Jmat.Real.erf

Percentage Accurate: 79.0% → 99.9%

Time: 22.9s

Alternatives: 13

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: 79.0% 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, 0.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := e^{x \cdot \left(-x\right)}\\ t_1 := {\left(e^{x}\right)}^{x}\\ t_2 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{1.453152027}{t_1}}{{t_2}^{4}} + \mathsf{fma}\left(0.284496736, \frac{t_0}{{t_2}^{2}}, 1\right)\right) - \frac{\frac{1.421413741}{t_1}}{{t_2}^{3}}\right) - \mathsf{fma}\left(1.061405429, \frac{t_0}{{t_2}^{5}}, \frac{0.254829592}{t_1 \cdot t_2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* x (- x))))
        (t_1 (pow (exp x) x))
        (t_2 (fma (fabs x) 0.3275911 1.0)))
   (if (<= (fabs x) 0.0001)
     (+
      (fma x 1.128386358070218 1e-9)
      (* (* x x) (+ -0.00011824294398844343 (* x -0.37545125292247583))))
     (-
      (-
       (+
        (/ (/ 1.453152027 t_1) (pow t_2 4.0))
        (fma 0.284496736 (/ t_0 (pow t_2 2.0)) 1.0))
       (/ (/ 1.421413741 t_1) (pow t_2 3.0)))
      (fma 1.061405429 (/ t_0 (pow t_2 5.0)) (/ 0.254829592 (* t_1 t_2)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = exp((x * -x));
	double t_1 = pow(exp(x), x);
	double t_2 = fma(fabs(x), 0.3275911, 1.0);
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = fma(x, 1.128386358070218, 1e-9) + ((x * x) * (-0.00011824294398844343 + (x * -0.37545125292247583)));
	} else {
		tmp = ((((1.453152027 / t_1) / pow(t_2, 4.0)) + fma(0.284496736, (t_0 / pow(t_2, 2.0)), 1.0)) - ((1.421413741 / t_1) / pow(t_2, 3.0))) - fma(1.061405429, (t_0 / pow(t_2, 5.0)), (0.254829592 / (t_1 * t_2)));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = exp(Float64(x * Float64(-x)))
	t_1 = exp(x) ^ x
	t_2 = fma(abs(x), 0.3275911, 1.0)
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64(fma(x, 1.128386358070218, 1e-9) + Float64(Float64(x * x) * Float64(-0.00011824294398844343 + Float64(x * -0.37545125292247583))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.453152027 / t_1) / (t_2 ^ 4.0)) + fma(0.284496736, Float64(t_0 / (t_2 ^ 2.0)), 1.0)) - Float64(Float64(1.421413741 / t_1) / (t_2 ^ 3.0))) - fma(1.061405429, Float64(t_0 / (t_2 ^ 5.0)), Float64(0.254829592 / Float64(t_1 * t_2))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(N[(x * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.00011824294398844343 + N[(x * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.453152027 / t$95$1), $MachinePrecision] / N[Power[t$95$2, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.284496736 * N[(t$95$0 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(1.421413741 / t$95$1), $MachinePrecision] / N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.061405429 * N[(t$95$0 / N[Power[t$95$2, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.254829592 / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   1. Initial program 57.8%

