Jmat.Real.erf

Percentage Accurate: 79.1% → 99.6%

Time: 1.3min

Alternatives: 8

Speedup: 279.5×

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

real(8) function code(x)
    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

real(8) function code(x)
    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: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \frac{1}{1 + x \cdot 0.3275911}\\ t_1 := 1 + \left|x\right| \cdot 0.3275911\\ t_2 := \frac{1}{t_1}\\ \mathbf{if}\;\left|x\right| \leq 4 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{1 + e^{x \cdot \left(-x\right)} \cdot \left(t_0 \cdot \left(t_0 \cdot \left(t_2 \cdot \left(t_2 \cdot \left(1.061405429 \cdot \frac{-1}{t_1} - -1.453152027\right) - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* x 0.3275911))))
        (t_1 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_2 (/ 1.0 t_1)))
   (if (<= (fabs x) 4e-10)
     (+ 1e-9 (* x (+ 1.128386358070218 (* x -0.00011824294398844343))))
     (log
      (exp
       (+
        1.0
        (*
         (exp (* x (- x)))
         (*
          t_0
          (-
           (*
            t_0
            (-
             (*
              t_2
              (-
               (* t_2 (- (* 1.061405429 (/ -1.0 t_1)) -1.453152027))
               1.421413741))
             -0.284496736))
           0.254829592)))))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 / (1.0 + (x * 0.3275911));
	double t_1 = 1.0 + (fabs(x) * 0.3275911);
	double t_2 = 1.0 / t_1;
	double tmp;
	if (fabs(x) <= 4e-10) {
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)));
	} else {
		tmp = log(exp((1.0 + (exp((x * -x)) * (t_0 * ((t_0 * ((t_2 * ((t_2 * ((1.061405429 * (-1.0 / t_1)) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592))))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / (1.0d0 + (x * 0.3275911d0))
    t_1 = 1.0d0 + (abs(x) * 0.3275911d0)
    t_2 = 1.0d0 / t_1
    if (abs(x) <= 4d-10) then
        tmp = 1d-9 + (x * (1.128386358070218d0 + (x * (-0.00011824294398844343d0))))
    else
        tmp = log(exp((1.0d0 + (exp((x * -x)) * (t_0 * ((t_0 * ((t_2 * ((t_2 * ((1.061405429d0 * ((-1.0d0) / t_1)) - (-1.453152027d0))) - 1.421413741d0)) - (-0.284496736d0))) - 0.254829592d0))))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (x * 0.3275911));
	double t_1 = 1.0 + (Math.abs(x) * 0.3275911);
	double t_2 = 1.0 / t_1;
	double tmp;
	if (Math.abs(x) <= 4e-10) {
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)));
	} else {
		tmp = Math.log(Math.exp((1.0 + (Math.exp((x * -x)) * (t_0 * ((t_0 * ((t_2 * ((t_2 * ((1.061405429 * (-1.0 / t_1)) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592))))));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = 1.0 / (1.0 + (x * 0.3275911))
	t_1 = 1.0 + (math.fabs(x) * 0.3275911)
	t_2 = 1.0 / t_1
	tmp = 0
	if math.fabs(x) <= 4e-10:
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)))
	else:
		tmp = math.log(math.exp((1.0 + (math.exp((x * -x)) * (t_0 * ((t_0 * ((t_2 * ((t_2 * ((1.061405429 * (-1.0 / t_1)) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592))))))
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(x * 0.3275911)))
	t_1 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_2 = Float64(1.0 / t_1)
	tmp = 0.0
	if (abs(x) <= 4e-10)
		tmp = Float64(1e-9 + Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343))));
	else
		tmp = log(exp(Float64(1.0 + Float64(exp(Float64(x * Float64(-x))) * Float64(t_0 * Float64(Float64(t_0 * Float64(Float64(t_2 * Float64(Float64(t_2 * Float64(Float64(1.061405429 * Float64(-1.0 / t_1)) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592))))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = 1.0 / (1.0 + (x * 0.3275911));
	t_1 = 1.0 + (abs(x) * 0.3275911);
	t_2 = 1.0 / t_1;
	tmp = 0.0;
	if (abs(x) <= 4e-10)
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)));
	else
		tmp = log(exp((1.0 + (exp((x * -x)) * (t_0 * ((t_0 * ((t_2 * ((t_2 * ((1.061405429 * (-1.0 / t_1)) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 4e-10], N[(1e-9 + N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(1.0 + N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(N[(t$95$0 * N[(N[(t$95$2 * N[(N[(t$95$2 * N[(N[(1.061405429 * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - -1.453152027), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.00000000000000015e-10

