Jmat.Real.erf

Percentage Accurate: 79.1% → 99.9%

Time: 7.2s

Alternatives: 15

Speedup: 37.3×

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: 79.1% 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: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ \mathbf{if}\;x\_m \leq 0.00038:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + x\_m \cdot \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - \left|x\_m\right| \cdot 0.3275911, \frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{\mathsf{fma}\left(x\_m \cdot x\_m, -0.10731592879921, 1\right)}, -0.284496736\right), {t\_0}^{-1}, 0.254829592\right)}{t\_0} \cdot e^{-\left|x\_m\right| \cdot \left|x\_m\right|}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (if (<= x_m 0.00038)
     (+
      1e-9
      (*
       x_m
       (+
        1.128386358070218
        (* x_m (- (* -0.37545125292247583 x_m) 0.00011824294398844343)))))
     (-
      1.0
      (*
       (/
        (*
         1.0
         (fma
          (fma
           (- 1.0 (* (fabs x_m) 0.3275911))
           (/
            (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741)
            (fma (* x_m x_m) -0.10731592879921 1.0))
           -0.284496736)
          (pow t_0 -1.0)
          0.254829592))
        t_0)
       (exp (- (* (fabs x_m) (fabs x_m)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double tmp;
	if (x_m <= 0.00038) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((-0.37545125292247583 * x_m) - 0.00011824294398844343))));
	} else {
		tmp = 1.0 - (((1.0 * fma(fma((1.0 - (fabs(x_m) * 0.3275911)), (((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / fma((x_m * x_m), -0.10731592879921, 1.0)), -0.284496736), pow(t_0, -1.0), 0.254829592)) / t_0) * exp(-(fabs(x_m) * fabs(x_m))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	tmp = 0.0
	if (x_m <= 0.00038)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(x_m * Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343)))));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 * fma(fma(Float64(1.0 - Float64(abs(x_m) * 0.3275911)), Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / fma(Float64(x_m * x_m), -0.10731592879921, 1.0)), -0.284496736), (t_0 ^ -1.0), 0.254829592)) / t_0) * exp(Float64(-Float64(abs(x_m) * abs(x_m))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00038], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(x$95$m * N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(1.0 * N[(N[(N[(1.0 - N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $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[(x$95$m * x$95$m), $MachinePrecision] * -0.10731592879921 + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] * N[Power[t$95$0, -1.0], $MachinePrecision] + 0.254829592), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[Exp[(-N[(N[Abs[x$95$m], $MachinePrecision] * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.8000000000000002e-4

   1. Initial program 57.9%

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

   3. Applied rewrites 57.9%

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

   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 57.6%

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

   8. Applied rewrites 57.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \color{blue}{x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \color{blue}{\left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \color{blue}{\frac{2364858879768868679}{20000000000000000000000}}\right)\right) \]

       6. lower-*.f64 100.0

          \[\leadsto 10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right)} \]

