Jmat.Real.erf

Percentage Accurate: 79.1% → 99.9%

Time: 18.1s

Alternatives: 9

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: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ \mathbf{if}\;\left|x\right| \leq 10^{-5}:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 - \log \left(1 + \mathsf{expm1}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{t_0}}{t_0}\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{t_0}, 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0)))
   (if (<= (fabs x) 1e-5)
     (+
      1e-9
      (fma
       (pow x 3.0)
       -0.37545125292247583
       (* x (+ 1.128386358070218 (* x -0.00011824294398844343)))))
     (fma
      (-
       -0.254829592
       (log
        (+
         1.0
         (expm1
          (/
           (+
            -0.284496736
            (/
             (+
              1.421413741
              (/
               (+ (/ 1.061405429 (fma 0.3275911 x 1.0)) -1.453152027)
               (fma 0.3275911 x 1.0)))
             t_0))
           t_0)))))
      (/ (pow (exp x) (- x)) t_0)
      1.0))))

C

x = abs(x);
double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double tmp;
	if (fabs(x) <= 1e-5) {
		tmp = 1e-9 + fma(pow(x, 3.0), -0.37545125292247583, (x * (1.128386358070218 + (x * -0.00011824294398844343))));
	} else {
		tmp = fma((-0.254829592 - log((1.0 + expm1(((-0.284496736 + ((1.421413741 + (((1.061405429 / fma(0.3275911, x, 1.0)) + -1.453152027) / fma(0.3275911, x, 1.0))) / t_0)) / t_0))))), (pow(exp(x), -x) / t_0), 1.0);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	tmp = 0.0
	if (abs(x) <= 1e-5)
		tmp = Float64(1e-9 + fma((x ^ 3.0), -0.37545125292247583, Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343)))));
	else
		tmp = fma(Float64(-0.254829592 - log(Float64(1.0 + expm1(Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(Float64(1.061405429 / fma(0.3275911, x, 1.0)) + -1.453152027) / fma(0.3275911, x, 1.0))) / t_0)) / t_0))))), Float64((exp(x) ^ Float64(-x)) / t_0), 1.0);
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1e-5], N[(1e-9 + N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.254829592 - N[Log[N[(1.0 + N[(Exp[N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000008e-5

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   5. Applied egg-rr 57.8%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in x around 0 99.0%

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

      1. *-un-lft-identity 99.0%

         \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      2. fma-def 99.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)}\right) \]

      3. fma-def 99.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\mathsf{fma}\left(-0.00011824294398844343, {x}^{2}, 1.128386358070218 \cdot x\right)}\right)\right) \]

      4. pow2 99.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(-0.00011824294398844343, \color{blue}{x \cdot x}, 1.128386358070218 \cdot x\right)\right)\right) \]

      5. *-commutative 99.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \color{blue}{x \cdot 1.128386358070218}\right)\right)\right) \]

   10. Applied egg-rr 99.0%

       \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, x \cdot 1.128386358070218\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 99.0%

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

       2. fma-udef 99.0%

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

       3. *-commutative 99.0%

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

       4. fma-def 99.0%

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

       5. fma-def 99.0%

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

       6. +-commutative 99.0%

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

       7. *-commutative 99.0%

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

       8. associate-*r* 99.0%

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

       9. distribute-rgt-out 99.0%

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

       10. *-commutative 99.0%

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

   12. Simplified 99.0%

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

   if 1.00000000000000008e-5 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. log1p-expm1-u 99.9%

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

      2. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. sub-neg 100.0%

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

      3. un-div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-udef 100.0%

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

      6. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. unpow1 100.0%

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

      3. sqr-pow 53.4%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \log \left(1 + \mathsf{expm1}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      4. fabs-sqr 53.4%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \log \left(1 + \mathsf{expm1}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      5. sqr-pow 99.4%

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

      6. unpow1 99.4%

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

      7. unpow1 99.4%

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

      8. sqr-pow 53.4%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \log \left(1 + \mathsf{expm1}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      9. fabs-sqr 53.4%

         \[\leadsto \mathsf{fma}\left(-0.254829592 - \log \left(1 + \mathsf{expm1}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]