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

   3. Applied egg-rr 57.8%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in x around 0 96.9%

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

      1. +-commutative 96.9%

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

      2. associate-+r+ 96.9%

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

      3. associate-+l+ 96.9%

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

      4. *-commutative 96.9%

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

      5. fma-def 96.9%

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

      6. unpow2 96.9%

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

      7. *-commutative 96.9%

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

      8. *-commutative 96.9%

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

      9. fma-def 96.9%

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

   8. Simplified 96.9%

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

   9. Taylor expanded in x around 0 96.9%

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

       1. associate-+r+ 96.9%

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

       2. *-commutative 96.9%

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

       3. unpow2 96.9%

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

       4. *-commutative 96.9%

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

       5. fma-udef 96.9%

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

       6. associate-+r+ 96.9%

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

       7. +-commutative 96.9%

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

       8. associate-+l+ 96.9%

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

       9. +-commutative 96.9%

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

       10. *-commutative 96.9%

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

       11. +-commutative 96.9%

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

       12. fma-def 96.9%

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

       13. fma-udef 96.9%

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

       14. unpow2 96.9%

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

       15. unpow3 96.9%

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

       16. unpow2 96.9%

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

       17. associate-*l* 96.9%

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

       18. distribute-lft-out 96.9%

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

   11. Simplified 96.9%

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

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{\left(1 + \left(1.453152027 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{4} \cdot e^{{x}^{2}}} + 0.284496736 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2} \cdot e^{{x}^{2}}}\right)\right) - \left(1.421413741 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3} \cdot e^{{x}^{2}}} + \left(1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{5} \cdot e^{{x}^{2}}} + 0.254829592 \cdot \frac{1}{\left(0.3275911 \cdot \left|x\right| + 1\right) \cdot e^{{x}^{2}}}\right)\right)} \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\left(\left(\frac{\frac{1.453152027}{{\left(e^{x}\right)}^{x}}}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{4}} + \mathsf{fma}\left(0.284496736, \frac{e^{x \cdot \left(-x\right)}}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}}, 1\right)\right) - \frac{\frac{1.421413741}{{\left(e^{x}\right)}^{x}}}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{3}}\right) - \mathsf{fma}\left(1.061405429, \frac{e^{x \cdot \left(-x\right)}}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{5}}, \frac{0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{1.453152027}{{\left(e^{x}\right)}^{x}}}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{4}} + \mathsf{fma}\left(0.284496736, \frac{e^{x \cdot \left(-x\right)}}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}}, 1\right)\right) - \frac{\frac{1.421413741}{{\left(e^{x}\right)}^{x}}}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{3}}\right) - \mathsf{fma}\left(1.061405429, \frac{e^{x \cdot \left(-x\right)}}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{5}}, \frac{0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := {\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(x, 0.3275911, 1\right)\\ t_1 := \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\\ t_2 := t_1 - -0.254829592\\ \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(-0.254829592 - t_1\right) \cdot t_2}{t_0 \cdot t_0}}{1 + \frac{t_2}{t_0}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (pow (exp x) x) (fma x 0.3275911 1.0)))
        (t_1
         (/
          (+
           -0.284496736
           (/
            (+
             1.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (fma x 0.3275911 1.0)))
              (fma x 0.3275911 1.0)))
            (fma x 0.3275911 1.0)))
          (fma x 0.3275911 1.0)))
        (t_2 (- t_1 -0.254829592)))
   (if (<= (fabs x) 0.0001)
     (+
      (fma x 1.128386358070218 1e-9)
      (* (* x x) (+ -0.00011824294398844343 (* x -0.37545125292247583))))
     (/
      (+ 1.0 (/ (* (- -0.254829592 t_1) t_2) (* t_0 t_0)))
      (+ 1.0 (/ t_2 t_0))))))

C

x = abs(x);
double code(double x) {
	double t_0 = pow(exp(x), x) * fma(x, 0.3275911, 1.0);
	double t_1 = (-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0);
	double t_2 = t_1 - -0.254829592;
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = fma(x, 1.128386358070218, 1e-9) + ((x * x) * (-0.00011824294398844343 + (x * -0.37545125292247583)));
	} else {
		tmp = (1.0 + (((-0.254829592 - t_1) * t_2) / (t_0 * t_0))) / (1.0 + (t_2 / t_0));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64((exp(x) ^ x) * fma(x, 0.3275911, 1.0))
	t_1 = Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))
	t_2 = Float64(t_1 - -0.254829592)
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64(fma(x, 1.128386358070218, 1e-9) + Float64(Float64(x * x) * Float64(-0.00011824294398844343 + Float64(x * -0.37545125292247583))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.254829592 - t_1) * t_2) / Float64(t_0 * t_0))) / Float64(1.0 + Float64(t_2 / t_0)));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - -0.254829592), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(N[(x * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.00011824294398844343 + N[(x * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(-0.254829592 - t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   1. Initial program 57.8%