   if 3.8000000000000002e-4 < x

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   5. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ t_1 := \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\\ \mathbf{if}\;x\_m \leq 0.00066:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + x\_m \cdot \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(\frac{\frac{\frac{t\_1 \cdot t\_1 - 2.111650813574209}{t\_1 + 1.453152027}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} - \frac{-1.421413741}{t\_0}\right) + -0.284496736}{t\_0} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)}, e^{\left(-x\_m\right) \cdot x\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0))
        (t_1 (/ 1.061405429 (fma 0.3275911 x_m 1.0))))
   (if (<= x_m 0.00066)
     (+
      1e-9
      (*
       x_m
       (+
        1.128386358070218
        (* x_m (- (* -0.37545125292247583 x_m) 0.00011824294398844343)))))
     (fma
      (/
       (+
        (/
         (+
          (-
           (/
            (/
             (/ (- (* t_1 t_1) 2.111650813574209) (+ t_1 1.453152027))
             (fma 0.3275911 x_m 1.0))
            (fma 0.3275911 x_m 1.0))
           (/ -1.421413741 t_0))
          -0.284496736)
         t_0)
        0.254829592)
       (fma -0.3275911 (fabs x_m) -1.0))
      (exp (* (- x_m) x_m))
      1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double t_1 = 1.061405429 / fma(0.3275911, x_m, 1.0);
	double tmp;
	if (x_m <= 0.00066) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((-0.37545125292247583 * x_m) - 0.00011824294398844343))));
	} else {
		tmp = fma(((((((((((t_1 * t_1) - 2.111650813574209) / (t_1 + 1.453152027)) / fma(0.3275911, x_m, 1.0)) / fma(0.3275911, x_m, 1.0)) - (-1.421413741 / t_0)) + -0.284496736) / t_0) + 0.254829592) / fma(-0.3275911, fabs(x_m), -1.0)), exp((-x_m * x_m)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	t_1 = Float64(1.061405429 / fma(0.3275911, x_m, 1.0))
	tmp = 0.0
	if (x_m <= 0.00066)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(x_m * Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343)))));
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(t_1 * t_1) - 2.111650813574209) / Float64(t_1 + 1.453152027)) / fma(0.3275911, x_m, 1.0)) / fma(0.3275911, x_m, 1.0)) - Float64(-1.421413741 / t_0)) + -0.284496736) / t_0) + 0.254829592) / fma(-0.3275911, abs(x_m), -1.0)), exp(Float64(Float64(-x_m) * x_m)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00066], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(x$95$m * N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - 2.111650813574209), $MachinePrecision] / N[(t$95$1 + 1.453152027), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.421413741 / t$95$0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.6e-4

   1. Initial program 57.9%

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

   3. Applied rewrites 57.9%

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

   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 57.5%

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

   8. Applied rewrites 57.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \color{blue}{x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \color{blue}{\left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \color{blue}{\frac{2364858879768868679}{20000000000000000000000}}\right)\right) \]

       6. lower-*.f64 99.9

          \[\leadsto 10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right)} \]

   if 6.6e-4 < x

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fabs.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{3275911}{10000000} \cdot x + 1} \cdot \frac{\frac{1061405429}{1000000000}}{\frac{3275911}{10000000} \cdot x + 1} - \color{blue}{\frac{2111650813574208729}{1000000000000000000}}}{\frac{\frac{1061405429}{1000000000}}{\frac{3275911}{10000000} \cdot x + 1} + \frac{1453152027}{1000000000}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)} - \frac{\frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}\right) + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, e^{\left(-x\right) \cdot x}, 1\right) \]

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)} \cdot \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)} - \color{blue}{\frac{2111650813574208729}{1000000000000000000}}}{\frac{\frac{1061405429}{1000000000}}{\frac{3275911}{10000000} \cdot x + 1} + \frac{1453152027}{1000000000}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)} - \frac{\frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}\right) + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, e^{\left(-x\right) \cdot x}, 1\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)} \cdot \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)} - \frac{2111650813574208729}{1000000000000000000}}{\color{blue}{\frac{\frac{1061405429}{1000000000}}{\frac{3275911}{10000000} \cdot x + 1} + \frac{1453152027}{1000000000}}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)} - \frac{\frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}\right) + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, e^{\left(-x\right) \cdot x}, 1\right) \]

      16. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)} \cdot \frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)} - \frac{2111650813574208729}{1000000000000000000}}{\frac{\frac{1061405429}{1000000000}}{\color{blue}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)}} + \frac{1453152027}{1000000000}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, x, 1\right)} - \frac{\frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}\right) + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, e^{\left(-x\right) \cdot x}, 1\right) \]

      17. lift-/.f64 99.9

          \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} - 2.111650813574209}{\color{blue}{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}} + 1.453152027}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} - \frac{-1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right) + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, e^{\left(-x\right) \cdot x}, 1\right) \]