   1. Initial program 57.6%

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

   2. Taylor expanded in x around 0 54.1%

      \[\leadsto \color{blue}{1 - \frac{e^{-{\left(\left|x\right|\right)}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   4. Simplified 57.5%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{3}}\right)\right)\right)}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}} \]

   5. Taylor expanded in x around 0 98.1%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 98.1%

         \[\leadsto 10^{-9} + \left(\color{blue}{{x}^{2} \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right) \]

      2. fma-def 98.1%

         \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left({x}^{2}, -0.00011824294398844343, 1.128386358070218 \cdot x\right)} \]

      3. unpow2 98.1%

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

      4. *-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in x around 0 98.1%

      \[\leadsto 10^{-9} + \color{blue}{\left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 98.1%

         \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

      2. *-commutative 98.1%

         \[\leadsto 10^{-9} + \left(\color{blue}{x \cdot 1.128386358070218} + -0.00011824294398844343 \cdot {x}^{2}\right) \]

      3. *-commutative 98.1%

         \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) \]

      4. unpow2 98.1%

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

      5. associate-*r* 98.1%

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

      6. distribute-lft-out 98.1%

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

   10. Simplified 98.1%

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

   if 4.00000000000000015e-10 < (fabs.f64 x)

   1. Initial program 99.9%

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

      1. log1p-expm1-u_binary64 99.9%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + \mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \]

   3. Applied rewrite-once 99.9%

      \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   4. Step-by-step derivation

      1. log1p-expm1 99.9%

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

      2. *-commutative 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 53.1%

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

      5. fabs-sqr 53.1%

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

      6. sqr-pow 99.9%

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

      7. unpow1 99.9%

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

   5. Simplified 99.9%

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

      1. log1p-expm1-u_binary64 99.9%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + \mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \]

   7. Applied rewrite-once 99.9%

      \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   8. Step-by-step derivation

