Jmat.Real.erf

Percentage Accurate: 79.5% → 83.1%

Time: 24.9s

Alternatives: 10

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))
    code = 1.0d0 - ((t_0 * (0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (t_0 * 1.061405429d0))))))))) * exp(-(abs(x) * abs(x))))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * Math.abs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * Math.exp(-(Math.abs(x) * Math.abs(x))));
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + (0.3275911 * math.fabs(x)))
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * math.exp(-(math.fabs(x) * math.fabs(x))))

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * Float64(-1.453152027 + Float64(t_0 * 1.061405429))))))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / (1.0 + (0.3275911 * abs(x)));
	tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(abs(x) * abs(x))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(-1.453152027 + N[(t$95$0 * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 79.5% 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: 83.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right)\\ \mathbf{if}\;\left|x\_m\right| \leq 10^{-17}:\\ \;\;\;\;\frac{\frac{1 - {t\_1}^{-4} \cdot 0.999999996}{\mathsf{fma}\left(0.999999998, {t\_1}^{-2}, 1\right)}}{1 + \frac{0.999999999}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(\frac{1}{1 + {\left(\sqrt[3]{x\_m \cdot 0.3275911}\right)}^{3}} \cdot \left(e^{-\mathsf{log1p}\left(t\_0\right)} \cdot \left(\left(1.421413741 + \frac{1}{1 + t\_0} \cdot \left(-1.453152027 - \frac{1.061405429}{-1 - x\_m \cdot 0.3275911}\right)\right) \cdot \frac{1}{-1 - t\_0} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)) (t_1 (fma 0.3275911 (fabs x_m) 1.0)))
   (if (<= (fabs x_m) 1e-17)
     (/
      (/
       (- 1.0 (* (pow t_1 -4.0) 0.999999996))
       (fma 0.999999998 (pow t_1 -2.0) 1.0))
      (+ 1.0 (/ 0.999999999 t_1)))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (/ 1.0 (+ 1.0 (pow (cbrt (* x_m 0.3275911)) 3.0)))
        (-
         (*
          (exp (- (log1p t_0)))
          (-
           (*
            (+
             1.421413741
             (*
              (/ 1.0 (+ 1.0 t_0))
              (- -1.453152027 (/ 1.061405429 (- -1.0 (* x_m 0.3275911))))))
            (/ 1.0 (- -1.0 t_0)))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double t_1 = fma(0.3275911, fabs(x_m), 1.0);
	double tmp;
	if (fabs(x_m) <= 1e-17) {
		tmp = ((1.0 - (pow(t_1, -4.0) * 0.999999996)) / fma(0.999999998, pow(t_1, -2.0), 1.0)) / (1.0 + (0.999999999 / t_1));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((1.0 / (1.0 + pow(cbrt((x_m * 0.3275911)), 3.0))) * ((exp(-log1p(t_0)) * (((1.421413741 + ((1.0 / (1.0 + t_0)) * (-1.453152027 - (1.061405429 / (-1.0 - (x_m * 0.3275911)))))) * (1.0 / (-1.0 - t_0))) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	t_1 = fma(0.3275911, abs(x_m), 1.0)
	tmp = 0.0
	if (abs(x_m) <= 1e-17)
		tmp = Float64(Float64(Float64(1.0 - Float64((t_1 ^ -4.0) * 0.999999996)) / fma(0.999999998, (t_1 ^ -2.0), 1.0)) / Float64(1.0 + Float64(0.999999999 / t_1)));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(1.0 / Float64(1.0 + (cbrt(Float64(x_m * 0.3275911)) ^ 3.0))) * Float64(Float64(exp(Float64(-log1p(t_0))) * Float64(Float64(Float64(1.421413741 + Float64(Float64(1.0 / Float64(1.0 + t_0)) * Float64(-1.453152027 - Float64(1.061405429 / Float64(-1.0 - Float64(x_m * 0.3275911)))))) * Float64(1.0 / Float64(-1.0 - t_0))) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-17], N[(N[(N[(1.0 - N[(N[Power[t$95$1, -4.0], $MachinePrecision] * 0.999999996), $MachinePrecision]), $MachinePrecision] / N[(0.999999998 * N[Power[t$95$1, -2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.999999999 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / N[(1.0 + N[Power[N[Power[N[(x$95$m * 0.3275911), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Exp[(-N[Log[1 + t$95$0], $MachinePrecision])], $MachinePrecision] * N[(N[(N[(1.421413741 + N[(N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(-1.453152027 - N[(1.061405429 / N[(-1.0 - N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000007e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