      10. sqr-pow 99.4%

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

      11. unpow1 99.4%

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

   10. Simplified 99.4%

       \[\leadsto \mathsf{fma}\left(-0.254829592 - \log \left(1 + \mathsf{expm1}\left(\frac{-0.284496736 + \frac{\color{blue}{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-5}:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 - \log \left(1 + \mathsf{expm1}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := 1 + \left|x\right| \cdot 0.3275911\\ t_2 := \frac{1}{t_1}\\ \mathbf{if}\;\left|x\right| \leq 10^{-5}:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(0.254829592 + t_2 \cdot \left(-0.284496736 + t_2 \cdot \left(1.421413741 + \frac{1}{t_0} \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\right) \cdot \frac{-1}{t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x 0.3275911)))
        (t_1 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_2 (/ 1.0 t_1)))
   (if (<= (fabs x) 1e-5)
     (+
      1e-9
      (fma
       (pow x 3.0)
       -0.37545125292247583
       (* x (+ 1.128386358070218 (* x -0.00011824294398844343)))))
     (+
      1.0
      (*
       (*
        (+
         0.254829592
         (*
          t_2
          (+
           -0.284496736
           (*
            t_2
            (+
             1.421413741
             (* (/ 1.0 t_0) (+ -1.453152027 (/ 1.061405429 t_0))))))))
        (exp (* x (- x))))
       (/ -1.0 t_1))))))

C

x = abs(x);
double code(double x) {
	double t_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) <= 1e-5) {
		tmp = 1e-9 + fma(pow(x, 3.0), -0.37545125292247583, (x * (1.128386358070218 + (x * -0.00011824294398844343))));
	} else {
		tmp = 1.0 + (((0.254829592 + (t_2 * (-0.284496736 + (t_2 * (1.421413741 + ((1.0 / t_0) * (-1.453152027 + (1.061405429 / t_0)))))))) * exp((x * -x))) * (-1.0 / t_1));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_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) <= 1e-5)
		tmp = Float64(1e-9 + fma((x ^ 3.0), -0.37545125292247583, Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343)))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(0.254829592 + Float64(t_2 * Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(Float64(1.0 / t_0) * Float64(-1.453152027 + Float64(1.061405429 / t_0)))))))) * exp(Float64(x * Float64(-x)))) * Float64(-1.0 / t_1)));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * 0.3275911), $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], 1e-5], N[(1e-9 + N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(0.254829592 + N[(t$95$2 * N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000008e-5

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   5. Applied egg-rr 57.8%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in x around 0 99.0%

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

      1. *-un-lft-identity 99.0%

         \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      2. fma-def 99.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)}\right) \]

      3. fma-def 99.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\mathsf{fma}\left(-0.00011824294398844343, {x}^{2}, 1.128386358070218 \cdot x\right)}\right)\right) \]

      4. pow2 99.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(-0.00011824294398844343, \color{blue}{x \cdot x}, 1.128386358070218 \cdot x\right)\right)\right) \]

      5. *-commutative 99.0%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \color{blue}{x \cdot 1.128386358070218}\right)\right)\right) \]

   10. Applied egg-rr 99.0%

       \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, x \cdot 1.128386358070218\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 99.0%

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

       2. fma-udef 99.0%

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

       3. *-commutative 99.0%

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

       4. fma-def 99.0%

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

       5. fma-def 99.0%

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

       6. +-commutative 99.0%

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

       7. *-commutative 99.0%

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

       8. associate-*r* 99.0%

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

       9. distribute-rgt-out 99.0%

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

       10. *-commutative 99.0%

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

   12. Simplified 99.0%

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

   if 1.00000000000000008e-5 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 99.9%

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

      2. expm1-udef 99.9%

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

      3. log1p-udef 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-udef 99.9%

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

      6. add-exp-log 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. associate--l+ 99.9%

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

      3. metadata-eval 99.9%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot \left|x\right| + \color{blue}{0}\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. +-rgt-identity 99.9%