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

   3. Applied egg-rr 57.8%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in x around 0 96.9%

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

      1. +-commutative 96.9%

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

      2. associate-+r+ 96.9%

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

      3. associate-+l+ 96.9%

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

      4. *-commutative 96.9%

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

      5. fma-def 96.9%

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

      6. unpow2 96.9%

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

      7. *-commutative 96.9%

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

      8. *-commutative 96.9%

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

      9. fma-def 96.9%

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

   8. Simplified 96.9%

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

   9. Taylor expanded in x around 0 96.9%

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

       1. associate-+r+ 96.9%

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

       2. *-commutative 96.9%

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

       3. unpow2 96.9%

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

       4. *-commutative 96.9%

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

       5. fma-udef 96.9%

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

       6. associate-+r+ 96.9%

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

       7. +-commutative 96.9%

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

       8. associate-+l+ 96.9%

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

       9. +-commutative 96.9%

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

       10. *-commutative 96.9%

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

       11. +-commutative 96.9%

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

       12. fma-def 96.9%

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

       13. fma-udef 96.9%

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

       14. unpow2 96.9%

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

       15. unpow3 96.9%

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

       16. unpow2 96.9%

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

       17. associate-*l* 96.9%

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

       18. distribute-lft-out 96.9%

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

   11. Simplified 96.9%

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

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Applied egg-rr 99.8%

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

   5. Simplified 99.1%

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

      1. add-log-exp 99.1%

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

      2. log1p-udef 99.1%

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

      3. add-exp-log 99.1%

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

      4. pow-exp 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. fma-udef 99.1%

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

   9. Applied egg-rr 99.1%

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

       1. add-log-exp 99.2%

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

       2. flip-+ 99.1%

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

   11. Applied egg-rr 99.1%

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

   13. Simplified 99.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right) \cdot \left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} - -0.254829592\right)}{\left({\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(x, 0.3275911, 1\right)\right) \cdot \left({\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(x, 0.3275911, 1\right)\right)}}{1 + \frac{\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)} - -0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(x, 0.3275911, 1\right)}}\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0001)
   (+
    (fma x 1.128386358070218 1e-9)
    (* (* x x) (+ -0.00011824294398844343 (* x -0.37545125292247583))))
   (+
    1.0
    (/
     (-
      -0.254829592
      (/
       (+
        -0.284496736
        (/
         (+
          1.421413741
          (/
           (+ -1.453152027 (/ 1.061405429 (fma x 0.3275911 1.0)))
           (fma x 0.3275911 1.0)))
         (fma x 0.3275911 1.0)))
       (fma x 0.3275911 1.0)))
     (* (fma x 0.3275911 1.0) (exp (* x x)))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = fma(x, 1.128386358070218, 1e-9) + ((x * x) * (-0.00011824294398844343 + (x * -0.37545125292247583)));
	} else {
		tmp = 1.0 + ((-0.254829592 - ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / (fma(x, 0.3275911, 1.0) * exp((x * x))));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64(fma(x, 1.128386358070218, 1e-9) + Float64(Float64(x * x) * Float64(-0.00011824294398844343 + Float64(x * -0.37545125292247583))));
	else
		tmp = Float64(1.0 + Float64(Float64(-0.254829592 - Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / Float64(fma(x, 0.3275911, 1.0) * exp(Float64(x * x)))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(N[(x * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.00011824294398844343 + N[(x * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-0.254829592 - N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   1. Initial program 57.8%

