Jmat.Real.erf

Percentage Accurate: 78.9% → 99.7%

Time: 26.2s

Alternatives: 10

Speedup: 142.3×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (+
         -0.284496736
         (*
          t_0
          (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(fabs(x) * fabs(x))));
}

Fortran

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: 78.9% 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: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.3275911, x\_m, 1\right)\right)}^{3}\\ t_1 := \frac{1}{1 + \left|x\_m\right| \cdot 0.3275911}\\ \mathbf{if}\;\left|x\_m\right| \leq 10^{-8}:\\ \;\;\;\;10^{-9} + \left(x\_m \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot t\_1\right) + {x\_m}^{2} \cdot \left(t\_1 \cdot 0.36953108532122814 - 0.10731592869189407\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \left(\frac{1.453152027}{t\_0} - \frac{\frac{1.061405429}{t\_0} + \left(\frac{1.421413741}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)} + -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right), \frac{{\left(e^{x\_m}\right)}^{\left(-x\_m\right)}}{\mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (pow (fma 0.3275911 x_m 1.0) 3.0))
        (t_1 (/ 1.0 (+ 1.0 (* (fabs x_m) 0.3275911)))))
   (if (<= (fabs x_m) 1e-8)
     (+
      1e-9
      (+
       (* x_m (+ 0.3275910996724089 (* 0.8007952583978091 t_1)))
       (* (pow x_m 2.0) (- (* t_1 0.36953108532122814) 0.10731592869189407))))
     (fma
      (+
       -0.254829592
       (-
        (/ 1.453152027 t_0)
        (/
         (+
          (/ 1.061405429 t_0)
          (+ (/ 1.421413741 (fma 0.3275911 x_m 1.0)) -0.284496736))
         (fma 0.3275911 x_m 1.0))))
      (/ (pow (exp x_m) (- x_m)) (fma 0.3275911 (fabs x_m) 1.0))
      1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = pow(fma(0.3275911, x_m, 1.0), 3.0);
	double t_1 = 1.0 / (1.0 + (fabs(x_m) * 0.3275911));
	double tmp;
	if (fabs(x_m) <= 1e-8) {
		tmp = 1e-9 + ((x_m * (0.3275910996724089 + (0.8007952583978091 * t_1))) + (pow(x_m, 2.0) * ((t_1 * 0.36953108532122814) - 0.10731592869189407)));
	} else {
		tmp = fma((-0.254829592 + ((1.453152027 / t_0) - (((1.061405429 / t_0) + ((1.421413741 / fma(0.3275911, x_m, 1.0)) + -0.284496736)) / fma(0.3275911, x_m, 1.0)))), (pow(exp(x_m), -x_m) / fma(0.3275911, fabs(x_m), 1.0)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(0.3275911, x_m, 1.0) ^ 3.0
	t_1 = Float64(1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))
	tmp = 0.0
	if (abs(x_m) <= 1e-8)
		tmp = Float64(1e-9 + Float64(Float64(x_m * Float64(0.3275910996724089 + Float64(0.8007952583978091 * t_1))) + Float64((x_m ^ 2.0) * Float64(Float64(t_1 * 0.36953108532122814) - 0.10731592869189407))));
	else
		tmp = fma(Float64(-0.254829592 + Float64(Float64(1.453152027 / t_0) - Float64(Float64(Float64(1.061405429 / t_0) + Float64(Float64(1.421413741 / fma(0.3275911, x_m, 1.0)) + -0.284496736)) / fma(0.3275911, x_m, 1.0)))), Float64((exp(x_m) ^ Float64(-x_m)) / fma(0.3275911, abs(x_m), 1.0)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-8], N[(1e-9 + N[(N[(x$95$m * N[(0.3275910996724089 + N[(0.8007952583978091 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$1 * 0.36953108532122814), $MachinePrecision] - 0.10731592869189407), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.254829592 + N[(N[(1.453152027 / t$95$0), $MachinePrecision] - N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] + N[(N[(1.421413741 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1e-8

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

      1. add-cube-cbrt 57.7%

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

      2. pow3 57.7%

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

   6. Applied egg-rr 57.2%

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

   7. Taylor expanded in x around 0 57.2%

      \[\leadsto \color{blue}{\left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

