Jmat.Real.erf

Percentage Accurate: 79.1% → 99.9%

Time: 12.0s

Alternatives: 10

Speedup: 8.4×

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, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00056:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 1.453152027}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right) \cdot \mathsf{fma}\left(-0.3275911, x\_m, -1\right)} + \left(\frac{-1.421413741}{\mathsf{fma}\left(-0.3275911, x\_m, -1\right)} + -0.284496736\right)}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(x\_m, 0.3275911, 1\right)} \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00056)
   (fma
    (fma
     (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
     x_m
     1.128386358070218)
    x_m
    1e-9)
   (-
    1.0
    (*
     (/
      (+
       (/
        (+
         (/
          (+ (/ -1.061405429 (fma x_m 0.3275911 1.0)) 1.453152027)
          (* (fma x_m 0.3275911 1.0) (fma -0.3275911 x_m -1.0)))
         (+ (/ -1.421413741 (fma -0.3275911 x_m -1.0)) -0.284496736))
        (fma x_m 0.3275911 1.0))
       0.254829592)
      (fma x_m 0.3275911 1.0))
     (exp (* (- x_m) x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00056) {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = 1.0 - ((((((((-1.061405429 / fma(x_m, 0.3275911, 1.0)) + 1.453152027) / (fma(x_m, 0.3275911, 1.0) * fma(-0.3275911, x_m, -1.0))) + ((-1.421413741 / fma(-0.3275911, x_m, -1.0)) + -0.284496736)) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / fma(x_m, 0.3275911, 1.0)) * exp((-x_m * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00056)
		tmp = fma(fma(Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1.061405429 / fma(x_m, 0.3275911, 1.0)) + 1.453152027) / Float64(fma(x_m, 0.3275911, 1.0) * fma(-0.3275911, x_m, -1.0))) + Float64(Float64(-1.421413741 / fma(-0.3275911, x_m, -1.0)) + -0.284496736)) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / fma(x_m, 0.3275911, 1.0)) * exp(Float64(Float64(-x_m) * x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.00056], N[(N[(N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(-1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.453152027), $MachinePrecision] / N[(N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision] * N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.421413741 / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999995e-4

   1. Initial program 72.0%

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

   4. Applied rewrites 69.3%

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

   6. Applied rewrites 69.3%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 65.3

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

   9. Applied rewrites 65.3%

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

   if 5.5999999999999995e-4 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. div-add N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f64 N/A

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

   6. Applied rewrites 99.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00056:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x - 0.00011824294398844343, x, 1.128386358070218\right), x, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 1.453152027}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot \mathsf{fma}\left(-0.3275911, x, -1\right)} + \left(\frac{-1.421413741}{\mathsf{fma}\left(-0.3275911, x, -1\right)} + -0.284496736\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(x, 0.3275911, 1\right)} \cdot e^{\left(-x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|}\\ \mathbf{if}\;\left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\_m\right) \cdot x\_m} \leq 2 \cdot 10^{-176}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m))))))
   (if (<=
        (*
         (*
          t_0
          (+
           0.254829592
           (*
            t_0
            (+
             -0.284496736
             (*
              t_0
              (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
         (exp (* (- x_m) x_m)))
        2e-176)
     1e-9
     (fma
      (fma
       (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
       x_m
       1.128386358070218)
      x_m
      1e-9))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x_m)));
	double tmp;
	if (((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp((-x_m * x_m))) <= 2e-176) {
		tmp = 1e-9;
	} else {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x_m))))
	tmp = 0.0
	if (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(-x_m) * x_m))) <= 2e-176)
		tmp = 1e-9;
	else
		tmp = fma(fma(Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-176], 1e-9, N[(N[(N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 2e-176

   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. Add Preprocessing

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 11.1%

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

   if 2e-176 < (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))