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

      5. unpow1 99.9%

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

      6. sqr-pow 53.4%

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

      7. fabs-sqr 53.4%

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

      8. sqr-pow 99.4%

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

      9. unpow1 99.4%

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

   7. Simplified 99.4%

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

      1. expm1-log1p-u 99.9%

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

      2. expm1-udef 99.9%

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

      3. log1p-udef 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-udef 99.9%

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

      6. add-exp-log 99.9%

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

   9. Applied egg-rr 99.4%

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

       1. fma-udef 99.9%

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

       2. associate--l+ 99.9%

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

       3. metadata-eval 99.9%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot \left|x\right| + \color{blue}{0}\right)}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. +-rgt-identity 99.9%

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

       5. unpow1 99.9%

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

       6. sqr-pow 53.4%

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

       7. fabs-sqr 53.4%

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

       8. sqr-pow 99.4%

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

       9. unpow1 99.4%

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

   11. Simplified 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-5}:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 3: 99.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.7778892405807117 \cdot \frac{e^{x \cdot \left(-x\right)}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.1)
   (+
    1e-9
    (fma
     (pow x 3.0)
     -0.37545125292247583
     (* x (+ 1.128386358070218 (* x -0.00011824294398844343)))))
   (+ 1.0 (* -0.7778892405807117 (/ (exp (* x (- x))) x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.1) {
		tmp = 1e-9 + fma(pow(x, 3.0), -0.37545125292247583, (x * (1.128386358070218 + (x * -0.00011824294398844343))));
	} else {
		tmp = 1.0 + (-0.7778892405807117 * (exp((x * -x)) / x));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.1)
		tmp = Float64(1e-9 + fma((x ^ 3.0), -0.37545125292247583, Float64(x * Float64(1.128386358070218 + Float64(x * -0.00011824294398844343)))));
	else
		tmp = Float64(1.0 + Float64(-0.7778892405807117 * Float64(exp(Float64(x * Float64(-x))) / x)));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.1], N[(1e-9 + N[(N[Power[x, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(x * N[(1.128386358070218 + N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.7778892405807117 * N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   5. Applied egg-rr 71.6%

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

   7. Simplified 71.0%

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

   8. Taylor expanded in x around 0 67.3%

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

      1. *-un-lft-identity 67.3%

         \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)\right)} \]

      2. fma-def 67.3%

         \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{fma}\left(-0.37545125292247583, {x}^{3}, -0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)}\right) \]

      3. fma-def 67.3%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \color{blue}{\mathsf{fma}\left(-0.00011824294398844343, {x}^{2}, 1.128386358070218 \cdot x\right)}\right)\right) \]

      4. pow2 67.3%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(-0.00011824294398844343, \color{blue}{x \cdot x}, 1.128386358070218 \cdot x\right)\right)\right) \]

      5. *-commutative 67.3%

         \[\leadsto 1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, \color{blue}{x \cdot 1.128386358070218}\right)\right)\right) \]

   10. Applied egg-rr 67.3%

       \[\leadsto \color{blue}{1 \cdot \left(10^{-9} + \mathsf{fma}\left(-0.37545125292247583, {x}^{3}, \mathsf{fma}\left(-0.00011824294398844343, x \cdot x, x \cdot 1.128386358070218\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 67.3%

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

       2. fma-udef 67.3%

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

       3. *-commutative 67.3%

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

       4. fma-def 67.3%

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

       5. fma-def 67.3%

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

       6. +-commutative 67.3%

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

       7. *-commutative 67.3%

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

       8. associate-*r* 67.3%

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

       9. distribute-rgt-out 67.3%

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

       10. *-commutative 67.3%

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

   12. Simplified 67.3%

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

   if 1.1000000000000001 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 98.9%

      \[\leadsto \color{blue}{1 + -0.7778892405807117 \cdot \frac{e^{-1 \cdot {x}^{2}}}{x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 98.9%

         \[\leadsto 1 + -0.7778892405807117 \cdot \frac{e^{\color{blue}{-{x}^{2}}}}{x} \]