      2. log1p-define 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

      3. +-commutative 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

      4. fma-undefine 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

      5. expm1-undefine 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

      6. add-exp-log 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

      7. add-sqr-sqrt 28.6%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

      8. fabs-sqr 28.6%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   9. Applied egg-rr 57.2%

      \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   10. Step-by-step derivation

       1. fma-undefine 57.2%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 57.2%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 57.2%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 57.2%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   11. Simplified 57.2%

       \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Taylor expanded in x around 0 98.7%

       \[\leadsto \color{blue}{10^{-9} + \left(x \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + {x}^{2} \cdot \left(0.36953108532122814 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 0.10731592869189407\right)\right)} \]

   if 1e-8 < (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%

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

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \mathsf{fma}\left(-0.254829592 - \color{blue}{\frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right) \]
   6. Step-by-step derivation

      1. associate--r+ 99.8%

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

      2. div-sub 99.9%

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

   7. Simplified 99.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-8}:\\ \;\;\;\;10^{-9} + \left(x \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right) + {x}^{2} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.36953108532122814 - 0.10731592869189407\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \left(\frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \frac{\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} + \left(\frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x\_m\right| \cdot 0.3275911}\\ \mathbf{if}\;\left|x\_m\right| \leq 10^{-8}:\\ \;\;\;\;10^{-9} + \left(x\_m \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot t\_0\right) + {x\_m}^{2} \cdot \left(t\_0 \cdot 0.36953108532122814 - 0.10731592869189407\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + {\left(\sqrt[3]{\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}\right)}^{3}}{{\left(e^{x\_m}\right)}^{x\_m}}}{\mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x_m) 0.3275911)))))
   (if (<= (fabs x_m) 1e-8)
     (+
      1e-9
      (+
       (* x_m (+ 0.3275910996724089 (* 0.8007952583978091 t_0)))
       (* (pow x_m 2.0) (- (* t_0 0.36953108532122814) 0.10731592869189407))))
     (-
      1.0
      (/
       (/
        (+
         0.254829592
         (pow
          (cbrt
           (/
            (+
             -0.284496736
             (/
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
                (fma 0.3275911 x_m 1.0)))
              (fma 0.3275911 x_m 1.0)))
            (fma 0.3275911 x_m 1.0)))
          3.0))
        (pow (exp x_m) x_m))
       (fma 0.3275911 (fabs x_m) 1.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (fabs(x_m) * 0.3275911));
	double tmp;
	if (fabs(x_m) <= 1e-8) {
		tmp = 1e-9 + ((x_m * (0.3275910996724089 + (0.8007952583978091 * t_0))) + (pow(x_m, 2.0) * ((t_0 * 0.36953108532122814) - 0.10731592869189407)));
	} else {
		tmp = 1.0 - (((0.254829592 + pow(cbrt(((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))), 3.0)) / pow(exp(x_m), x_m)) / fma(0.3275911, fabs(x_m), 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))
	tmp = 0.0
	if (abs(x_m) <= 1e-8)
		tmp = Float64(1e-9 + Float64(Float64(x_m * Float64(0.3275910996724089 + Float64(0.8007952583978091 * t_0))) + Float64((x_m ^ 2.0) * Float64(Float64(t_0 * 0.36953108532122814) - 0.10731592869189407))));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.254829592 + (cbrt(Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) ^ 3.0)) / (exp(x_m) ^ x_m)) / fma(0.3275911, abs(x_m), 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-8], N[(1e-9 + N[(N[(x$95$m * N[(0.3275910996724089 + N[(0.8007952583978091 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$0 * 0.36953108532122814), $MachinePrecision] - 0.10731592869189407), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(0.254829592 + N[Power[N[Power[N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Exp[x$95$m], $MachinePrecision], x$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1e-8

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

      1. add-cube-cbrt 57.7%

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

      2. pow3 57.7%

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

   6. Applied egg-rr 57.2%

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

   7. Taylor expanded in x around 0 57.2%

      \[\leadsto \color{blue}{\left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

      2. log1p-define 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

      3. +-commutative 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

      4. fma-undefine 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

      5. expm1-undefine 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

      6. add-exp-log 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

      7. add-sqr-sqrt 28.6%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

      8. fabs-sqr 28.6%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   9. Applied egg-rr 57.2%