      1. log1p-expm1 99.9%

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

      2. *-commutative 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 53.1%

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

      5. fabs-sqr 53.1%

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

      6. sqr-pow 99.9%

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

      7. unpow1 99.9%

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

   9. Simplified 99.9%

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

       1. add-log-exp_binary64 99.9%

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

   11. Applied rewrite-once 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.061405429 \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - -1.453152027\right) - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)}\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \frac{1}{1 + x \cdot 0.3275911}\\ t_1 := 1 + \left|x\right| \cdot 0.3275911\\ \mathbf{if}\;x \leq 5.7 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(t_0 \cdot \left(t_0 \cdot \left(t_0 \cdot \left(\frac{1}{t_1} \cdot \left(1.061405429 \cdot \frac{-1}{t_1} - -1.453152027\right) - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* x 0.3275911))))
        (t_1 (+ 1.0 (* (fabs x) 0.3275911))))
   (if (<= x 5.7e-6)
     (+ 1e-9 (* x (+ 1.128386358070218 (* x -0.00011824294398844343))))
     (+
      1.0
      (*
       (exp (* x (- x)))
       (*
        t_0
        (-
         (*
          t_0
          (-
           (*
            t_0
            (-
             (* (/ 1.0 t_1) (- (* 1.061405429 (/ -1.0 t_1)) -1.453152027))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 / (1.0 + (x * 0.3275911));
	double t_1 = 1.0 + (fabs(x) * 0.3275911);
	double tmp;
	if (x <= 5.7e-6) {
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)));
	} else {
		tmp = 1.0 + (exp((x * -x)) * (t_0 * ((t_0 * ((t_0 * (((1.0 / t_1) * ((1.061405429 * (-1.0 / t_1)) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (1.0d0 + (x * 0.3275911d0))
    t_1 = 1.0d0 + (abs(x) * 0.3275911d0)
    if (x <= 5.7d-6) then
        tmp = 1d-9 + (x * (1.128386358070218d0 + (x * (-0.00011824294398844343d0))))
    else
        tmp = 1.0d0 + (exp((x * -x)) * (t_0 * ((t_0 * ((t_0 * (((1.0d0 / t_1) * ((1.061405429d0 * ((-1.0d0) / t_1)) - (-1.453152027d0))) - 1.421413741d0)) - (-0.284496736d0))) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (x * 0.3275911));
	double t_1 = 1.0 + (Math.abs(x) * 0.3275911);
	double tmp;
	if (x <= 5.7e-6) {
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)));
	} else {
		tmp = 1.0 + (Math.exp((x * -x)) * (t_0 * ((t_0 * ((t_0 * (((1.0 / t_1) * ((1.061405429 * (-1.0 / t_1)) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = 1.0 / (1.0 + (x * 0.3275911))
	t_1 = 1.0 + (math.fabs(x) * 0.3275911)
	tmp = 0
	if x <= 5.7e-6:
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)))
	else:
		tmp = 1.0 + (math.exp((x * -x)) * (t_0 * ((t_0 * ((t_0 * (((1.0 / t_1) * ((1.061405429 * (-1.0 / t_1)) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592)))
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(x * 0.3275911)))
	t_1 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	tmp = 0.0
	if (x <= 5.7e-6)
		tmp = Float64(1e-9 + Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x * Float64(-x))) * Float64(t_0 * Float64(Float64(t_0 * Float64(Float64(t_0 * Float64(Float64(Float64(1.0 / t_1) * Float64(Float64(1.061405429 * Float64(-1.0 / t_1)) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = 1.0 / (1.0 + (x * 0.3275911));
	t_1 = 1.0 + (abs(x) * 0.3275911);
	tmp = 0.0;
	if (x <= 5.7e-6)
		tmp = 1e-9 + (x * (1.128386358070218 + (x * -0.00011824294398844343)));
	else
		tmp = 1.0 + (exp((x * -x)) * (t_0 * ((t_0 * ((t_0 * (((1.0 / t_1) * ((1.061405429 * (-1.0 / t_1)) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.7e-6], N[(1e-9 + N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(N[(t$95$0 * N[(N[(t$95$0 * N[(N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(N[(1.061405429 * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - -1.453152027), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.6999999999999996e-6

   1. Initial program 72.8%

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

   2. Taylor expanded in x around 0 70.5%

      \[\leadsto \color{blue}{1 - \frac{e^{-{\left(\left|x\right|\right)}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   4. Simplified 72.7%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{3}}\right)\right)\right)}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}} \]

   5. Taylor expanded in x around 0 63.4%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 63.4%

         \[\leadsto 10^{-9} + \left(\color{blue}{{x}^{2} \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right) \]

      2. fma-def 63.4%

         \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left({x}^{2}, -0.00011824294398844343, 1.128386358070218 \cdot x\right)} \]

      3. unpow2 63.4%

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

      4. *-commutative 63.4%

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

   7. Simplified 63.4%

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

   8. Taylor expanded in x around 0 63.4%

      \[\leadsto 10^{-9} + \color{blue}{\left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 63.4%

         \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

      2. *-commutative 63.4%

         \[\leadsto 10^{-9} + \left(\color{blue}{x \cdot 1.128386358070218} + -0.00011824294398844343 \cdot {x}^{2}\right) \]

      3. *-commutative 63.4%

         \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) \]

      4. unpow2 63.4%

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

      5. associate-*r* 63.4%

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

      6. distribute-lft-out 63.4%

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

   10. Simplified 63.4%

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

   if 5.6999999999999996e-6 < x

   1. Initial program 99.8%

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

      1. log1p-expm1-u_binary64 99.8%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + \mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \]

   3. Applied rewrite-once 99.8%

      \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   4. Step-by-step derivation