      2. unpow2 98.9%

         \[\leadsto 1 + -0.7778892405807117 \cdot \frac{e^{-\color{blue}{x \cdot x}}}{x} \]

      3. distribute-rgt-neg-out 98.9%

         \[\leadsto 1 + -0.7778892405807117 \cdot \frac{e^{\color{blue}{x \cdot \left(-x\right)}}}{x} \]

      4. exp-prod 98.9%

         \[\leadsto 1 + -0.7778892405807117 \cdot \frac{\color{blue}{{\left(e^{x}\right)}^{\left(-x\right)}}}{x} \]

   10. Simplified 98.9%

       \[\leadsto \color{blue}{1 + -0.7778892405807117 \cdot \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{x}} \]

   11. Taylor expanded in x around inf 98.9%

       \[\leadsto 1 + \color{blue}{-0.7778892405807117 \cdot \frac{e^{-1 \cdot {x}^{2}}}{x}} \]
   12. Step-by-step derivation

       1. associate-*r/ 98.9%

          \[\leadsto 1 + \color{blue}{\frac{-0.7778892405807117 \cdot e^{-1 \cdot {x}^{2}}}{x}} \]

       2. *-commutative 98.9%

          \[\leadsto 1 + \frac{\color{blue}{e^{-1 \cdot {x}^{2}} \cdot -0.7778892405807117}}{x} \]

       3. mul-1-neg 98.9%

          \[\leadsto 1 + \frac{e^{\color{blue}{-{x}^{2}}} \cdot -0.7778892405807117}{x} \]

       4. unpow2 98.9%

          \[\leadsto 1 + \frac{e^{-\color{blue}{x \cdot x}} \cdot -0.7778892405807117}{x} \]

       5. distribute-rgt-neg-in 98.9%

          \[\leadsto 1 + \frac{e^{\color{blue}{x \cdot \left(-x\right)}} \cdot -0.7778892405807117}{x} \]

       6. exp-prod 98.9%

          \[\leadsto 1 + \frac{\color{blue}{{\left(e^{x}\right)}^{\left(-x\right)}} \cdot -0.7778892405807117}{x} \]

       7. associate-*l/ 98.9%

          \[\leadsto 1 + \color{blue}{\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{x} \cdot -0.7778892405807117} \]

       8. *-commutative 98.9%

          \[\leadsto 1 + \color{blue}{-0.7778892405807117 \cdot \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{x}} \]

       9. exp-prod 98.9%

          \[\leadsto 1 + -0.7778892405807117 \cdot \frac{\color{blue}{e^{x \cdot \left(-x\right)}}}{x} \]

   13. Simplified 98.9%

       \[\leadsto 1 + \color{blue}{-0.7778892405807117 \cdot \frac{e^{x \cdot \left(-x\right)}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.7778892405807117 \cdot \frac{e^{x \cdot \left(-x\right)}}{x}\\ \end{array} \]