      \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   10. Step-by-step derivation

       1. fma-undefine 57.2%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 57.2%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 57.2%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 57.2%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   11. Simplified 57.2%

       \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Taylor expanded in x around 0 98.7%

       \[\leadsto \color{blue}{10^{-9} + \left(x \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + {x}^{2} \cdot \left(0.36953108532122814 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 0.10731592869189407\right)\right)} \]

   if 1e-8 < (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. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 99.8%

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

      2. pow3 99.8%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-8}:\\ \;\;\;\;10^{-9} + \left(x \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right) + {x}^{2} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.36953108532122814 - 0.10731592869189407\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + {\left(\sqrt[3]{\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}^{3}}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x\_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;\left|x\_m\right| \leq 10^{-8}:\\ \;\;\;\;10^{-9} + \left(x\_m \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot t\_1\right) + {x\_m}^{2} \cdot \left(t\_1 \cdot 0.36953108532122814 - 0.10731592869189407\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(t\_1 \cdot \left(\left(-0.284496736 + t\_1 \cdot \left(1.421413741 + t\_1 \cdot {\left(\sqrt[3]{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}\right)}^{3}\right)\right) \cdot \frac{-1}{t\_0} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x_m) 1e-8)
     (+
      1e-9
      (+
       (* x_m (+ 0.3275910996724089 (* 0.8007952583978091 t_1)))
       (* (pow x_m 2.0) (- (* t_1 0.36953108532122814) 0.10731592869189407))))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        t_1
        (-
         (*
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (*
              t_1
              (pow
               (cbrt (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0))))
               3.0)))))
          (/ -1.0 t_0))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x_m) <= 1e-8) {
		tmp = 1e-9 + ((x_m * (0.3275910996724089 + (0.8007952583978091 * t_1))) + (pow(x_m, 2.0) * ((t_1 * 0.36953108532122814) - 0.10731592869189407)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * (t_1 * (((-0.284496736 + (t_1 * (1.421413741 + (t_1 * pow(cbrt((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0)))), 3.0))))) * (-1.0 / t_0)) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x_m) <= 1e-8)
		tmp = Float64(1e-9 + Float64(Float64(x_m * Float64(0.3275910996724089 + Float64(0.8007952583978091 * t_1))) + Float64((x_m ^ 2.0) * Float64(Float64(t_1 * 0.36953108532122814) - 0.10731592869189407))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(t_1 * Float64(Float64(Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(t_1 * (cbrt(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0)))) ^ 3.0))))) * Float64(-1.0 / t_0)) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-8], N[(1e-9 + N[(N[(x$95$m * N[(0.3275910996724089 + N[(0.8007952583978091 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$1 * 0.36953108532122814), $MachinePrecision] - 0.10731592869189407), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(N[(N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[Power[N[Power[N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1e-8

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

      1. add-cube-cbrt 57.7%

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

      2. pow3 57.7%

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

   6. Applied egg-rr 57.2%

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

   7. Taylor expanded in x around 0 57.2%

      \[\leadsto \color{blue}{\left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

      2. log1p-define 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

      3. +-commutative 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

      4. fma-undefine 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

      5. expm1-undefine 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

      6. add-exp-log 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

      7. add-sqr-sqrt 28.6%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

      8. fabs-sqr 28.6%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 57.2%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   9. Applied egg-rr 57.2%

      \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   10. Step-by-step derivation

       1. fma-undefine 57.2%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 57.2%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 57.2%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 57.2%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   11. Simplified 57.2%