   1. Initial program 59.0%

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

   4. Applied rewrites 55.1%

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

   6. Applied rewrites 55.1%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 94.1

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

   9. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x - 0.00011824294398844343, x, 1.128386358070218\right), x, 10^{-9}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x} \leq 2 \cdot 10^{-176}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x - 0.00011824294398844343, x, 1.128386358070218\right), x, 10^{-9}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\_m\right|}\\ \mathbf{if}\;\left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\_m\right) \cdot x\_m} \leq 0:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x_m))))))
   (if (<=
        (*
         (*
          t_0
          (+
           0.254829592
           (*
            t_0
            (+
             -0.284496736
             (*
              t_0
              (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
         (exp (* (- x_m) x_m)))
        0.0)
     1e-9
     (fma (fma -0.00011824294398844343 x_m 1.128386358070218) x_m 1e-9))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x_m)));
	double tmp;
	if (((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp((-x_m * x_m))) <= 0.0) {
		tmp = 1e-9;
	} else {
		tmp = fma(fma(-0.00011824294398844343, x_m, 1.128386358070218), x_m, 1e-9);
	}
	return tmp;
}

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], 1e-9, N[(N[(-0.00011824294398844343 * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 0.0

   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. Add Preprocessing

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 11.1%

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

   if 0.0 < (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))

   1. Initial program 59.3%

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

   4. Applied rewrites 55.4%

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

   6. Applied rewrites 55.5%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 93.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.00011824294398844343, x, 1.128386358070218\right)}, x, 10^{-9}\right) \]

   9. Applied rewrites 93.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.00011824294398844343, x, 1.128386358070218\right), x, 10^{-9}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x} \leq 0:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.00011824294398844343, x, 1.128386358070218\right), x, 10^{-9}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00055)
   (fma
    (fma
     (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
     x_m
     1.128386358070218)
    x_m
    1e-9)
   (fma
    (/
     (+
      (/
       (+
        (/
         (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
         (* (fma -0.3275911 x_m -1.0) (fma -0.3275911 x_m -1.0)))
        (+ (/ -1.421413741 (fma -0.3275911 x_m -1.0)) -0.284496736))
       (fma x_m 0.3275911 1.0))
      0.254829592)
     (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 <= 0.00055) {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = fma(((((((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / (fma(-0.3275911, x_m, -1.0) * fma(-0.3275911, x_m, -1.0))) + ((-1.421413741 / fma(-0.3275911, x_m, -1.0)) + -0.284496736)) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / 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 <= 0.00055)
		tmp = fma(fma(Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / Float64(fma(-0.3275911, x_m, -1.0) * fma(-0.3275911, x_m, -1.0))) + Float64(Float64(-1.421413741 / fma(-0.3275911, x_m, -1.0)) + -0.284496736)) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / 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, 0.00055], N[(N[(N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision] * N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.421413741 / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $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 < 5.50000000000000033e-4

   1. Initial program 72.0%

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

   4. Applied rewrites 69.3%

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

   6. Applied rewrites 69.3%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 65.3

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

   9. Applied rewrites 65.3%

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

   if 5.50000000000000033e-4 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   8. Applied rewrites 99.7%

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

Alternative 5: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00055)
		tmp = fma(fma(Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1.061405429 / fma(x_m, 0.3275911, 1.0)) + 1.453152027) / fma(-0.3275911, x_m, -1.0)) + 1.421413741) / fma(x_m, 0.3275911, 1.0)) + -0.284496736) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) * exp(Float64(Float64(-x_m) * x_m))) / fma(x_m, 0.3275911, 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[(N[(N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(-1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.453152027), $MachinePrecision] / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + 1.421413741), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $MachinePrecision] * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000033e-4