      1. expm1-log1p-u 57.8%

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

      2. expm1-undefine 57.8%

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

   6. Applied egg-rr 57.8%

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

      1. expm1-define 57.8%

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

   8. Simplified 57.8%

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

   9. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. flip-- 57.8%

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

       2. metadata-eval 57.8%

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

       3. pow2 57.8%

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

       4. un-div-inv 57.8%

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

       5. +-commutative 57.8%

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

       6. fma-undefine 57.8%

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

       7. un-div-inv 57.8%

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

       8. +-commutative 57.8%

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

       9. fma-undefine 57.8%

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

   11. Applied egg-rr 57.8%

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

       1. unpow2 57.8%

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

       2. metadata-eval 57.8%

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

       3. associate-*l/ 57.8%

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

       4. metadata-eval 57.8%

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

       5. associate-*l/ 57.8%

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

       6. swap-sqr 57.8%

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

       7. unpow-1 57.8%

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

       8. unpow-1 57.8%

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

       9. pow-sqr 57.8%

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

       10. metadata-eval 57.8%

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

       11. metadata-eval 57.8%

           \[\leadsto \frac{1 - {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot \color{blue}{0.999999998}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

   13. Simplified 57.8%

       \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}} \]
   14. Step-by-step derivation

       1. flip-- 57.7%

          \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \left({\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998\right) \cdot \left({\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998\right)}{1 + {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998}}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

       2. metadata-eval 57.7%

          \[\leadsto \frac{\frac{\color{blue}{1} - \left({\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998\right) \cdot \left({\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998\right)}{1 + {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

       3. pow2 57.7%

          \[\leadsto \frac{\frac{1 - \color{blue}{{\left({\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998\right)}^{2}}}{1 + {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

   15. Applied egg-rr 57.7%

       \[\leadsto \frac{\color{blue}{\frac{1 - {\left({\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998\right)}^{2}}{1 + {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998}}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]
   16. Step-by-step derivation

       1. unpow2 57.7%

          \[\leadsto \frac{\frac{1 - \color{blue}{\left({\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998\right) \cdot \left({\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998\right)}}{1 + {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

       2. swap-sqr 57.7%

          \[\leadsto \frac{\frac{1 - \color{blue}{\left({\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}\right) \cdot \left(0.999999998 \cdot 0.999999998\right)}}{1 + {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

       3. pow-sqr 57.7%

          \[\leadsto \frac{\frac{1 - \color{blue}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{\left(2 \cdot -2\right)}} \cdot \left(0.999999998 \cdot 0.999999998\right)}{1 + {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

       4. metadata-eval 57.7%

          \[\leadsto \frac{\frac{1 - {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{\color{blue}{-4}} \cdot \left(0.999999998 \cdot 0.999999998\right)}{1 + {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

       5. metadata-eval 65.3%

          \[\leadsto \frac{\frac{1 - {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-4} \cdot \color{blue}{0.999999996}}{1 + {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

       6. +-commutative 65.3%

          \[\leadsto \frac{\frac{1 - {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-4} \cdot 0.999999996}{\color{blue}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2} \cdot 0.999999998 + 1}}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

       7. *-commutative 65.3%

          \[\leadsto \frac{\frac{1 - {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-4} \cdot 0.999999996}{\color{blue}{0.999999998 \cdot {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}} + 1}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

       8. fma-define 65.3%

          \[\leadsto \frac{\frac{1 - {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-4} \cdot 0.999999996}{\color{blue}{\mathsf{fma}\left(0.999999998, {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, 1\right)}}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

   17. Simplified 65.3%

       \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-4} \cdot 0.999999996}{\mathsf{fma}\left(0.999999998, {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, 1\right)}}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \]

   if 1.00000000000000007e-17 < (fabs.f64 x)

   1. Initial program 98.5%

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

   3. Simplified 98.5%

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

      1. add-cube-cbrt 98.5%

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

      2. pow3 98.5%

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

      3. add-sqr-sqrt 44.3%

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

      4. fabs-sqr 44.3%

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

      5. add-sqr-sqrt 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. add-exp-log 97.2%