Alternative 4: 99.4% accurate, 6.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + \frac{1.3981393803054172 \cdot 10^{-8} \cdot {x}^{4} - \left(x \cdot x\right) \cdot 1.2732557730789702}{-0.00011824294398844343 \cdot \left(x \cdot x\right) - x \cdot 1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.7778892405807117 \cdot \frac{e^{x \cdot \left(-x\right)}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.9)
   (+
    1e-9
    (/
     (- (* 1.3981393803054172e-8 (pow x 4.0)) (* (* x x) 1.2732557730789702))
     (- (* -0.00011824294398844343 (* x x)) (* x 1.128386358070218))))
   (+ 1.0 (* -0.7778892405807117 (/ (exp (* x (- x))) x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.9) {
		tmp = 1e-9 + (((1.3981393803054172e-8 * pow(x, 4.0)) - ((x * x) * 1.2732557730789702)) / ((-0.00011824294398844343 * (x * x)) - (x * 1.128386358070218)));
	} else {
		tmp = 1.0 + (-0.7778892405807117 * (exp((x * -x)) / x));
	}
	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 + (((1.3981393803054172d-8 * (x ** 4.0d0)) - ((x * x) * 1.2732557730789702d0)) / (((-0.00011824294398844343d0) * (x * x)) - (x * 1.128386358070218d0)))
    else
        tmp = 1.0d0 + ((-0.7778892405807117d0) * (exp((x * -x)) / x))
    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 + (((1.3981393803054172e-8 * Math.pow(x, 4.0)) - ((x * x) * 1.2732557730789702)) / ((-0.00011824294398844343 * (x * x)) - (x * 1.128386358070218)));
	} else {
		tmp = 1.0 + (-0.7778892405807117 * (Math.exp((x * -x)) / x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.9:
		tmp = 1e-9 + (((1.3981393803054172e-8 * math.pow(x, 4.0)) - ((x * x) * 1.2732557730789702)) / ((-0.00011824294398844343 * (x * x)) - (x * 1.128386358070218)))
	else:
		tmp = 1.0 + (-0.7778892405807117 * (math.exp((x * -x)) / x))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.9)
		tmp = Float64(1e-9 + Float64(Float64(Float64(1.3981393803054172e-8 * (x ^ 4.0)) - Float64(Float64(x * x) * 1.2732557730789702)) / Float64(Float64(-0.00011824294398844343 * Float64(x * x)) - Float64(x * 1.128386358070218))));
	else
		tmp = Float64(1.0 + Float64(-0.7778892405807117 * Float64(exp(Float64(x * Float64(-x))) / x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.9)
		tmp = 1e-9 + (((1.3981393803054172e-8 * (x ^ 4.0)) - ((x * x) * 1.2732557730789702)) / ((-0.00011824294398844343 * (x * x)) - (x * 1.128386358070218)));
	else
		tmp = 1.0 + (-0.7778892405807117 * (exp((x * -x)) / x));
	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[(N[(N[(1.3981393803054172e-8 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision] / N[(N[(-0.00011824294398844343 * N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.7778892405807117 * N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   5. Applied egg-rr 71.6%

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

   7. Simplified 71.0%

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

   8. Taylor expanded in x around 0 66.7%

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

      1. fma-def 66.7%

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

      2. unpow2 66.7%

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

      3. *-commutative 66.7%

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

   10. Simplified 66.7%

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

       1. pow2 66.7%

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

       2. *-commutative 66.7%

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

       3. fma-def 66.7%

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

       4. flip-+ 66.6%

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

       5. pow2 66.6%

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

       6. pow2 66.6%

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

       7. *-commutative 66.6%

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

       8. *-commutative 66.6%

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

       9. pow2 66.6%

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

       10. *-commutative 66.6%

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

   12. Applied egg-rr 66.6%

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

       1. unpow2 66.6%

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

       2. unpow2 66.6%

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

       3. swap-sqr 66.6%

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

       4. metadata-eval 66.6%

          \[\leadsto 10^{-9} + \frac{\color{blue}{1.3981393803054172 \cdot 10^{-8}} \cdot \left({x}^{2} \cdot {x}^{2}\right) - \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)}{-0.00011824294398844343 \cdot \left(x \cdot x\right) - x \cdot 1.128386358070218} \]

       5. pow-sqr 66.6%

          \[\leadsto 10^{-9} + \frac{1.3981393803054172 \cdot 10^{-8} \cdot \color{blue}{{x}^{\left(2 \cdot 2\right)}} - \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)}{-0.00011824294398844343 \cdot \left(x \cdot x\right) - x \cdot 1.128386358070218} \]

       6. metadata-eval 66.6%

          \[\leadsto 10^{-9} + \frac{1.3981393803054172 \cdot 10^{-8} \cdot {x}^{\color{blue}{4}} - \left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)}{-0.00011824294398844343 \cdot \left(x \cdot x\right) - x \cdot 1.128386358070218} \]