   1. Initial program 72.0%

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

   4. Applied rewrites 69.3%

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

   6. Applied rewrites 69.3%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 65.3

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

   9. Applied rewrites 65.3%

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

   if 5.50000000000000033e-4 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

Alternative 6: 99.9% accurate, 1.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00055)
   (fma
    (fma
     (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
     x_m
     1.128386358070218)
    x_m
    1e-9)
   (fma
    (/
     (+
      (/
       (+
        (/
         (+
          (/
           (+ (/ -1.061405429 (fma x_m 0.3275911 1.0)) 1.453152027)
           (fma -0.3275911 x_m -1.0))
          1.421413741)
         (fma x_m 0.3275911 1.0))
        -0.284496736)
       (fma x_m 0.3275911 1.0))
      0.254829592)
     (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 <= 0.00055) {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = fma((((((((((-1.061405429 / fma(x_m, 0.3275911, 1.0)) + 1.453152027) / fma(-0.3275911, x_m, -1.0)) + 1.421413741) / fma(x_m, 0.3275911, 1.0)) + -0.284496736) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / 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 <= 0.00055)
		tmp = fma(fma(Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1.061405429 / fma(x_m, 0.3275911, 1.0)) + 1.453152027) / fma(-0.3275911, x_m, -1.0)) + 1.421413741) / fma(x_m, 0.3275911, 1.0)) + -0.284496736) / fma(x_m, 0.3275911, 1.0)) + 0.254829592) / 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, 0.00055], N[(N[(N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(-1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.453152027), $MachinePrecision] / N[(-0.3275911 * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + 1.421413741), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + 0.254829592), $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 < 5.50000000000000033e-4

   1. Initial program 72.0%

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

   4. Applied rewrites 69.3%

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

   6. Applied rewrites 69.3%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 65.3

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

   9. Applied rewrites 65.3%

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

   if 5.50000000000000033e-4 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

Alternative 7: 99.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - 0.999999999 \cdot e^{\left(-x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.15)
   (fma
    (fma
     (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
     x_m
     1.128386358070218)
    x_m
    1e-9)
   (- 1.0 (* 0.999999999 (exp (* (- x_m) x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.15) {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = 1.0 - (0.999999999 * exp((-x_m * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.15)
		tmp = fma(fma(Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	else
		tmp = Float64(1.0 - Float64(0.999999999 * exp(Float64(Float64(-x_m) * x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.15], N[(N[(N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], N[(1.0 - N[(0.999999999 * N[Exp[N[((-x$95$m) * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 72.0%

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

   4. Applied rewrites 69.3%

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

   6. Applied rewrites 69.4%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 65.4

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

   9. Applied rewrites 65.4%

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

   if 1.1499999999999999 < x

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{\frac{999999999}{1000000000} \cdot e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{999999999}{1000000000} \cdot e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - \frac{999999999}{1000000000} \cdot e^{\color{blue}{-1 \cdot {\left(\left|x\right|\right)}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto 1 - \frac{999999999}{1000000000} \cdot e^{-1 \cdot \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}} \]

      4. sqr-abs-rev N/A

         \[\leadsto 1 - \frac{999999999}{1000000000} \cdot e^{-1 \cdot \color{blue}{\left(x \cdot x\right)}} \]

      5. unpow2 N/A

         \[\leadsto 1 - \frac{999999999}{1000000000} \cdot e^{-1 \cdot \color{blue}{{x}^{2}}} \]

      6. lower-exp.f64 N/A

         \[\leadsto 1 - \frac{999999999}{1000000000} \cdot \color{blue}{e^{-1 \cdot {x}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto 1 - \frac{999999999}{1000000000} \cdot e^{-1 \cdot \color{blue}{\left(x \cdot x\right)}} \]

      8. sqr-abs-rev N/A

         \[\leadsto 1 - \frac{999999999}{1000000000} \cdot e^{-1 \cdot \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}} \]

      9. unpow2 N/A

         \[\leadsto 1 - \frac{999999999}{1000000000} \cdot e^{-1 \cdot \color{blue}{{\left(\left|x\right|\right)}^{2}}} \]

      10. mul-1-neg N/A

          \[\leadsto 1 - \frac{999999999}{1000000000} \cdot e^{\color{blue}{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)}} \]

      11. unpow2 N/A

          \[\leadsto 1 - \frac{999999999}{1000000000} \cdot e^{\mathsf{neg}\left(\color{blue}{\left|x\right| \cdot \left|x\right|}\right)} \]

      12. sqr-abs-rev N/A

          \[\leadsto 1 - \frac{999999999}{1000000000} \cdot e^{\mathsf{neg}\left(\color{blue}{x \cdot x}\right)} \]