       \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Taylor expanded in x around 0 98.7%

       \[\leadsto \color{blue}{10^{-9} + \left(x \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + {x}^{2} \cdot \left(0.36953108532122814 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 0.10731592869189407\right)\right)} \]

   if 1e-8 < (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%

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

      1. +-commutative 99.8%

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

      2. fma-undefine 99.8%

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

      3. add-cube-cbrt 99.8%

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

      4. pow3 99.8%

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

      5. add-sqr-sqrt 48.9%

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

      6. fabs-sqr 48.9%

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

      7. add-sqr-sqrt 99.4%

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

   6. Applied egg-rr 99.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-8}:\\ \;\;\;\;10^{-9} + \left(x \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right) + {x}^{2} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.36953108532122814 - 0.10731592869189407\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot {\left(\sqrt[3]{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}^{3}\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x\_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;x\_m \leq 8.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \left(x\_m \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot t\_1\right) + {x\_m}^{2} \cdot \left(t\_1 \cdot 0.36953108532122814 - 0.10731592869189407\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(t\_1 \cdot \left(\left(-0.284496736 + t\_1 \cdot \left(1.421413741 + t\_1 \cdot \mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}, -1.453152027\right)\right)\right) \cdot \frac{-1}{t\_0} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= x_m 8.5e-6)
     (+
      1e-9
      (+
       (* x_m (+ 0.3275910996724089 (* 0.8007952583978091 t_1)))
       (* (pow x_m 2.0) (- (* t_1 0.36953108532122814) 0.10731592869189407))))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        t_1
        (-
         (*
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (*
              t_1
              (fma
               1.061405429
               (/ 1.0 (fma 0.3275911 x_m 1.0))
               -1.453152027)))))
          (/ -1.0 t_0))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x_m <= 8.5e-6) {
		tmp = 1e-9 + ((x_m * (0.3275910996724089 + (0.8007952583978091 * t_1))) + (pow(x_m, 2.0) * ((t_1 * 0.36953108532122814) - 0.10731592869189407)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * (t_1 * (((-0.284496736 + (t_1 * (1.421413741 + (t_1 * fma(1.061405429, (1.0 / fma(0.3275911, x_m, 1.0)), -1.453152027))))) * (-1.0 / t_0)) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x_m <= 8.5e-6)
		tmp = Float64(1e-9 + Float64(Float64(x_m * Float64(0.3275910996724089 + Float64(0.8007952583978091 * t_1))) + Float64((x_m ^ 2.0) * Float64(Float64(t_1 * 0.36953108532122814) - 0.10731592869189407))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(t_1 * Float64(Float64(Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(t_1 * fma(1.061405429, Float64(1.0 / fma(0.3275911, x_m, 1.0)), -1.453152027))))) * Float64(-1.0 / t_0)) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 8.5e-6], N[(1e-9 + N[(N[(x$95$m * N[(0.3275910996724089 + N[(0.8007952583978091 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$1 * 0.36953108532122814), $MachinePrecision] - 0.10731592869189407), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(N[(N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[(1.061405429 * N[(1.0 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.4999999999999999e-6

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

      1. add-cube-cbrt 73.2%

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

      2. pow3 73.2%

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

   6. Applied egg-rr 72.5%

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

   7. Taylor expanded in x around 0 37.9%

      \[\leadsto \color{blue}{\left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 37.9%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

      2. log1p-define 37.9%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

      3. +-commutative 37.9%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

      4. fma-undefine 37.9%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

      5. expm1-undefine 37.9%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

      6. add-exp-log 37.9%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

      7. add-sqr-sqrt 18.0%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

      8. fabs-sqr 18.0%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 37.9%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   9. Applied egg-rr 37.9%

      \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   10. Step-by-step derivation

       1. fma-undefine 37.9%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 37.9%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 37.9%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 37.9%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   11. Simplified 37.9%

       \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Taylor expanded in x around 0 62.5%

       \[\leadsto \color{blue}{10^{-9} + \left(x \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + {x}^{2} \cdot \left(0.36953108532122814 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 0.10731592869189407\right)\right)} \]

   if 8.4999999999999999e-6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. div-inv 100.0%

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

      3. fma-define 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-undefine 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. fabs-sqr 100.0%