      1. log1p-expm1 99.8%

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

      2. *-commutative 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. fabs-sqr 99.8%

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

      6. sqr-pow 99.8%

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

      7. unpow1 99.8%

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

   5. Simplified 99.8%

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

      1. log1p-expm1-u_binary64 99.8%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + \mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \]

   7. Applied rewrite-once 99.8%

      \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   8. Step-by-step derivation

      1. log1p-expm1 99.8%

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

      2. *-commutative 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. fabs-sqr 99.8%

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

      6. sqr-pow 99.8%

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

      7. unpow1 99.8%

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

   9. Simplified 99.8%

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

       1. log1p-expm1-u_binary64 99.8%

          \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \]

   11. Applied rewrite-once 99.8%

       \[\leadsto 1 - \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   12. Step-by-step derivation

       1. log1p-expm1 99.8%

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

       2. *-commutative 99.8%

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

       3. unpow1 99.8%

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

       4. sqr-pow 99.8%

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

       5. fabs-sqr 99.8%

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

       6. sqr-pow 99.8%

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

       7. unpow1 99.8%

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

   13. Simplified 99.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.7 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.061405429 \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - -1.453152027\right) - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.05)
   (+
    1e-9
    (+
     (* -0.37545125292247583 (pow x 3.0))
     (+ (* -0.00011824294398844343 (pow x 2.0)) (* x 1.128386358070218))))
   1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.05) {
		tmp = 1e-9 + ((-0.37545125292247583 * pow(x, 3.0)) + ((-0.00011824294398844343 * pow(x, 2.0)) + (x * 1.128386358070218)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.05d0) then
        tmp = 1d-9 + (((-0.37545125292247583d0) * (x ** 3.0d0)) + (((-0.00011824294398844343d0) * (x ** 2.0d0)) + (x * 1.128386358070218d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.05) {
		tmp = 1e-9 + ((-0.37545125292247583 * Math.pow(x, 3.0)) + ((-0.00011824294398844343 * Math.pow(x, 2.0)) + (x * 1.128386358070218)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.05:
		tmp = 1e-9 + ((-0.37545125292247583 * math.pow(x, 3.0)) + ((-0.00011824294398844343 * math.pow(x, 2.0)) + (x * 1.128386358070218)))
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.05)
		tmp = Float64(1e-9 + Float64(Float64(-0.37545125292247583 * (x ^ 3.0)) + Float64(Float64(-0.00011824294398844343 * (x ^ 2.0)) + Float64(x * 1.128386358070218))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.05)
		tmp = 1e-9 + ((-0.37545125292247583 * (x ^ 3.0)) + ((-0.00011824294398844343 * (x ^ 2.0)) + (x * 1.128386358070218)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.05], N[(1e-9 + N[(N[(-0.37545125292247583 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.00011824294398844343 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 72.8%

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

   2. Taylor expanded in x around 0 70.6%

      \[\leadsto \color{blue}{1 - \frac{e^{-{\left(\left|x\right|\right)}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   4. Simplified 72.8%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{3}}\right)\right)\right)}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}} \]

   5. Taylor expanded in x around 0 64.1%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]

   if 1.05000000000000004 < x

   1. Initial program 100.0%

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

      1. log1p-expm1-u_binary64 100.0%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + \mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \]

   3. Applied rewrite-once 100.0%

      \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   4. Step-by-step derivation

      1. log1p-expm1 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow1 100.0%

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

      4. sqr-pow 100.0%

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

      5. fabs-sqr 100.0%

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

      6. sqr-pow 100.0%

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

      7. unpow1 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05:\\ \;\;\;\;10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 99.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + \sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (+ 1.128386358070218 (* x -0.00011824294398844343)))))
   (if (<= x 0.9) (+ 1e-9 (cbrt (* t_0 (* t_0 t_0)))) 1.0)))