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

   3. Applied egg-rr 57.8%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in x around 0 96.9%

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

      1. +-commutative 96.9%

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

      2. associate-+r+ 96.9%

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

      3. associate-+l+ 96.9%

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

      4. *-commutative 96.9%

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

      5. fma-def 96.9%

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

      6. unpow2 96.9%

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

      7. *-commutative 96.9%

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

      8. *-commutative 96.9%

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

      9. fma-def 96.9%

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

   8. Simplified 96.9%

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

   9. Taylor expanded in x around 0 96.9%

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

       1. associate-+r+ 96.9%

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

       2. *-commutative 96.9%

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

       3. unpow2 96.9%

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

       4. *-commutative 96.9%

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

       5. fma-udef 96.9%

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

       6. associate-+r+ 96.9%

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

       7. +-commutative 96.9%

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

       8. associate-+l+ 96.9%

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

       9. +-commutative 96.9%

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

       10. *-commutative 96.9%

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

       11. +-commutative 96.9%

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

       12. fma-def 96.9%

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

       13. fma-udef 96.9%

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

       14. unpow2 96.9%

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

       15. unpow3 96.9%

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

       16. unpow2 96.9%

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

       17. associate-*l* 96.9%

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

       18. distribute-lft-out 96.9%

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

   11. Simplified 96.9%

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

   if 1.00000000000000005e-4 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Applied egg-rr 99.8%

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

   5. Simplified 99.1%

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

      1. log1p-udef 99.1%

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

      2. add-exp-log 99.2%

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

      3. pow-exp 99.2%

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

   7. Applied egg-rr 99.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := 1 + x \cdot 0.3275911\\ t_2 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq 0.0006:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{t_1} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\left(-0.284496736 + t_2 \cdot \left(1.421413741 + t_2 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right) \cdot \frac{-1}{t_1} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_1 (+ 1.0 (* x 0.3275911)))
        (t_2 (/ 1.0 t_0)))
   (if (<= x 0.0006)
     (+
      (fma x 1.128386358070218 1e-9)
      (* (* x x) (+ -0.00011824294398844343 (* x -0.37545125292247583))))
     (+
      1.0
      (*
       (/ 1.0 t_1)
       (*
        (exp (* x (- x)))
        (-
         (*
          (+
           -0.284496736
           (*
            t_2
            (+ 1.421413741 (* t_2 (+ -1.453152027 (/ 1.061405429 t_0))))))
          (/ -1.0 t_1))
         0.254829592)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 + (x * 0.3275911);
	double t_2 = 1.0 / t_0;
	double tmp;
	if (x <= 0.0006) {
		tmp = fma(x, 1.128386358070218, 1e-9) + ((x * x) * (-0.00011824294398844343 + (x * -0.37545125292247583)));
	} else {
		tmp = 1.0 + ((1.0 / t_1) * (exp((x * -x)) * (((-0.284496736 + (t_2 * (1.421413741 + (t_2 * (-1.453152027 + (1.061405429 / t_0)))))) * (-1.0 / t_1)) - 0.254829592)));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 + Float64(x * 0.3275911))
	t_2 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x <= 0.0006)
		tmp = Float64(fma(x, 1.128386358070218, 1e-9) + Float64(Float64(x * x) * Float64(-0.00011824294398844343 + Float64(x * -0.37545125292247583))));
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 / t_1) * Float64(exp(Float64(x * Float64(-x))) * Float64(Float64(Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(t_2 * Float64(-1.453152027 + Float64(1.061405429 / t_0)))))) * Float64(-1.0 / t_1)) - 0.254829592))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x, 0.0006], N[(N[(x * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.00011824294398844343 + N[(x * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(t$95$2 * N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999947e-4