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

      2. log-rec 97.2%

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

      3. log1p-define 97.2%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\color{blue}{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   8. Applied egg-rr 97.2%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + \color{blue}{e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 97.2%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. log1p-define 97.2%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. +-commutative 97.2%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. fma-undefine 97.2%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. expm1-undefine 97.2%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      6. add-exp-log 97.2%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      7. add-sqr-sqrt 44.3%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      8. fabs-sqr 44.3%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      9. add-sqr-sqrt 97.0%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   10. Applied egg-rr 97.0%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   11. Step-by-step derivation

       1. fma-undefine 97.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. associate--l+ 97.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. metadata-eval 97.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. +-rgt-identity 97.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   12. Simplified 97.0%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-17}:\\ \;\;\;\;\frac{\frac{1 - {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-4} \cdot 0.999999996}{\mathsf{fma}\left(0.999999998, {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, 1\right)}}{1 + \frac{0.999999999}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + {\left(\sqrt[3]{x \cdot 0.3275911}\right)}^{3}} \cdot \left(e^{-\mathsf{log1p}\left(\left|x\right| \cdot 0.3275911\right)} \cdot \left(\left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 - \frac{1.061405429}{-1 - x \cdot 0.3275911}\right)\right) \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ 1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(\frac{1}{1 + {\left(\sqrt[3]{x\_m \cdot 0.3275911}\right)}^{3}} \cdot \left(e^{-\mathsf{log1p}\left(t\_0\right)} \cdot \left(\left(1.421413741 + \frac{1}{1 + t\_0} \cdot \left(-1.453152027 - \frac{1.061405429}{-1 - x\_m \cdot 0.3275911}\right)\right) \cdot \frac{1}{-1 - t\_0} - -0.284496736\right) - 0.254829592\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)))
   (+
    1.0
    (*
     (exp (* x_m (- x_m)))
     (*
      (/ 1.0 (+ 1.0 (pow (cbrt (* x_m 0.3275911)) 3.0)))
      (-
       (*
        (exp (- (log1p t_0)))
        (-
         (*
          (+
           1.421413741
           (*
            (/ 1.0 (+ 1.0 t_0))
            (- -1.453152027 (/ 1.061405429 (- -1.0 (* x_m 0.3275911))))))
          (/ 1.0 (- -1.0 t_0)))
         -0.284496736))
       0.254829592))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	return 1.0 + (exp((x_m * -x_m)) * ((1.0 / (1.0 + pow(cbrt((x_m * 0.3275911)), 3.0))) * ((exp(-log1p(t_0)) * (((1.421413741 + ((1.0 / (1.0 + t_0)) * (-1.453152027 - (1.061405429 / (-1.0 - (x_m * 0.3275911)))))) * (1.0 / (-1.0 - t_0))) - -0.284496736)) - 0.254829592)));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.abs(x_m) * 0.3275911;
	return 1.0 + (Math.exp((x_m * -x_m)) * ((1.0 / (1.0 + Math.pow(Math.cbrt((x_m * 0.3275911)), 3.0))) * ((Math.exp(-Math.log1p(t_0)) * (((1.421413741 + ((1.0 / (1.0 + t_0)) * (-1.453152027 - (1.061405429 / (-1.0 - (x_m * 0.3275911)))))) * (1.0 / (-1.0 - t_0))) - -0.284496736)) - 0.254829592)));
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	return Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(1.0 / Float64(1.0 + (cbrt(Float64(x_m * 0.3275911)) ^ 3.0))) * Float64(Float64(exp(Float64(-log1p(t_0))) * Float64(Float64(Float64(1.421413741 + Float64(Float64(1.0 / Float64(1.0 + t_0)) * Float64(-1.453152027 - Float64(1.061405429 / Float64(-1.0 - Float64(x_m * 0.3275911)))))) * Float64(1.0 / Float64(-1.0 - t_0))) - -0.284496736)) - 0.254829592))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / N[(1.0 + N[Power[N[Power[N[(x$95$m * 0.3275911), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Exp[(-N[Log[1 + t$95$0], $MachinePrecision])], $MachinePrecision] * N[(N[(N[(1.421413741 + N[(N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(-1.453152027 - N[(1.061405429 / N[(-1.0 - N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.9%