C

x = abs(x);
double code(double x) {
	double t_0 = x * (1.128386358070218 + (x * -0.00011824294398844343));
	double tmp;
	if (x <= 0.9) {
		tmp = 1e-9 + cbrt((t_0 * (t_0 * t_0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = x * (1.128386358070218 + (x * -0.00011824294398844343));
	double tmp;
	if (x <= 0.9) {
		tmp = 1e-9 + Math.cbrt((t_0 * (t_0 * t_0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343)))
	tmp = 0.0
	if (x <= 0.9)
		tmp = Float64(1e-9 + cbrt(Float64(t_0 * Float64(t_0 * t_0))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.9], N[(1e-9 + N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 72.8%

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

   2. Taylor expanded in x around 0 70.6%

      \[\leadsto \color{blue}{1 - \frac{e^{-{\left(\left|x\right|\right)}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   4. Simplified 72.8%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{3}}\right)\right)\right)}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}} \]

   5. Taylor expanded in x around 0 63.4%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 63.4%

         \[\leadsto 10^{-9} + \left(\color{blue}{{x}^{2} \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right) \]

      2. fma-def 63.4%

         \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left({x}^{2}, -0.00011824294398844343, 1.128386358070218 \cdot x\right)} \]

      3. unpow2 63.4%

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

      4. *-commutative 63.4%

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

   7. Simplified 63.4%

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

   8. Taylor expanded in x around 0 63.4%

      \[\leadsto 10^{-9} + \color{blue}{\left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 63.4%

         \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

      2. *-commutative 63.4%

         \[\leadsto 10^{-9} + \left(\color{blue}{x \cdot 1.128386358070218} + -0.00011824294398844343 \cdot {x}^{2}\right) \]

      3. *-commutative 63.4%

         \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) \]

      4. unpow2 63.4%

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

      5. associate-*r* 63.4%

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

      6. distribute-lft-out 63.4%

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

   10. Simplified 63.4%

       \[\leadsto 10^{-9} + \color{blue}{x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)} \]
   11. Step-by-step derivation

       1. add-cbrt-cube_binary64 63.3%

          \[\leadsto \color{blue}{10^{-9} + \sqrt[3]{\left(\left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right) \cdot \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\right) \cdot \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)}} \]

   12. Applied rewrite-once 63.3%

       \[\leadsto 10^{-9} + \color{blue}{\sqrt[3]{\left(\left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right) \cdot \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\right) \cdot \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)}} \]

   if 0.900000000000000022 < x

   1. Initial program 100.0%

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

      1. log1p-expm1-u_binary64 100.0%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + \mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \]

   3. Applied rewrite-once 100.0%

      \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   4. Step-by-step derivation

      1. log1p-expm1 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow1 100.0%

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

      4. sqr-pow 100.0%

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

      5. fabs-sqr 100.0%

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

      6. sqr-pow 100.0%

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

      7. unpow1 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + \sqrt[3]{\left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right) \cdot \left(\left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right) \cdot \left(x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 99.4% accurate, 77.4× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.9)
   (+ 1e-9 (* x (+ 1.128386358070218 (* x -0.00011824294398844343))))
   1.0))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.9d0) then
        tmp = 1d-9 + (x * (1.128386358070218d0 + (x * (-0.00011824294398844343d0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.9], N[(1e-9 + N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 72.8%

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

   2. Taylor expanded in x around 0 70.6%

      \[\leadsto \color{blue}{1 - \frac{e^{-{\left(\left|x\right|\right)}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   4. Simplified 72.8%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{3}}\right)\right)\right)}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}} \]

   5. Taylor expanded in x around 0 63.4%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 63.4%

         \[\leadsto 10^{-9} + \left(\color{blue}{{x}^{2} \cdot -0.00011824294398844343} + 1.128386358070218 \cdot x\right) \]

      2. fma-def 63.4%

         \[\leadsto 10^{-9} + \color{blue}{\mathsf{fma}\left({x}^{2}, -0.00011824294398844343, 1.128386358070218 \cdot x\right)} \]

      3. unpow2 63.4%

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

      4. *-commutative 63.4%

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

   7. Simplified 63.4%

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

   8. Taylor expanded in x around 0 63.4%

      \[\leadsto 10^{-9} + \color{blue}{\left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   9. Step-by-step derivation