      13. unpow2 N/A

          \[\leadsto 1 - \frac{999999999}{1000000000} \cdot e^{\mathsf{neg}\left(\color{blue}{{x}^{2}}\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto 1 - \frac{999999999}{1000000000} \cdot e^{\color{blue}{-{x}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto 1 - \frac{999999999}{1000000000} \cdot e^{-\color{blue}{x \cdot x}} \]

      16. lower-*.f64 100.0

          \[\leadsto 1 - 0.999999999 \cdot e^{-\color{blue}{x \cdot x}} \]

   7. Applied rewrites 100.0%

      \[\leadsto 1 - \color{blue}{0.999999999 \cdot e^{-x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x - 0.00011824294398844343, x, 1.128386358070218\right), x, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - 0.999999999 \cdot e^{\left(-x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.37545125292247583 \cdot x\_m - 0.00011824294398844343, x\_m, 1.128386358070218\right), x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.999999999}{-0.3275911 \cdot \left|x\_m\right| - 1} + 1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.55)
   (fma
    (fma
     (- (* -0.37545125292247583 x_m) 0.00011824294398844343)
     x_m
     1.128386358070218)
    x_m
    1e-9)
   (+ (/ 0.999999999 (- (* -0.3275911 (fabs x_m)) 1.0)) 1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = fma(fma(((-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	} else {
		tmp = (0.999999999 / ((-0.3275911 * fabs(x_m)) - 1.0)) + 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.55)
		tmp = fma(fma(Float64(Float64(-0.37545125292247583 * x_m) - 0.00011824294398844343), x_m, 1.128386358070218), x_m, 1e-9);
	else
		tmp = Float64(Float64(0.999999999 / Float64(Float64(-0.3275911 * abs(x_m)) - 1.0)) + 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.55], N[(N[(N[(N[(-0.37545125292247583 * x$95$m), $MachinePrecision] - 0.00011824294398844343), $MachinePrecision] * x$95$m + 1.128386358070218), $MachinePrecision] * x$95$m + 1e-9), $MachinePrecision], N[(N[(0.999999999 / N[(N[(-0.3275911 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 72.0%

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

   4. Applied rewrites 69.3%

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

   6. Applied rewrites 69.4%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 65.4

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

   9. Applied rewrites 65.4%

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

   if 1.55000000000000004 < 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. Add Preprocessing

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{999999999}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{999999999}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{999999999}{1000000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + 1} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{999999999}{1000000000} \cdot 1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}} + 1 \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{999999999}{1000000000}}}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + 1 \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{999999999}{1000000000}}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}} + 1 \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\frac{999999999}{1000000000}}{\color{blue}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}} + 1 \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{999999999}{1000000000}}{\color{blue}{\frac{-3275911}{10000000} \cdot \left|x\right|} - 1} + 1 \]

      8. lower-fabs.f64 96.1

         \[\leadsto \frac{0.999999999}{-0.3275911 \cdot \color{blue}{\left|x\right|} - 1} + 1 \]

   9. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\frac{0.999999999}{-0.3275911 \cdot \left|x\right| - 1} + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 55.0% accurate, 20.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 920000000:\\ \;\;\;\;\mathsf{fma}\left(1.128386358070218, x\_m, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 920000000.0) (fma 1.128386358070218 x_m 1e-9) 1e-9))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 920000000.0) {
		tmp = fma(1.128386358070218, x_m, 1e-9);
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 920000000.0)
		tmp = fma(1.128386358070218, x_m, 1e-9);
	else
		tmp = 1e-9;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 920000000.0], N[(1.128386358070218 * x$95$m + 1e-9), $MachinePrecision], 1e-9]

Derivation

1. Split input into 2 regimes

2. if x < 9.2e8

   1. Initial program 72.4%

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

   4. Applied rewrites 69.8%

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

   6. Applied rewrites 69.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 63.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(1.128386358070218, x, 10^{-9}\right)} \]

   9. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.128386358070218, x, 10^{-9}\right)} \]

   if 9.2e8 < 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. Add Preprocessing

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 11.1%

      \[\leadsto \color{blue}{10^{-9}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 53.4% 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 79.0%

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

4. Applied rewrites 77.0%

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

6. Applied rewrites 77.0%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 52.4%

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

Reproduce

herbie shell --seed 2025017 
(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)))))))