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

3. Simplified 78.9%

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

   1. add-cube-cbrt 78.9%

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

   2. pow3 78.9%

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

   3. add-sqr-sqrt 36.8%

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

   4. fabs-sqr 36.8%

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

   5. add-sqr-sqrt 78.3%

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

6. Applied egg-rr 78.3%

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

   1. add-exp-log 78.3%

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

   2. log-rec 78.3%

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

   3. log1p-define 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\color{blue}{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

8. Applied egg-rr 78.3%

   \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + \color{blue}{e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
9. Step-by-step derivation

   1. expm1-log1p-u 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   2. log1p-define 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   3. +-commutative 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   4. fma-undefine 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   5. expm1-undefine 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   6. add-exp-log 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   7. add-sqr-sqrt 36.8%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   8. fabs-sqr 36.8%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   9. add-sqr-sqrt 78.2%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

10. Applied egg-rr 78.2%

    \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
11. Step-by-step derivation

    1. fma-undefine 78.2%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

    2. associate--l+ 78.2%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

    3. metadata-eval 78.2%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

    4. +-rgt-identity 78.2%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

12. Simplified 78.2%

    \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

13. Final simplification 78.2%

    \[\leadsto 1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + {\left(\sqrt[3]{x \cdot 0.3275911}\right)}^{3}} \cdot \left(e^{-\mathsf{log1p}\left(\left|x\right| \cdot 0.3275911\right)} \cdot \left(\left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 - \frac{1.061405429}{-1 - x \cdot 0.3275911}\right)\right) \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911} - -0.284496736\right) - 0.254829592\right)\right) \]
14. Add Preprocessing

Alternative 3: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{-1 - t\_0}\\ 1 - e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(\frac{1}{1 + {\left(\sqrt[3]{x\_m \cdot 0.3275911}\right)}^{3}} \cdot \left(0.254829592 + t\_1 \cdot \left(\left(1.421413741 + \frac{1}{1 + t\_0} \cdot \left(-1.453152027 - \frac{1.061405429}{-1 - x\_m \cdot 0.3275911}\right)\right) \cdot t\_1 - -0.284496736\right)\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)) (t_1 (/ 1.0 (- -1.0 t_0))))
   (-
    1.0
    (*
     (exp (* x_m (- x_m)))
     (*
      (/ 1.0 (+ 1.0 (pow (cbrt (* x_m 0.3275911)) 3.0)))
      (+
       0.254829592
       (*
        t_1
        (-
         (*
          (+
           1.421413741
           (*
            (/ 1.0 (+ 1.0 t_0))
            (- -1.453152027 (/ 1.061405429 (- -1.0 (* x_m 0.3275911))))))
          t_1)
         -0.284496736))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double t_1 = 1.0 / (-1.0 - t_0);
	return 1.0 - (exp((x_m * -x_m)) * ((1.0 / (1.0 + pow(cbrt((x_m * 0.3275911)), 3.0))) * (0.254829592 + (t_1 * (((1.421413741 + ((1.0 / (1.0 + t_0)) * (-1.453152027 - (1.061405429 / (-1.0 - (x_m * 0.3275911)))))) * t_1) - -0.284496736)))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.abs(x_m) * 0.3275911;
	double t_1 = 1.0 / (-1.0 - t_0);
	return 1.0 - (Math.exp((x_m * -x_m)) * ((1.0 / (1.0 + Math.pow(Math.cbrt((x_m * 0.3275911)), 3.0))) * (0.254829592 + (t_1 * (((1.421413741 + ((1.0 / (1.0 + t_0)) * (-1.453152027 - (1.061405429 / (-1.0 - (x_m * 0.3275911)))))) * t_1) - -0.284496736)))));
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	t_1 = Float64(1.0 / Float64(-1.0 - t_0))
	return Float64(1.0 - Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(1.0 / Float64(1.0 + (cbrt(Float64(x_m * 0.3275911)) ^ 3.0))) * Float64(0.254829592 + Float64(t_1 * Float64(Float64(Float64(1.421413741 + Float64(Float64(1.0 / Float64(1.0 + t_0)) * Float64(-1.453152027 - Float64(1.061405429 / Float64(-1.0 - Float64(x_m * 0.3275911)))))) * t_1) - -0.284496736))))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / N[(1.0 + N[Power[N[Power[N[(x$95$m * 0.3275911), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.254829592 + N[(t$95$1 * N[(N[(N[(1.421413741 + N[(N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(-1.453152027 - N[(1.061405429 / N[(-1.0 - N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 78.9%