   9. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\frac{\frac{\color{blue}{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} - 2.111650813574209}{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} + 1.453152027}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} - \frac{-1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right) + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, e^{\left(-x\right) \cdot x}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ \mathbf{if}\;x\_m \leq 0.00052:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + x\_m \cdot \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} - \frac{-1.421413741}{t\_0}}{t\_0} + \frac{-0.284496736}{t\_0}\right) + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)}, e^{\left(-x\_m\right) \cdot x\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (if (<= x_m 0.00052)
     (+
      1e-9
      (*
       x_m
       (+
        1.128386358070218
        (* x_m (- (* -0.37545125292247583 x_m) 0.00011824294398844343)))))
     (fma
      (/
       (+
        (+
         (/
          (-
           (/
            (/
             (- (/ 1.061405429 (fma 0.3275911 x_m 1.0)) 1.453152027)
             (fma 0.3275911 x_m 1.0))
            (fma 0.3275911 x_m 1.0))
           (/ -1.421413741 t_0))
          t_0)
         (/ -0.284496736 t_0))
        0.254829592)
       (fma -0.3275911 (fabs x_m) -1.0))
      (exp (* (- x_m) x_m))
      1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double tmp;
	if (x_m <= 0.00052) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((-0.37545125292247583 * x_m) - 0.00011824294398844343))));
	} else {
		tmp = fma((((((((((1.061405429 / fma(0.3275911, x_m, 1.0)) - 1.453152027) / fma(0.3275911, x_m, 1.0)) / fma(0.3275911, x_m, 1.0)) - (-1.421413741 / t_0)) / t_0) + (-0.284496736 / t_0)) + 0.254829592) / fma(-0.3275911, fabs(x_m), -1.0)), exp((-x_m * x_m)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	tmp = 0.0
	if (x_m <= 0.00052)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(x_m * Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343)))));
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / fma(0.3275911, x_m, 1.0)) - 1.453152027) / fma(0.3275911, x_m, 1.0)) / fma(0.3275911, x_m, 1.0)) - Float64(-1.421413741 / t_0)) / t_0) + Float64(-0.284496736 / t_0)) + 0.254829592) / fma(-0.3275911, abs(x_m), -1.0)), exp(Float64(Float64(-x_m) * x_m)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00052], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(x$95$m * N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - 1.453152027), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.421413741 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(-0.284496736 / t$95$0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.19999999999999954e-4

   1. Initial program 57.9%

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

   3. Applied rewrites 57.9%

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

   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 57.5%

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

   8. Applied rewrites 57.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \color{blue}{x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \color{blue}{\left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \color{blue}{\frac{2364858879768868679}{20000000000000000000000}}\right)\right) \]

       6. lower-*.f64 99.9

          \[\leadsto 10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right)} \]

   if 5.19999999999999954e-4 < x

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fabs.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

Alternative 4: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ \mathbf{if}\;x\_m \leq 0.00052:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + x\_m \cdot \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} - \left(\frac{-1.421413741}{t\_0} - -0.284496736\right)}{t\_0} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)} \cdot e^{\left(-x\_m\right) \cdot x\_m} + 1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (if (<= x_m 0.00052)
     (+
      1e-9
      (*
       x_m
       (+
        1.128386358070218
        (* x_m (- (* -0.37545125292247583 x_m) 0.00011824294398844343)))))
     (+
      (*
       (/
        (+
         (/
          (-
           (/
            (/
             (- (/ 1.061405429 (fma 0.3275911 x_m 1.0)) 1.453152027)
             (fma 0.3275911 x_m 1.0))
            (fma 0.3275911 x_m 1.0))
           (- (/ -1.421413741 t_0) -0.284496736))
          t_0)
         0.254829592)
        (fma -0.3275911 (fabs x_m) -1.0))
       (exp (* (- x_m) x_m)))
      1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double tmp;
	if (x_m <= 0.00052) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((-0.37545125292247583 * x_m) - 0.00011824294398844343))));
	} else {
		tmp = (((((((((1.061405429 / fma(0.3275911, x_m, 1.0)) - 1.453152027) / fma(0.3275911, x_m, 1.0)) / fma(0.3275911, x_m, 1.0)) - ((-1.421413741 / t_0) - -0.284496736)) / t_0) + 0.254829592) / fma(-0.3275911, fabs(x_m), -1.0)) * exp((-x_m * x_m))) + 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	tmp = 0.0
	if (x_m <= 0.00052)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(x_m * Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / fma(0.3275911, x_m, 1.0)) - 1.453152027) / fma(0.3275911, x_m, 1.0)) / fma(0.3275911, x_m, 1.0)) - Float64(Float64(-1.421413741 / t_0) - -0.284496736)) / t_0) + 0.254829592) / fma(-0.3275911, abs(x_m), -1.0)) * exp(Float64(Float64(-x_m) * x_m))) + 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00052], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(x$95$m * N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - 1.453152027), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.421413741 / t$95$0), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.19999999999999954e-4