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

      8. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \left(x \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right) + {x}^{2} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.36953108532122814 - 0.10731592869189407\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + x\_m \cdot 0.3275911\\ t_1 := \frac{1}{t\_0}\\ t_2 := 1 + \left|x\_m\right| \cdot 0.3275911\\ t_3 := \frac{1}{t\_2}\\ \mathbf{if}\;x\_m \leq 9 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \left(x\_m \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot t\_3\right) + {x\_m}^{2} \cdot \left(t\_3 \cdot 0.36953108532122814 - 0.10731592869189407\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(\left(0.254829592 + t\_1 \cdot \left(-0.284496736 + t\_1 \cdot \left(1.421413741 + t\_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t\_0}\right)\right)\right)\right) \cdot \frac{-1}{t\_2}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x_m 0.3275911)))
        (t_1 (/ 1.0 t_0))
        (t_2 (+ 1.0 (* (fabs x_m) 0.3275911)))
        (t_3 (/ 1.0 t_2)))
   (if (<= x_m 9e-6)
     (+
      1e-9
      (+
       (* x_m (+ 0.3275910996724089 (* 0.8007952583978091 t_3)))
       (* (pow x_m 2.0) (- (* t_3 0.36953108532122814) 0.10731592869189407))))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+ 1.421413741 (* t_1 (+ -1.453152027 (/ 1.061405429 t_0))))))))
        (/ -1.0 t_2)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (x_m * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_3 = 1.0 / t_2;
	double tmp;
	if (x_m <= 9e-6) {
		tmp = 1e-9 + ((x_m * (0.3275910996724089 + (0.8007952583978091 * t_3))) + (pow(x_m, 2.0) * ((t_3 * 0.36953108532122814) - 0.10731592869189407)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0)))))))) * (-1.0 / t_2)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 1.0d0 + (x_m * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    t_2 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    t_3 = 1.0d0 / t_2
    if (x_m <= 9d-6) then
        tmp = 1d-9 + ((x_m * (0.3275910996724089d0 + (0.8007952583978091d0 * t_3))) + ((x_m ** 2.0d0) * ((t_3 * 0.36953108532122814d0) - 0.10731592869189407d0)))
    else
        tmp = 1.0d0 + (exp((x_m * -x_m)) * ((0.254829592d0 + (t_1 * ((-0.284496736d0) + (t_1 * (1.421413741d0 + (t_1 * ((-1.453152027d0) + (1.061405429d0 / t_0)))))))) * ((-1.0d0) / t_2)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (x_m * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double t_3 = 1.0 / t_2;
	double tmp;
	if (x_m <= 9e-6) {
		tmp = 1e-9 + ((x_m * (0.3275910996724089 + (0.8007952583978091 * t_3))) + (Math.pow(x_m, 2.0) * ((t_3 * 0.36953108532122814) - 0.10731592869189407)));
	} else {
		tmp = 1.0 + (Math.exp((x_m * -x_m)) * ((0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0)))))))) * (-1.0 / t_2)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (x_m * 0.3275911)
	t_1 = 1.0 / t_0
	t_2 = 1.0 + (math.fabs(x_m) * 0.3275911)
	t_3 = 1.0 / t_2
	tmp = 0
	if x_m <= 9e-6:
		tmp = 1e-9 + ((x_m * (0.3275910996724089 + (0.8007952583978091 * t_3))) + (math.pow(x_m, 2.0) * ((t_3 * 0.36953108532122814) - 0.10731592869189407)))
	else:
		tmp = 1.0 + (math.exp((x_m * -x_m)) * ((0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0)))))))) * (-1.0 / t_2)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(x_m * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_3 = Float64(1.0 / t_2)
	tmp = 0.0
	if (x_m <= 9e-6)
		tmp = Float64(1e-9 + Float64(Float64(x_m * Float64(0.3275910996724089 + Float64(0.8007952583978091 * t_3))) + Float64((x_m ^ 2.0) * Float64(Float64(t_3 * 0.36953108532122814) - 0.10731592869189407))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(t_1 * Float64(-1.453152027 + Float64(1.061405429 / t_0)))))))) * Float64(-1.0 / t_2))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (x_m * 0.3275911);
	t_1 = 1.0 / t_0;
	t_2 = 1.0 + (abs(x_m) * 0.3275911);
	t_3 = 1.0 / t_2;
	tmp = 0.0;
	if (x_m <= 9e-6)
		tmp = 1e-9 + ((x_m * (0.3275910996724089 + (0.8007952583978091 * t_3))) + ((x_m ^ 2.0) * ((t_3 * 0.36953108532122814) - 0.10731592869189407)));
	else
		tmp = 1.0 + (exp((x_m * -x_m)) * ((0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0)))))))) * (-1.0 / t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / t$95$2), $MachinePrecision]}, If[LessEqual[x$95$m, 9e-6], N[(1e-9 + N[(N[(x$95$m * N[(0.3275910996724089 + N[(0.8007952583978091 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$3 * 0.36953108532122814), $MachinePrecision] - 0.10731592869189407), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.00000000000000023e-6