       7. swap-sqr 66.6%

          \[\leadsto 10^{-9} + \frac{1.3981393803054172 \cdot 10^{-8} \cdot {x}^{4} - \color{blue}{\left(x \cdot x\right) \cdot \left(1.128386358070218 \cdot 1.128386358070218\right)}}{-0.00011824294398844343 \cdot \left(x \cdot x\right) - x \cdot 1.128386358070218} \]

       8. metadata-eval 66.6%

          \[\leadsto 10^{-9} + \frac{1.3981393803054172 \cdot 10^{-8} \cdot {x}^{4} - \left(x \cdot x\right) \cdot \color{blue}{1.2732557730789702}}{-0.00011824294398844343 \cdot \left(x \cdot x\right) - x \cdot 1.128386358070218} \]

       9. unpow2 66.6%

          \[\leadsto 10^{-9} + \frac{1.3981393803054172 \cdot 10^{-8} \cdot {x}^{4} - \left(x \cdot x\right) \cdot 1.2732557730789702}{-0.00011824294398844343 \cdot \color{blue}{{x}^{2}} - x \cdot 1.128386358070218} \]

       10. *-commutative 66.6%

           \[\leadsto 10^{-9} + \frac{1.3981393803054172 \cdot 10^{-8} \cdot {x}^{4} - \left(x \cdot x\right) \cdot 1.2732557730789702}{\color{blue}{{x}^{2} \cdot -0.00011824294398844343} - x \cdot 1.128386358070218} \]

       11. unpow2 66.6%

           \[\leadsto 10^{-9} + \frac{1.3981393803054172 \cdot 10^{-8} \cdot {x}^{4} - \left(x \cdot x\right) \cdot 1.2732557730789702}{\color{blue}{\left(x \cdot x\right)} \cdot -0.00011824294398844343 - x \cdot 1.128386358070218} \]

   14. Simplified 66.6%

       \[\leadsto 10^{-9} + \color{blue}{\frac{1.3981393803054172 \cdot 10^{-8} \cdot {x}^{4} - \left(x \cdot x\right) \cdot 1.2732557730789702}{\left(x \cdot x\right) \cdot -0.00011824294398844343 - x \cdot 1.128386358070218}} \]

   if 0.900000000000000022 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 98.9%

      \[\leadsto \color{blue}{1 + -0.7778892405807117 \cdot \frac{e^{-1 \cdot {x}^{2}}}{x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 98.9%

         \[\leadsto 1 + -0.7778892405807117 \cdot \frac{e^{\color{blue}{-{x}^{2}}}}{x} \]

      2. unpow2 98.9%

         \[\leadsto 1 + -0.7778892405807117 \cdot \frac{e^{-\color{blue}{x \cdot x}}}{x} \]

      3. distribute-rgt-neg-out 98.9%

         \[\leadsto 1 + -0.7778892405807117 \cdot \frac{e^{\color{blue}{x \cdot \left(-x\right)}}}{x} \]

      4. exp-prod 98.9%

         \[\leadsto 1 + -0.7778892405807117 \cdot \frac{\color{blue}{{\left(e^{x}\right)}^{\left(-x\right)}}}{x} \]

   10. Simplified 98.9%

       \[\leadsto \color{blue}{1 + -0.7778892405807117 \cdot \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{x}} \]

   11. Taylor expanded in x around inf 98.9%

       \[\leadsto 1 + \color{blue}{-0.7778892405807117 \cdot \frac{e^{-1 \cdot {x}^{2}}}{x}} \]
   12. Step-by-step derivation

       1. associate-*r/ 98.9%

          \[\leadsto 1 + \color{blue}{\frac{-0.7778892405807117 \cdot e^{-1 \cdot {x}^{2}}}{x}} \]

       2. *-commutative 98.9%

          \[\leadsto 1 + \frac{\color{blue}{e^{-1 \cdot {x}^{2}} \cdot -0.7778892405807117}}{x} \]

       3. mul-1-neg 98.9%

          \[\leadsto 1 + \frac{e^{\color{blue}{-{x}^{2}}} \cdot -0.7778892405807117}{x} \]

       4. unpow2 98.9%

          \[\leadsto 1 + \frac{e^{-\color{blue}{x \cdot x}} \cdot -0.7778892405807117}{x} \]