   1. Initial program 71.7%

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

   3. Applied egg-rr 71.6%

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

   5. Simplified 70.2%

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

   6. Taylor expanded in x around 0 65.8%

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

      1. +-commutative 65.8%

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

      2. associate-+r+ 65.8%

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

      3. associate-+l+ 65.8%

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

      4. *-commutative 65.8%

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

      5. fma-def 65.8%

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

      6. unpow2 65.8%

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

      7. *-commutative 65.8%

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

      8. *-commutative 65.8%

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

      9. fma-def 65.8%

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

   8. Simplified 65.8%

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

   9. Taylor expanded in x around 0 65.8%

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

       1. associate-+r+ 65.8%

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

       2. *-commutative 65.8%

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

       3. unpow2 65.8%

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

       4. *-commutative 65.8%

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

       5. fma-udef 65.8%

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

       6. associate-+r+ 65.8%

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

       7. +-commutative 65.8%

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

       8. associate-+l+ 65.8%

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

       9. +-commutative 65.8%

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

       10. *-commutative 65.8%

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

       11. +-commutative 65.8%

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

       12. fma-def 65.8%

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

       13. fma-udef 65.8%

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

       14. unpow2 65.8%

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

       15. unpow3 65.8%

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

       16. unpow2 65.8%

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

       17. associate-*l* 65.8%

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

       18. distribute-lft-out 66.3%

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

   11. Simplified 66.3%

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

   if 5.99999999999999947e-4 < x

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 99.8%

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

      2. expm1-udef 99.8%

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

      3. log1p-udef 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-udef 99.8%

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

      6. add-exp-log 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-def 99.8%

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

      2. associate--l+ 99.8%

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

      3. metadata-eval 99.8%

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

      4. +-rgt-identity 99.8%

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

      5. *-commutative 99.8%

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

      6. unpow1 99.8%

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

      7. sqr-pow 99.8%

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

      8. fabs-sqr 99.8%

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

      9. sqr-pow 99.8%

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

      10. unpow1 99.8%

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

   7. Simplified 99.8%

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

      1. expm1-log1p-u 99.8%

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

      2. expm1-udef 99.8%

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

      3. log1p-udef 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-udef 99.8%

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

      6. add-exp-log 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. fma-def 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

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

       4. +-rgt-identity 99.8%

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

       5. *-commutative 99.8%

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

       6. unpow1 99.8%

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

       7. sqr-pow 99.8%

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

       8. fabs-sqr 99.8%

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

       9. sqr-pow 99.8%

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

       10. unpow1 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0006:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right) \cdot \frac{-1}{1 + x \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]

Alternative 5: 99.8% accurate, 2.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (+
    (fma x 1.128386358070218 1e-9)
    (* (* x x) (+ -0.00011824294398844343 (* x -0.37545125292247583))))
   (-
    1.0
    (log (exp (/ 0.254829592 (* (fma x 0.3275911 1.0) (exp (* x x)))))))))

C

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

Julia

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.0], N[(N[(x * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.00011824294398844343 + N[(x * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[Exp[N[(0.254829592 / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 71.7%