   1. Initial program 57.9%

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

   3. Applied rewrites 57.9%

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

   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 57.5%

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

   8. Applied rewrites 57.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \color{blue}{x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \color{blue}{\left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \color{blue}{\frac{2364858879768868679}{20000000000000000000000}}\right)\right) \]

       6. lower-*.f64 99.9

          \[\leadsto 10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right)} \]

   if 5.19999999999999954e-4 < x

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fabs.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

Alternative 5: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ \mathbf{if}\;x\_m \leq 0.00066:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + x\_m \cdot \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} - \frac{-1.421413741}{t\_0}\right) + -0.284496736}{t\_0} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)}, e^{\left(-x\_m\right) \cdot x\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (if (<= x_m 0.00066)
     (+
      1e-9
      (*
       x_m
       (+
        1.128386358070218
        (* x_m (- (* -0.37545125292247583 x_m) 0.00011824294398844343)))))
     (fma
      (/
       (+
        (/
         (+
          (-
           (/
            (/
             (- (/ 1.061405429 (fma 0.3275911 x_m 1.0)) 1.453152027)
             (fma 0.3275911 x_m 1.0))
            (fma 0.3275911 x_m 1.0))
           (/ -1.421413741 t_0))
          -0.284496736)
         t_0)
        0.254829592)
       (fma -0.3275911 (fabs x_m) -1.0))
      (exp (* (- x_m) x_m))
      1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double tmp;
	if (x_m <= 0.00066) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((-0.37545125292247583 * x_m) - 0.00011824294398844343))));
	} else {
		tmp = fma((((((((((1.061405429 / fma(0.3275911, x_m, 1.0)) - 1.453152027) / fma(0.3275911, x_m, 1.0)) / fma(0.3275911, x_m, 1.0)) - (-1.421413741 / t_0)) + -0.284496736) / t_0) + 0.254829592) / fma(-0.3275911, fabs(x_m), -1.0)), exp((-x_m * x_m)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	tmp = 0.0
	if (x_m <= 0.00066)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(x_m * Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343)))));
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / fma(0.3275911, x_m, 1.0)) - 1.453152027) / fma(0.3275911, x_m, 1.0)) / fma(0.3275911, x_m, 1.0)) - Float64(-1.421413741 / t_0)) + -0.284496736) / t_0) + 0.254829592) / fma(-0.3275911, abs(x_m), -1.0)), exp(Float64(Float64(-x_m) * x_m)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00066], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(x$95$m * N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - 1.453152027), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.421413741 / t$95$0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.6e-4

   1. Initial program 57.9%

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

   3. Applied rewrites 57.9%

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

   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 57.5%

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

   8. Applied rewrites 57.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \color{blue}{x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \color{blue}{\left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \color{blue}{\frac{2364858879768868679}{20000000000000000000000}}\right)\right) \]

       6. lower-*.f64 99.9

          \[\leadsto 10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right)} \]

   if 6.6e-4 < x

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fabs.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