       5. distribute-rgt-neg-in 98.9%

          \[\leadsto 1 + \frac{e^{\color{blue}{x \cdot \left(-x\right)}} \cdot -0.7778892405807117}{x} \]

       6. exp-prod 98.9%

          \[\leadsto 1 + \frac{\color{blue}{{\left(e^{x}\right)}^{\left(-x\right)}} \cdot -0.7778892405807117}{x} \]

       7. associate-*l/ 98.9%

          \[\leadsto 1 + \color{blue}{\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{x} \cdot -0.7778892405807117} \]

       8. *-commutative 98.9%

          \[\leadsto 1 + \color{blue}{-0.7778892405807117 \cdot \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{x}} \]

       9. exp-prod 98.9%

          \[\leadsto 1 + -0.7778892405807117 \cdot \frac{\color{blue}{e^{x \cdot \left(-x\right)}}}{x} \]

   13. Simplified 98.9%

       \[\leadsto 1 + \color{blue}{-0.7778892405807117 \cdot \frac{e^{x \cdot \left(-x\right)}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + \frac{1.3981393803054172 \cdot 10^{-8} \cdot {x}^{4} - \left(x \cdot x\right) \cdot 1.2732557730789702}{-0.00011824294398844343 \cdot \left(x \cdot x\right) - x \cdot 1.128386358070218}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.7778892405807117 \cdot \frac{e^{x \cdot \left(-x\right)}}{x}\\ \end{array} \]

Alternative 5: 99.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.7778892405807117 \cdot \frac{e^{x \cdot \left(-x\right)}}{x}\\ \end{array} \end{array} \]

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 (* -0.7778892405807117 (/ (exp (* x (- x))) x)))))

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 + (-0.7778892405807117 * (exp((x * -x)) / x));
	}
	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 + ((-0.7778892405807117d0) * (exp((x * -x)) / x))
    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 + (-0.7778892405807117 * (Math.exp((x * -x)) / x));
	}
	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 + (-0.7778892405807117 * (math.exp((x * -x)) / x))
	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 = Float64(1.0 + Float64(-0.7778892405807117 * Float64(exp(Float64(x * Float64(-x))) / x)));
	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 + (-0.7778892405807117 * (exp((x * -x)) / x));
	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], N[(1.0 + N[(-0.7778892405807117 * N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   5. Applied egg-rr 71.6%

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

   7. Simplified 71.0%

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

   8. Taylor expanded in x around 0 66.7%

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

      1. fma-def 66.7%

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

      2. unpow2 66.7%

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

      3. *-commutative 66.7%

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

   10. Simplified 66.7%

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

   11. Taylor expanded in x around 0 66.7%

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

       1. +-commutative 66.7%

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

       2. unpow2 66.7%

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

       3. associate-*r* 66.7%

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

       4. distribute-rgt-out 66.7%

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

       5. *-commutative 66.7%

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

   13. Simplified 66.7%

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

   if 0.900000000000000022 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 98.9%

      \[\leadsto \color{blue}{1 + -0.7778892405807117 \cdot \frac{e^{-1 \cdot {x}^{2}}}{x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 98.9%

         \[\leadsto 1 + -0.7778892405807117 \cdot \frac{e^{\color{blue}{-{x}^{2}}}}{x} \]

      2. unpow2 98.9%

         \[\leadsto 1 + -0.7778892405807117 \cdot \frac{e^{-\color{blue}{x \cdot x}}}{x} \]

      3. distribute-rgt-neg-out 98.9%

         \[\leadsto 1 + -0.7778892405807117 \cdot \frac{e^{\color{blue}{x \cdot \left(-x\right)}}}{x} \]

      4. exp-prod 98.9%

         \[\leadsto 1 + -0.7778892405807117 \cdot \frac{\color{blue}{{\left(e^{x}\right)}^{\left(-x\right)}}}{x} \]

   10. Simplified 98.9%

       \[\leadsto \color{blue}{1 + -0.7778892405807117 \cdot \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{x}} \]