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

   3. Applied egg-rr 71.7%

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

   5. Simplified 70.3%

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

   6. Taylor expanded in x around 0 65.9%

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

      1. +-commutative 65.9%

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

      2. associate-+r+ 65.9%

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

      3. associate-+l+ 65.9%

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

      4. *-commutative 65.9%

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

      5. fma-def 65.9%

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

      6. unpow2 65.9%

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

      7. *-commutative 65.9%

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

      8. *-commutative 65.9%

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

      9. fma-def 65.9%

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

   8. Simplified 65.9%

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

   9. Taylor expanded in x around 0 65.9%

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

       1. associate-+r+ 65.9%

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

       2. *-commutative 65.9%

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

       3. unpow2 65.9%

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

       4. *-commutative 65.9%

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

       5. fma-udef 65.9%

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

       6. associate-+r+ 65.9%

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

       7. +-commutative 65.9%

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

       8. associate-+l+ 65.9%

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

       9. +-commutative 65.9%

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

       10. *-commutative 65.9%

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

       11. +-commutative 65.9%

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

       12. fma-def 65.9%

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

       13. fma-udef 65.9%

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

       14. unpow2 65.9%

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

       15. unpow3 65.9%

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

       16. unpow2 65.9%

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

       17. associate-*l* 65.9%

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

       18. distribute-lft-out 66.4%

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

   11. Simplified 66.4%

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

   if 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.3%

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

      1. add-log-exp 99.3%

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

      2. pow-exp 99.3%

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

   7. Applied egg-rr 99.3%

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

4. Final simplification 75.8%

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

Alternative 6: 99.8% accurate, 4.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (+
    (fma x 1.128386358070218 1e-9)
    (* (* x x) (+ -0.00011824294398844343 (* x -0.37545125292247583))))
   (- 1.0 (/ 0.254829592 (* (fma x 0.3275911 1.0) (exp (* x x)))))))

C

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

Julia

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.0], N[(N[(x * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.00011824294398844343 + N[(x * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.254829592 / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 71.7%

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

   3. Applied egg-rr 71.7%

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

   5. Simplified 70.3%

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

   6. Taylor expanded in x around 0 65.9%

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

      1. +-commutative 65.9%

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

      2. associate-+r+ 65.9%

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

      3. associate-+l+ 65.9%

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

      4. *-commutative 65.9%

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

      5. fma-def 65.9%

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

      6. unpow2 65.9%

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

      7. *-commutative 65.9%

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

      8. *-commutative 65.9%

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

      9. fma-def 65.9%

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

   8. Simplified 65.9%

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

   9. Taylor expanded in x around 0 65.9%

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

       1. associate-+r+ 65.9%

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

       2. *-commutative 65.9%

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

       3. unpow2 65.9%

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

       4. *-commutative 65.9%

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

       5. fma-udef 65.9%

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

       6. associate-+r+ 65.9%

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

       7. +-commutative 65.9%

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

       8. associate-+l+ 65.9%

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

       9. +-commutative 65.9%

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

       10. *-commutative 65.9%

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

       11. +-commutative 65.9%

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

       12. fma-def 65.9%

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

       13. fma-udef 65.9%

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

       14. unpow2 65.9%

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

       15. unpow3 65.9%

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

       16. unpow2 65.9%

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

       17. associate-*l* 65.9%

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

       18. distribute-lft-out 66.4%

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

   11. Simplified 66.4%

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

   if 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.3%

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

   6. Taylor expanded in x around inf 99.3%

      \[\leadsto 1 - \frac{0.254829592}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot \color{blue}{e^{{x}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 75.8%

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

Alternative 7: 99.8% accurate, 7.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.05)
   (+
    (fma x 1.128386358070218 1e-9)
    (* (* x x) (+ -0.00011824294398844343 (* x -0.37545125292247583))))
   (- 1.0 (/ 0.7778892405807117 (* x (exp (* x x)))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.05) {
		tmp = fma(x, 1.128386358070218, 1e-9) + ((x * x) * (-0.00011824294398844343 + (x * -0.37545125292247583)));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x * exp((x * x))));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.05)
		tmp = Float64(fma(x, 1.128386358070218, 1e-9) + Float64(Float64(x * x) * Float64(-0.00011824294398844343 + Float64(x * -0.37545125292247583))));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x * exp(Float64(x * x)))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.05], N[(N[(x * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.00011824294398844343 + N[(x * -0.37545125292247583), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 71.7%

