) x (fma 0.6551822 (fabs x) 1.0))))
       0.254829592)
      t_0)
     1.0))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 - (((((0.284496736 / fma(-0.3275911, fabs(x), -1.0)) + (((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / fma((0.10731592879921 * x), x, fma(0.6551822, fabs(x), 1.0)))) + 0.254829592) / t_0) * 1.0);
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.284496736 / fma(-0.3275911, abs(x), -1.0)) + Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / fma(Float64(0.10731592879921 * x), x, fma(0.6551822, abs(x), 1.0)))) + 0.254829592) / t_0) * 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[(N[(N[(0.284496736 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] - 1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(N[(0.10731592879921 * x), $MachinePrecision] * x + N[(0.6551822 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.4%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\left(\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921, x \cdot x, 0.6551822 \cdot \left|x\right|\right) + 1} + \frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
4. Step-by-step derivation

5. Applied rewrites 79.4%

   \[\leadsto 1 - \color{blue}{\frac{\left(\frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)}\right) + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \cdot e^{-x \cdot x}} \]

6. Taylor expanded in x around 0

   \[\leadsto 1 - \frac{\left(\frac{\frac{8890523}{31250000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} - \frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\frac{10731592879921}{100000000000000} \cdot x, x, \mathsf{fma}\left(\frac{3275911}{5000000}, \left|x\right|, 1\right)\right)}\right) + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 77.9%

   \[\leadsto 1 - \frac{\left(\frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921 \cdot x, x, \mathsf{fma}\left(0.6551822, \left|x\right|, 1\right)\right)}\right) + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \cdot \color{blue}{1} \]
9. Add Preprocessing

Alternative 8: 77.9% accurate, 1.6× speedup

Math

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

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))
    1.0
    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)), 1.0, 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)), 1.0, 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] * 1.0 + 1.0), $MachinePrecision]]

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.3%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 77.9%

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

Alternative 9: 56.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.4%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\left(\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921, x \cdot x, 0.6551822 \cdot \left|x\right|\right) + 1} + \frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} + \frac{8890523}{31250000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   4. lower-/.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} + \frac{8890523}{31250000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   5. lower--.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} + \frac{8890523}{31250000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   6. lower-*.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} + \frac{8890523}{31250000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   7. lower-fabs.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} + \frac{8890523}{31250000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-fabs.f64 56.0

       \[\leadsto 1 - \frac{0.254829592 + 0.284496736 \cdot \frac{1}{-0.3275911 \cdot \left|x\right| - 1}}{1 + 0.3275911 \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

6. Applied rewrites 56.0%

   \[\leadsto 1 - \color{blue}{\frac{0.254829592 + 0.284496736 \cdot \frac{1}{-0.3275911 \cdot \left|x\right| - 1}}{1 + 0.3275911 \cdot \left|x\right|}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

8. Applied rewrites 56.0%

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

Alternative 10: 55.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Applied rewrites 79.4%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\left(\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.10731592879921, x \cdot x, 0.6551822 \cdot \left|x\right|\right) + 1} + \frac{0.284496736}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} + \frac{8890523}{31250000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   4. lower-/.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} + \frac{8890523}{31250000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   5. lower--.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} + \frac{8890523}{31250000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   6. lower-*.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} + \frac{8890523}{31250000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   7. lower-fabs.f64 N/A

      \[\leadsto 1 - \frac{\frac{31853699}{125000000} + \frac{8890523}{31250000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-fabs.f64 56.0

       \[\leadsto 1 - \frac{0.254829592 + 0.284496736 \cdot \frac{1}{-0.3275911 \cdot \left|x\right| - 1}}{1 + 0.3275911 \cdot \left|x\right|} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

6. Applied rewrites 56.0%

   \[\leadsto 1 - \color{blue}{\frac{0.254829592 + 0.284496736 \cdot \frac{1}{-0.3275911 \cdot \left|x\right| - 1}}{1 + 0.3275911 \cdot \left|x\right|}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

8. Applied rewrites 56.0%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 55.1%

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

Reproduce

herbie shell --seed 2025141 
(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)))))))