   11. Taylor expanded in x around inf 98.9%

       \[\leadsto 1 + \color{blue}{-0.7778892405807117 \cdot \frac{e^{-1 \cdot {x}^{2}}}{x}} \]
   12. Step-by-step derivation

       1. associate-*r/ 98.9%

          \[\leadsto 1 + \color{blue}{\frac{-0.7778892405807117 \cdot e^{-1 \cdot {x}^{2}}}{x}} \]

       2. *-commutative 98.9%

          \[\leadsto 1 + \frac{\color{blue}{e^{-1 \cdot {x}^{2}} \cdot -0.7778892405807117}}{x} \]

       3. mul-1-neg 98.9%

          \[\leadsto 1 + \frac{e^{\color{blue}{-{x}^{2}}} \cdot -0.7778892405807117}{x} \]

       4. unpow2 98.9%

          \[\leadsto 1 + \frac{e^{-\color{blue}{x \cdot x}} \cdot -0.7778892405807117}{x} \]

       5. distribute-rgt-neg-in 98.9%

          \[\leadsto 1 + \frac{e^{\color{blue}{x \cdot \left(-x\right)}} \cdot -0.7778892405807117}{x} \]

       6. exp-prod 98.9%

          \[\leadsto 1 + \frac{\color{blue}{{\left(e^{x}\right)}^{\left(-x\right)}} \cdot -0.7778892405807117}{x} \]

       7. associate-*l/ 98.9%

          \[\leadsto 1 + \color{blue}{\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{x} \cdot -0.7778892405807117} \]

       8. *-commutative 98.9%

          \[\leadsto 1 + \color{blue}{-0.7778892405807117 \cdot \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{x}} \]

       9. exp-prod 98.9%

          \[\leadsto 1 + -0.7778892405807117 \cdot \frac{\color{blue}{e^{x \cdot \left(-x\right)}}}{x} \]

   13. Simplified 98.9%

       \[\leadsto 1 + \color{blue}{-0.7778892405807117 \cdot \frac{e^{x \cdot \left(-x\right)}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.7778892405807117 \cdot \frac{e^{x \cdot \left(-x\right)}}{x}\\ \end{array} \]

Alternative 6: 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 71.6%

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

   3. Simplified 71.6%

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

   5. Applied egg-rr 71.6%

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

   7. Simplified 71.0%

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

   8. Taylor expanded in x around 0 66.7%

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

      1. fma-def 66.7%

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

      2. unpow2 66.7%

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

      3. *-commutative 66.7%

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

   10. Simplified 66.7%

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

   11. Taylor expanded in x around 0 66.7%

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

       1. +-commutative 66.7%

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

       2. unpow2 66.7%

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

       3. associate-*r* 66.7%

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

       4. distribute-rgt-out 66.7%

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

       5. *-commutative 66.7%

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

   13. Simplified 66.7%

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

   if 0.900000000000000022 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 98.9%

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

4. Final simplification 75.5%

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

Alternative 7: 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 71.6%

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

   3. Simplified 71.6%

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

   5. Applied egg-rr 71.6%

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

   7. Simplified 71.0%

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

   8. Taylor expanded in x around 0 66.7%

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

      1. *-commutative 66.7%

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

   10. Simplified 66.7%

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

   if 0.900000000000000022 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 98.9%

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

4. Final simplification 75.5%

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

Alternative 8: 97.7% accurate, 279.5× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 2.8e-5) 1e-9 1.0))

C

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

Python

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

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.8e-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 <= 2.8e-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, 2.8e-5], 1e-9, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   5. Applied egg-rr 71.6%

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

   7. Simplified 71.0%

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

   8. Taylor expanded in x around 0 68.5%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 98.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 53.4% 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 79.4%

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

3. Simplified 79.4%

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

5. Applied egg-rr 79.4%

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

7. Simplified 78.9%

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

8. Taylor expanded in x around 0 52.8%

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

9. Final simplification 52.8%

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

Reproduce

herbie shell --seed 2023293 
(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)))))))