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

   3. Applied egg-rr 71.7%

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

   5. Simplified 70.3%

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

   6. Taylor expanded in x around 0 65.9%

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

      1. +-commutative 65.9%

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

      2. associate-+r+ 65.9%

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

      3. associate-+l+ 65.9%

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

      4. *-commutative 65.9%

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

      5. fma-def 65.9%

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

      6. unpow2 65.9%

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

      7. *-commutative 65.9%

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

      8. *-commutative 65.9%

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

      9. fma-def 65.9%

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

   8. Simplified 65.9%

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

   9. Taylor expanded in x around 0 65.9%

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

       1. associate-+r+ 65.9%

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

       2. *-commutative 65.9%

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

       3. unpow2 65.9%

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

       4. *-commutative 65.9%

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

       5. fma-udef 65.9%

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

       6. associate-+r+ 65.9%

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

       7. +-commutative 65.9%

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

       8. associate-+l+ 65.9%

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

       9. +-commutative 65.9%

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

       10. *-commutative 65.9%

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

       11. +-commutative 65.9%

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

       12. fma-def 65.9%

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

       13. fma-udef 65.9%

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

       14. unpow2 65.9%

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

       15. unpow3 65.9%

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

       16. unpow2 65.9%

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

       17. associate-*l* 65.9%

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

       18. distribute-lft-out 66.4%

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

   11. Simplified 66.4%

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

   if 1.05000000000000004 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.3%

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

   6. Taylor expanded in x around inf 99.3%

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

      1. *-commutative 99.3%

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

      2. unpow2 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \left(x \cdot x\right) \cdot \left(-0.00011824294398844343 + x \cdot -0.37545125292247583\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 8: 99.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + x \cdot \left(x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.88)
   (+ (fma x 1.128386358070218 1e-9) (* x (* x -0.00011824294398844343)))
   1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.88) {
		tmp = fma(x, 1.128386358070218, 1e-9) + (x * (x * -0.00011824294398844343));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(fma(x, 1.128386358070218, 1e-9) + Float64(x * Float64(x * -0.00011824294398844343)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.88], N[(N[(x * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(x * N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.7%

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

   3. Applied egg-rr 71.7%

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

   5. Simplified 70.3%

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

      1. add-log-exp 68.7%

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

      2. log1p-udef 68.7%

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

      3. add-exp-log 68.8%

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

      4. pow-exp 68.8%

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

   7. Applied egg-rr 68.8%

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

   8. Taylor expanded in x around 0 65.2%

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

      1. +-commutative 65.2%

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

      2. *-commutative 65.2%

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

      3. associate-+l+ 65.2%

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

      4. *-commutative 65.2%

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

      5. unpow2 65.2%

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

      6. associate-*l* 65.2%

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

      7. fma-def 65.2%

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

   10. Simplified 65.2%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

   5. Simplified 100.0%

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

      1. add-log-exp 100.0%

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

      2. log1p-udef 100.0%

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

      3. add-exp-log 100.0%

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

      4. pow-exp 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf 99.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + x \cdot \left(x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 99.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.85:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + x \cdot \left(x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.85)
   (+ (fma x 1.128386358070218 1e-9) (* x (* x -0.00011824294398844343)))
   (- 1.0 (/ 0.7778892405807117 (* x (exp (* x x)))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.85) {
		tmp = fma(x, 1.128386358070218, 1e-9) + (x * (x * -0.00011824294398844343));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x * exp((x * x))));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.85)
		tmp = Float64(fma(x, 1.128386358070218, 1e-9) + Float64(x * Float64(x * -0.00011824294398844343)));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x * exp(Float64(x * x)))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.85], N[(N[(x * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(x * N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.849999999999999978

   1. Initial program 71.7%

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

   3. Applied egg-rr 71.7%

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

   5. Simplified 70.3%

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

      1. add-log-exp 68.7%

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

      2. log1p-udef 68.7%

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

      3. add-exp-log 68.8%

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

      4. pow-exp 68.8%

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

   7. Applied egg-rr 68.8%

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

   8. Taylor expanded in x around 0 65.2%

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

      1. +-commutative 65.2%

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

      2. *-commutative 65.2%

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

      3. associate-+l+ 65.2%

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

      4. *-commutative 65.2%

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

      5. unpow2 65.2%

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

      6. associate-*l* 65.2%

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

      7. fma-def 65.2%

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

   10. Simplified 65.2%

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

   if 0.849999999999999978 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.3%

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

   6. Taylor expanded in x around inf 99.3%

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

      1. *-commutative 99.3%

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

      2. unpow2 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.85:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + x \cdot \left(x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{x \cdot x}}\\ \end{array} \]