Alternative 6: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00055:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + x\_m \cdot \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right) + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)}, e^{\left(-x\_m\right) \cdot x\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00055)
   (+
    1e-9
    (*
     x_m
     (+
      1.128386358070218
      (* x_m (- (* -0.37545125292247583 x_m) 0.00011824294398844343)))))
   (fma
    (/
     (+
      (+
       (/
        (/
         (-
          (/
           (- (/ 1.061405429 (fma 0.3275911 x_m 1.0)) 1.453152027)
           (fma 0.3275911 x_m 1.0))
          -1.421413741)
         (fma 0.3275911 x_m 1.0))
        (fma 0.3275911 x_m 1.0))
       (/ -0.284496736 (fma 0.3275911 x_m 1.0)))
      0.254829592)
     (fma -0.3275911 (fabs x_m) -1.0))
    (exp (* (- x_m) x_m))
    1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00055) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((-0.37545125292247583 * x_m) - 0.00011824294398844343))));
	} else {
		tmp = fma((((((((((1.061405429 / fma(0.3275911, x_m, 1.0)) - 1.453152027) / fma(0.3275911, x_m, 1.0)) - -1.421413741) / fma(0.3275911, x_m, 1.0)) / fma(0.3275911, x_m, 1.0)) + (-0.284496736 / fma(0.3275911, x_m, 1.0))) + 0.254829592) / fma(-0.3275911, fabs(x_m), -1.0)), exp((-x_m * x_m)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00055)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(x_m * Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343)))));
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / fma(0.3275911, x_m, 1.0)) - 1.453152027) / fma(0.3275911, x_m, 1.0)) - -1.421413741) / fma(0.3275911, x_m, 1.0)) / fma(0.3275911, x_m, 1.0)) + Float64(-0.284496736 / fma(0.3275911, x_m, 1.0))) + 0.254829592) / fma(-0.3275911, abs(x_m), -1.0)), exp(Float64(Float64(-x_m) * x_m)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.00055], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(x$95$m * N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - 1.453152027), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-0.284496736 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000033e-4

   1. Initial program 57.9%

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

   3. Applied rewrites 57.9%

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

   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 57.5%

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

   8. Applied rewrites 57.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \color{blue}{x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \color{blue}{\left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \color{blue}{\frac{2364858879768868679}{20000000000000000000000}}\right)\right) \]

       6. lower-*.f64 99.9

          \[\leadsto 10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right) \]

   11. Applied rewrites 99.9%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right)} \]

   if 5.50000000000000033e-4 < x

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fabs.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

Alternative 7: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ \mathbf{if}\;x\_m \leq 0.00038:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + x\_m \cdot \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_0} - 1.453152027}{t\_0} - -1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{t\_0} \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (if (<= x_m 0.00038)
     (+
      1e-9
      (*
       x_m
       (+
        1.128386358070218
        (* x_m (- (* -0.37545125292247583 x_m) 0.00011824294398844343)))))
     (-
      1.0
      (*
       (/
        (+
         (/
          (+
           (/ (- (/ (- (/ 1.061405429 t_0) 1.453152027) t_0) -1.421413741) t_0)
           -0.284496736)
          t_0)
         0.254829592)
        t_0)
       (exp (* (- x_m) x_m)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double tmp;
	if (x_m <= 0.00038) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((-0.37545125292247583 * x_m) - 0.00011824294398844343))));
	} else {
		tmp = 1.0 - ((((((((((1.061405429 / t_0) - 1.453152027) / t_0) - -1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_0) * exp((-x_m * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	tmp = 0.0
	if (x_m <= 0.00038)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(x_m * Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343)))));
	else
		tmp = 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) / t_0) + 0.254829592) / t_0) * exp(Float64(Float64(-x_m) * x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00038], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(x$95$m * N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.8000000000000002e-4

   1. Initial program 57.9%

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

   3. Applied rewrites 57.9%

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

   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 57.6%

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

   8. Applied rewrites 57.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \color{blue}{x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \color{blue}{\left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \color{blue}{\frac{2364858879768868679}{20000000000000000000000}}\right)\right) \]