      1. +-commutative 63.4%

         \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

      2. *-commutative 63.4%

         \[\leadsto 10^{-9} + \left(\color{blue}{x \cdot 1.128386358070218} + -0.00011824294398844343 \cdot {x}^{2}\right) \]

      3. *-commutative 63.4%

         \[\leadsto 10^{-9} + \left(x \cdot 1.128386358070218 + \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) \]

      4. unpow2 63.4%

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

      5. associate-*r* 63.4%

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

      6. distribute-lft-out 63.4%

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

   10. Simplified 63.4%

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

   if 0.900000000000000022 < x

   1. Initial program 100.0%

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

      1. log1p-expm1-u_binary64 100.0%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + \mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \]

   3. Applied rewrite-once 100.0%

      \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   4. Step-by-step derivation

      1. log1p-expm1 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow1 100.0%

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

      4. sqr-pow 100.0%

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

      5. fabs-sqr 100.0%

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

      6. sqr-pow 100.0%

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

      7. unpow1 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 73.8%

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

Alternative 6: 99.3% accurate, 121.2× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.9) (+ 1e-9 (* x 1.128386358070218)) 1.0))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.9d0) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.9], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 72.8%

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

   2. Taylor expanded in x around 0 70.6%

      \[\leadsto \color{blue}{1 - \frac{e^{-{\left(\left|x\right|\right)}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   4. Simplified 72.8%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{3}}\right)\right)\right)}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}} \]

   5. Taylor expanded in x around 0 63.4%

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 63.4%

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

   7. Simplified 63.4%

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

   if 0.900000000000000022 < x

   1. Initial program 100.0%

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

      1. log1p-expm1-u_binary64 100.0%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + \mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \]

   3. Applied rewrite-once 100.0%

      \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   4. Step-by-step derivation

      1. log1p-expm1 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow1 100.0%

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

      4. sqr-pow 100.0%

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

      5. fabs-sqr 100.0%

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

      6. sqr-pow 100.0%

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

      7. unpow1 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 97.6% accurate, 279.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.85 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 2.85e-5) 1e-9 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.85e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.85d-5) then
        tmp = 1d-9
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.85e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.85e-5:
		tmp = 1e-9
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.85e-5)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.85e-5)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.85e-5], 1e-9, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.8500000000000002e-5

   1. Initial program 72.8%

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

   2. Taylor expanded in x around 0 70.5%

      \[\leadsto \color{blue}{1 - \frac{e^{-{\left(\left|x\right|\right)}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

   4. Simplified 72.7%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{3}}\right)\right)\right)}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}} \]

   5. Taylor expanded in x around 0 65.9%

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

   if 2.8500000000000002e-5 < x

   1. Initial program 99.8%

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

      1. log1p-expm1-u_binary64 99.8%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{1 + \mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \]

   3. Applied rewrite-once 99.8%

      \[\leadsto 1 - \left(\frac{1}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   4. Step-by-step derivation

      1. log1p-expm1 99.8%

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

      2. *-commutative 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. fabs-sqr 99.8%

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

      6. sqr-pow 99.8%

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

      7. unpow1 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around inf 98.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.85 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 53.2% accurate, 856.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 1e-9)

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 1d-9
end function

Java

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

Python

x = abs(x)
def code(x):
	return 1e-9

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 1e-9

Derivation

1. Initial program 80.6%

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

2. Taylor expanded in x around 0 79.0%

   \[\leadsto \color{blue}{1 - \frac{e^{-{\left(\left|x\right|\right)}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]

4. Simplified 80.5%

   \[\leadsto \color{blue}{1 - \frac{\frac{0.254829592 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{4}} + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{2}} - \left(\frac{0.284496736}{\mathsf{fma}\left(x, 0.3275911, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(x, 0.3275911, 1\right)\right)}^{3}}\right)\right)\right)}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}} \]

5. Taylor expanded in x around 0 50.1%

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

6. Final simplification 50.1%

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

Reproduce

herbie shell --seed 2023297 
(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)))))))