Alternative 10: 99.5% accurate, 65.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;x \cdot \left(x \cdot -0.00011824294398844343\right) + \left(10^{-9} + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.88)
   (+ (* x (* x -0.00011824294398844343)) (+ 1e-9 (* x 1.128386358070218)))
   1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.88) {
		tmp = (x * (x * -0.00011824294398844343)) + (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 <= 0.88d0) then
        tmp = (x * (x * (-0.00011824294398844343d0))) + (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 <= 0.88) {
		tmp = (x * (x * -0.00011824294398844343)) + (1e-9 + (x * 1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.88:
		tmp = (x * (x * -0.00011824294398844343)) + (1e-9 + (x * 1.128386358070218))
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(Float64(x * Float64(x * -0.00011824294398844343)) + 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 <= 0.88)
		tmp = (x * (x * -0.00011824294398844343)) + (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, 0.88], N[(N[(x * N[(x * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision] + N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.7%

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

   3. Applied egg-rr 71.7%

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

   5. Simplified 70.3%

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

      1. add-log-exp 68.7%

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

      2. log1p-udef 68.7%

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

      3. add-exp-log 68.8%

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

      4. pow-exp 68.8%

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

   7. Applied egg-rr 68.8%

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

   8. Taylor expanded in x around 0 65.2%

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

      1. +-commutative 65.2%

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

      2. *-commutative 65.2%

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

      3. associate-+l+ 65.2%

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

      4. *-commutative 65.2%

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

      5. unpow2 65.2%

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

      6. associate-*l* 65.2%

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

      7. fma-def 65.2%

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

   10. Simplified 65.2%

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

       1. fma-udef 65.2%

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

   12. Applied egg-rr 65.2%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

   5. Simplified 100.0%

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

      1. add-log-exp 100.0%

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

      2. log1p-udef 100.0%

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

      3. add-exp-log 100.0%

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

      4. pow-exp 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf 99.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;x \cdot \left(x \cdot -0.00011824294398844343\right) + \left(10^{-9} + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 99.4% accurate, 121.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.88) (+ 1e-9 (* x 1.128386358070218)) 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.88) {
		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 <= 0.88d0) 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 <= 0.88) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.88:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.88)
		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 <= 0.88)
		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, 0.88], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.7%

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

   3. Applied egg-rr 71.7%

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

   5. Simplified 70.3%

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

   6. Taylor expanded in x around 0 65.3%

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

      1. *-commutative 65.3%

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

   8. Simplified 65.3%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

   5. Simplified 100.0%

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

      1. add-log-exp 100.0%

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

      2. log1p-udef 100.0%

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

      3. add-exp-log 100.0%

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

      4. pow-exp 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf 99.3%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 97.8% 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 71.7%

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

   3. Applied egg-rr 71.6%

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

   5. Simplified 70.2%

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

   6. Taylor expanded in x around 0 68.7%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 99.8%

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

   3. Applied egg-rr 99.8%

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

   5. Simplified 99.8%

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

      1. add-log-exp 99.8%

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

      2. log1p-udef 99.8%

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

      3. add-exp-log 99.8%

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

      4. pow-exp 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in x around inf 98.1%

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

4. Final simplification 77.2%

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

Alternative 13: 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 79.8%

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

3. Applied egg-rr 79.8%

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

5. Simplified 78.8%

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

6. Taylor expanded in x around 0 52.0%

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

7. Final simplification 52.0%

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

Reproduce

herbie shell --seed 2023208 
(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)))))))