       6. lower-*.f64 100.0

          \[\leadsto 10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right)} \]

   if 3.8000000000000002e-4 < x

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

Alternative 8: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00038:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + x\_m \cdot \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} - 1.453152027}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} - -1.421413741}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)}, e^{\left(-x\_m\right) \cdot x\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00038)
   (+
    1e-9
    (*
     x_m
     (+
      1.128386358070218
      (* x_m (- (* -0.37545125292247583 x_m) 0.00011824294398844343)))))
   (fma
    (/
     (+
      0.254829592
      (/
       (+
        -0.284496736
        (/
         (-
          (/
           (- (/ 1.061405429 (fma 0.3275911 x_m 1.0)) 1.453152027)
           (fma 0.3275911 x_m 1.0))
          -1.421413741)
         (fma 0.3275911 x_m 1.0)))
       (fma 0.3275911 x_m 1.0)))
     (fma -0.3275911 x_m -1.0))
    (exp (* (- x_m) x_m))
    1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00038) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((-0.37545125292247583 * x_m) - 0.00011824294398844343))));
	} else {
		tmp = fma(((0.254829592 + ((-0.284496736 + (((((1.061405429 / fma(0.3275911, x_m, 1.0)) - 1.453152027) / fma(0.3275911, x_m, 1.0)) - -1.421413741) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(-0.3275911, x_m, -1.0)), exp((-x_m * x_m)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00038)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(x_m * Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343)))));
	else
		tmp = fma(Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(Float64(Float64(Float64(1.061405429 / fma(0.3275911, x_m, 1.0)) - 1.453152027) / fma(0.3275911, x_m, 1.0)) - -1.421413741) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(-0.3275911, x_m, -1.0)), exp(Float64(Float64(-x_m) * x_m)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.00038], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(x$95$m * N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(N[(N[(N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - 1.453152027), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - -1.421413741), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.8000000000000002e-4

   1. Initial program 57.9%

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

   3. Applied rewrites 57.9%

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

   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 57.6%

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

   8. Applied rewrites 57.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \color{blue}{x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \color{blue}{\left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \color{blue}{\frac{2364858879768868679}{20000000000000000000000}}\right)\right) \]

       6. lower-*.f64 100.0

          \[\leadsto 10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right)} \]

   if 3.8000000000000002e-4 < x

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fabs.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

Alternative 9: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)\\ \mathbf{if}\;x\_m \leq 0.52:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + x\_m \cdot \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(-0.391746598 - \frac{-1.421413741}{t\_0}\right) + -0.284496736}{t\_0} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\_m\right|, -1\right)}, e^{\left(-x\_m\right) \cdot x\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (fabs x_m) 0.3275911 1.0)))
   (if (<= x_m 0.52)
     (+
      1e-9
      (*
       x_m
       (+
        1.128386358070218
        (* x_m (- (* -0.37545125292247583 x_m) 0.00011824294398844343)))))
     (fma
      (/
       (+
        (/ (+ (- -0.391746598 (/ -1.421413741 t_0)) -0.284496736) t_0)
        0.254829592)
       (fma -0.3275911 (fabs x_m) -1.0))
      (exp (* (- x_m) x_m))
      1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(fabs(x_m), 0.3275911, 1.0);
	double tmp;
	if (x_m <= 0.52) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((-0.37545125292247583 * x_m) - 0.00011824294398844343))));
	} else {
		tmp = fma((((((-0.391746598 - (-1.421413741 / t_0)) + -0.284496736) / t_0) + 0.254829592) / fma(-0.3275911, fabs(x_m), -1.0)), exp((-x_m * x_m)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(abs(x_m), 0.3275911, 1.0)
	tmp = 0.0
	if (x_m <= 0.52)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(x_m * Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343)))));
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(-0.391746598 - Float64(-1.421413741 / t_0)) + -0.284496736) / t_0) + 0.254829592) / fma(-0.3275911, abs(x_m), -1.0)), exp(Float64(Float64(-x_m) * x_m)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.52], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(x$95$m * N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.391746598 - N[(-1.421413741 / t$95$0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.52000000000000002

   1. Initial program 58.2%

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

   3. Applied rewrites 58.2%

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

   4. 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 57.2%

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

   8. Applied rewrites 57.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \color{blue}{x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \color{blue}{\left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \color{blue}{\frac{2364858879768868679}{20000000000000000000000}}\right)\right) \]

       6. lower-*.f64 99.5

          \[\leadsto 10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right) \]

   11. Applied rewrites 99.5%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right)} \]

   if 0.52000000000000002 < x

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-fabs.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fabs.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\color{blue}{\frac{-195873299}{500000000}} - \frac{\frac{-1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}\right) + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}, e^{\left(-x\right) \cdot x}, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 99.8%