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

      1. add-cube-cbrt 73.2%

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

      2. pow3 73.2%

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

   6. Applied egg-rr 72.5%

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

   7. Taylor expanded in x around 0 37.9%

      \[\leadsto \color{blue}{\left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 37.9%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

      2. log1p-define 37.9%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

      3. +-commutative 37.9%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

      4. fma-undefine 37.9%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

      5. expm1-undefine 37.9%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

      6. add-exp-log 37.9%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

      7. add-sqr-sqrt 18.0%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

      8. fabs-sqr 18.0%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 37.9%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   9. Applied egg-rr 37.9%

      \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   10. Step-by-step derivation

       1. fma-undefine 37.9%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 37.9%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 37.9%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 37.9%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   11. Simplified 37.9%

       \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Taylor expanded in x around 0 62.5%

       \[\leadsto \color{blue}{10^{-9} + \left(x \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + {x}^{2} \cdot \left(0.36953108532122814 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 0.10731592869189407\right)\right)} \]

   if 9.00000000000000023e-6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

      2. log1p-define 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

      3. +-commutative 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

      4. fma-undefine 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

      5. expm1-undefine 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

      6. add-exp-log 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

      7. add-sqr-sqrt 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

      8. fabs-sqr 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   6. Applied egg-rr 100.0%

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

      1. fma-undefine 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

      2. associate--l+ 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

      3. metadata-eval 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

      4. +-rgt-identity 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   8. Simplified 100.0%

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

      1. expm1-log1p-u 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

      2. log1p-define 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

      3. +-commutative 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

      4. fma-undefine 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

      5. expm1-undefine 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

      6. add-exp-log 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

      7. add-sqr-sqrt 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

      8. fabs-sqr 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 100.0%

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

       1. fma-undefine 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 100.0%

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

       1. expm1-log1p-u 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

       2. log1p-define 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

       3. +-commutative 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

       4. fma-undefine 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

       5. expm1-undefine 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

       6. add-exp-log 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

       7. add-sqr-sqrt 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

       8. fabs-sqr 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

       9. add-sqr-sqrt 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   14. Applied egg-rr 100.0%

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

       1. fma-undefine 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   16. Simplified 100.0%

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

       1. expm1-log1p-u 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

       2. log1p-define 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

       3. +-commutative 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

       4. fma-undefine 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

       5. expm1-undefine 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

       6. add-exp-log 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

       7. add-sqr-sqrt 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

       8. fabs-sqr 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

       9. add-sqr-sqrt 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   18. Applied egg-rr 100.0%

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

       1. fma-undefine 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   20. Simplified 100.0%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \left(x \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right) + {x}^{2} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.36953108532122814 - 0.10731592869189407\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\left(0.254829592 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + x\_m \cdot 0.3275911\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;x\_m \leq 10^{-6}:\\ \;\;\;\;10^{-9} + e^{\log \left(x\_m \cdot 1.128386358070218\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(\left(0.254829592 + t\_1 \cdot \left(-0.284496736 + t\_1 \cdot \left(1.421413741 + t\_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t\_0}\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\_m\right| \cdot 0.3275911}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x_m 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= x_m 1e-6)
     (+ 1e-9 (exp (log (* x_m 1.128386358070218))))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+ 1.421413741 (* t_1 (+ -1.453152027 (/ 1.061405429 t_0))))))))
        (/ -1.0 (+ 1.0 (* (fabs x_m) 0.3275911)))))))))

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x_m * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    if (x_m <= 1d-6) then
        tmp = 1d-9 + exp(log((x_m * 1.128386358070218d0)))
    else
        tmp = 1.0d0 + (exp((x_m * -x_m)) * ((0.254829592d0 + (t_1 * ((-0.284496736d0) + (t_1 * (1.421413741d0 + (t_1 * ((-1.453152027d0) + (1.061405429d0 / t_0)))))))) * ((-1.0d0) / (1.0d0 + (abs(x_m) * 0.3275911d0)))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 1e-6], N[(1e-9 + N[Exp[N[Log[N[(x$95$m * 1.128386358070218), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999955e-7