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\color{blue}{-0.391746598} - \frac{-1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right) + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)}, e^{\left(-x\right) \cdot x}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + x\_m \cdot \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{e^{\left(-x\_m\right) \cdot x\_m}}{\mathsf{fma}\left(\left|x\_m\right|, 0.3275911, 1\right)} \cdot 0.254829592\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.0)
   (+
    1e-9
    (*
     x_m
     (+
      1.128386358070218
      (* x_m (- (* -0.37545125292247583 x_m) 0.00011824294398844343)))))
   (-
    1.0
    (* (/ (exp (* (- x_m) x_m)) (fma (fabs x_m) 0.3275911 1.0)) 0.254829592))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((-0.37545125292247583 * x_m) - 0.00011824294398844343))));
	} else {
		tmp = 1.0 - ((exp((-x_m * x_m)) / fma(fabs(x_m), 0.3275911, 1.0)) * 0.254829592);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(x_m * Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343)))));
	else
		tmp = Float64(1.0 - Float64(Float64(exp(Float64(Float64(-x_m) * x_m)) / fma(abs(x_m), 0.3275911, 1.0)) * 0.254829592));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.0], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(x$95$m * N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.254829592), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 58.3%

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

   3. Applied rewrites 58.3%

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

   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 57.2%

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

   8. Applied rewrites 57.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \color{blue}{x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \color{blue}{\left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \color{blue}{\frac{2364858879768868679}{20000000000000000000000}}\right)\right) \]

       6. lower-*.f64 99.4

          \[\leadsto 10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right) \]

   11. Applied rewrites 99.4%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right)} \]

   if 1 < x

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. flip-+ N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 99.9%

      \[\leadsto 1 - \color{blue}{\frac{e^{\left(-x\right) \cdot x}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \cdot 0.254829592} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 99.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + x\_m \cdot \left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (+
    1e-9
    (*
     x_m
     (+
      1.128386358070218
      (* x_m (- (* -0.37545125292247583 x_m) 0.00011824294398844343)))))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((-0.37545125292247583 * x_m) - 0.00011824294398844343))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.1d0) then
        tmp = 1d-9 + (x_m * (1.128386358070218d0 + (x_m * (((-0.37545125292247583d0) * x_m) - 0.00011824294398844343d0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((-0.37545125292247583 * x_m) - 0.00011824294398844343))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.1:
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((-0.37545125292247583 * x_m) - 0.00011824294398844343))))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(x_m * Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.1)
		tmp = 1e-9 + (x_m * (1.128386358070218 + (x_m * ((-0.37545125292247583 * x_m) - 0.00011824294398844343))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(x$95$m * N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 58.3%

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

   3. Applied rewrites 58.3%

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

   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 57.2%

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

   8. Applied rewrites 57.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \color{blue}{x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \color{blue}{\left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \frac{2364858879768868679}{20000000000000000000000}\right)}\right) \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + x \cdot \left(\frac{-375451252922475856043509345477}{1000000000000000000000000000000} \cdot x - \color{blue}{\frac{2364858879768868679}{20000000000000000000000}}\right)\right) \]

       6. lower-*.f64 99.4

          \[\leadsto 10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right) \]

   11. Applied rewrites 99.4%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(1.128386358070218 + x \cdot \left(-0.37545125292247583 \cdot x - 0.00011824294398844343\right)\right)} \]

   if 1.1000000000000001 < x

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   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 97.9%

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

   8. Applied rewrites 97.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.9%

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

Alternative 12: 99.4% accurate, 11.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.9:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(1.128386358070218 + -0.00011824294398844343 \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.9)
   (+ 1e-9 (* x_m (+ 1.128386358070218 (* -0.00011824294398844343 x_m))))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (-0.00011824294398844343 * x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.9d0) then
        tmp = 1d-9 + (x_m * (1.128386358070218d0 + ((-0.00011824294398844343d0) * x_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = 1e-9 + (x_m * (1.128386358070218 + (-0.00011824294398844343 * x_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.9:
		tmp = 1e-9 + (x_m * (1.128386358070218 + (-0.00011824294398844343 * x_m)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.9)
		tmp = Float64(1e-9 + Float64(x_m * Float64(1.128386358070218 + Float64(-0.00011824294398844343 * x_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.9)
		tmp = 1e-9 + (x_m * (1.128386358070218 + (-0.00011824294398844343 * x_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.9], N[(1e-9 + N[(x$95$m * N[(1.128386358070218 + N[(-0.00011824294398844343 * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 58.2%

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

   3. Applied rewrites 58.3%

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

   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 57.2%

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

   8. Applied rewrites 57.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \frac{-2364858879768868679}{20000000000000000000000} \cdot x\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{x \cdot \left(\frac{564193179035109}{500000000000000} + \frac{-2364858879768868679}{20000000000000000000000} \cdot x\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \color{blue}{\left(\frac{564193179035109}{500000000000000} + \frac{-2364858879768868679}{20000000000000000000000} \cdot x\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + x \cdot \left(\frac{564193179035109}{500000000000000} + \color{blue}{\frac{-2364858879768868679}{20000000000000000000000} \cdot x}\right) \]

       4. lower-*.f64 99.0

          \[\leadsto 10^{-9} + x \cdot \left(1.128386358070218 + -0.00011824294398844343 \cdot \color{blue}{x}\right) \]

   11. Applied rewrites 99.0%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(1.128386358070218 + -0.00011824294398844343 \cdot x\right)} \]

   if 0.900000000000000022 < x

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   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 97.8%

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

   8. Applied rewrites 97.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.8%

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

Alternative 13: 99.3% accurate, 17.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.9:\\ \;\;\;\;10^{-9} + 1.128386358070218 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.9) (+ 1e-9 (* 1.128386358070218 x_m)) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = 1e-9 + (1.128386358070218 * x_m);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.9d0) then
        tmp = 1d-9 + (1.128386358070218d0 * x_m)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = 1e-9 + (1.128386358070218 * x_m);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.9:
		tmp = 1e-9 + (1.128386358070218 * x_m)
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.9)
		tmp = Float64(1e-9 + Float64(1.128386358070218 * x_m));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.9)
		tmp = 1e-9 + (1.128386358070218 * x_m);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.9], N[(1e-9 + N[(1.128386358070218 * x$95$m), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 58.2%

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

   3. Applied rewrites 58.3%

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

   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 57.2%

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

   8. Applied rewrites 57.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000} + \frac{564193179035109}{500000000000000} \cdot x} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{1}{1000000000} + \color{blue}{\frac{564193179035109}{500000000000000} \cdot x} \]

       2. lower-*.f64 98.7

          \[\leadsto 10^{-9} + 1.128386358070218 \cdot \color{blue}{x} \]

   11. Applied rewrites 98.7%

       \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]

   if 0.900000000000000022 < x

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   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 97.8%

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

   8. Applied rewrites 97.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.8%

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

Alternative 14: 97.7% accurate, 37.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.8e-5) 1e-9 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.8e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.8d-5) then
        tmp = 1d-9
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.8e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.8e-5:
		tmp = 1e-9
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.8e-5)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.8e-5)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.8e-5], 1e-9, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 57.8%

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

   3. Applied rewrites 57.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 57.6%

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

   8. Applied rewrites 57.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1000000000}} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.7%

       \[\leadsto \color{blue}{10^{-9}} \]

   if 2.79999999999999996e-5 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 96.8%

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

   8. Applied rewrites 96.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 98.6%

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

Alternative 15: 53.3% accurate, 262.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 10^{-9} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1e-9)

C

x_m = fabs(x);
double code(double x_m) {
	return 1e-9;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = 1d-9
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1e-9;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1e-9

Julia

x_m = abs(x)
function code(x_m)
	return 1e-9
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1e-9;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1e-9

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%

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

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 77.5%

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

8. Applied rewrites 77.5%

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

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{1000000000}} \]
10. Step-by-step derivation

11. Applied rewrites 53.3%

    \[\leadsto \color{blue}{10^{-9}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025096 
(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)))))))