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

   6. Applied egg-rr 35.7%

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

   7. Taylor expanded in x around 0 62.6%

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

      1. *-commutative 62.6%

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

   9. Simplified 62.6%

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

       1. add-exp-log 31.2%

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

   11. Applied egg-rr 31.2%

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

   if 9.99999999999999955e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

      2. log1p-define 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

      3. +-commutative 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

      4. fma-undefine 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

      5. expm1-undefine 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

      6. add-exp-log 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

      7. add-sqr-sqrt 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

      8. fabs-sqr 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   6. Applied egg-rr 100.0%

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

      1. fma-undefine 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

      2. associate--l+ 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

      3. metadata-eval 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

      4. +-rgt-identity 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   8. Simplified 100.0%

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

      1. expm1-log1p-u 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

      2. log1p-define 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

      3. +-commutative 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

      4. fma-undefine 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

      5. expm1-undefine 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

      6. add-exp-log 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

      7. add-sqr-sqrt 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

      8. fabs-sqr 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 5.4%

         \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 100.0%

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

       1. fma-undefine 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 100.0%

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

       1. expm1-log1p-u 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

       2. log1p-define 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

       3. +-commutative 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

       4. fma-undefine 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

       5. expm1-undefine 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

       6. add-exp-log 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

       7. add-sqr-sqrt 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

       8. fabs-sqr 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

       9. add-sqr-sqrt 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   14. Applied egg-rr 100.0%

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

       1. fma-undefine 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   16. Simplified 100.0%

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

       1. expm1-log1p-u 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

       2. log1p-define 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

       3. +-commutative 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

       4. fma-undefine 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

       5. expm1-undefine 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

       6. add-exp-log 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

       7. add-sqr-sqrt 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

       8. fabs-sqr 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

       9. add-sqr-sqrt 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   18. Applied egg-rr 100.0%

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

       1. fma-undefine 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 5.4%

          \[\leadsto \left(1 + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   20. Simplified 100.0%

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

4. Final simplification 49.2%

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

Alternative 7: 99.3% accurate, 4.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88) (+ 1e-9 (exp (log (* x_m 1.128386358070218)))) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = 1e-9 + exp(log((x_m * 1.128386358070218)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.88d0) then
        tmp = 1d-9 + exp(log((x_m * 1.128386358070218d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = 1e-9 + Math.exp(Math.log((x_m * 1.128386358070218)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.88:
		tmp = 1e-9 + math.exp(math.log((x_m * 1.128386358070218)))
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.88)
		tmp = 1e-9 + exp(log((x_m * 1.128386358070218)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.88], N[(1e-9 + N[Exp[N[Log[N[(x$95$m * 1.128386358070218), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

   6. Applied egg-rr 35.7%

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

   7. Taylor expanded in x around 0 62.6%

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

      1. *-commutative 62.6%

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

   9. Simplified 62.6%

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

       1. add-exp-log 31.2%

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

   11. Applied egg-rr 31.2%

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

   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. Simplified 100.0%

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

   6. Applied egg-rr 0.6%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 49.2%

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

Alternative 8: 99.3% accurate, 85.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88) (+ 1e-9 (* x_m 1.128386358070218)) 1.0))

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.88d0) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.88)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.88], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

   6. Applied egg-rr 35.7%

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

   7. Taylor expanded in x around 0 62.6%

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

      1. *-commutative 62.6%

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

   9. Simplified 62.6%

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

   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. Simplified 100.0%

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

   6. Applied egg-rr 0.6%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 72.4%

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

Alternative 9: 97.5% accurate, 142.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

   6. Applied egg-rr 35.7%

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

   7. Taylor expanded in x around 0 65.1%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 0.6%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 74.3%

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

Alternative 10: 53.6% accurate, 856.0× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1d-9
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

3. Simplified 80.2%

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

6. Applied egg-rr 26.6%

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

7. Taylor expanded in x around 0 51.0%

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

8. Final simplification 51.0%

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

Reproduce

herbie shell --seed 2024044 
(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)))))))