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

3. Simplified 78.9%

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

   1. add-cube-cbrt 78.9%

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

   2. pow3 78.9%

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

   3. add-sqr-sqrt 36.8%

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

   4. fabs-sqr 36.8%

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

   5. add-sqr-sqrt 78.3%

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

6. Applied egg-rr 78.3%

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

   1. expm1-log1p-u 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   2. log1p-define 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   3. +-commutative 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   4. fma-undefine 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   5. expm1-undefine 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   6. add-exp-log 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   7. add-sqr-sqrt 36.8%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   8. fabs-sqr 36.8%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   9. add-sqr-sqrt 78.2%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

8. Applied egg-rr 78.2%

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

   1. fma-undefine 78.2%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   2. associate--l+ 78.2%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   3. metadata-eval 78.2%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   4. +-rgt-identity 78.2%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

10. Simplified 78.2%

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

11. Final simplification 78.2%

    \[\leadsto 1 - e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + {\left(\sqrt[3]{x \cdot 0.3275911}\right)}^{3}} \cdot \left(0.254829592 + \frac{1}{-1 - \left|x\right| \cdot 0.3275911} \cdot \left(\left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 - \frac{1.061405429}{-1 - x \cdot 0.3275911}\right)\right) \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911} - -0.284496736\right)\right)\right) \]
12. Add Preprocessing

Alternative 4: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{-1 - t\_0}\\ t_2 := \frac{1}{1 + t\_0}\\ 1 - e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(t\_2 \cdot \left(0.254829592 + t\_1 \cdot \left(\left(1.421413741 + t\_2 \cdot \left(-1.453152027 - \frac{1.061405429}{-1 - x\_m \cdot 0.3275911}\right)\right) \cdot t\_1 - -0.284496736\right)\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911))
        (t_1 (/ 1.0 (- -1.0 t_0)))
        (t_2 (/ 1.0 (+ 1.0 t_0))))
   (-
    1.0
    (*
     (exp (* x_m (- x_m)))
     (*
      t_2
      (+
       0.254829592
       (*
        t_1
        (-
         (*
          (+
           1.421413741
           (* t_2 (- -1.453152027 (/ 1.061405429 (- -1.0 (* x_m 0.3275911))))))
          t_1)
         -0.284496736))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double t_1 = 1.0 / (-1.0 - t_0);
	double t_2 = 1.0 / (1.0 + t_0);
	return 1.0 - (exp((x_m * -x_m)) * (t_2 * (0.254829592 + (t_1 * (((1.421413741 + (t_2 * (-1.453152027 - (1.061405429 / (-1.0 - (x_m * 0.3275911)))))) * t_1) - -0.284496736)))));
}

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
    t_0 = abs(x_m) * 0.3275911d0
    t_1 = 1.0d0 / ((-1.0d0) - t_0)
    t_2 = 1.0d0 / (1.0d0 + t_0)
    code = 1.0d0 - (exp((x_m * -x_m)) * (t_2 * (0.254829592d0 + (t_1 * (((1.421413741d0 + (t_2 * ((-1.453152027d0) - (1.061405429d0 / ((-1.0d0) - (x_m * 0.3275911d0)))))) * t_1) - (-0.284496736d0))))))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.abs(x_m) * 0.3275911;
	double t_1 = 1.0 / (-1.0 - t_0);
	double t_2 = 1.0 / (1.0 + t_0);
	return 1.0 - (Math.exp((x_m * -x_m)) * (t_2 * (0.254829592 + (t_1 * (((1.421413741 + (t_2 * (-1.453152027 - (1.061405429 / (-1.0 - (x_m * 0.3275911)))))) * t_1) - -0.284496736)))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.fabs(x_m) * 0.3275911
	t_1 = 1.0 / (-1.0 - t_0)
	t_2 = 1.0 / (1.0 + t_0)
	return 1.0 - (math.exp((x_m * -x_m)) * (t_2 * (0.254829592 + (t_1 * (((1.421413741 + (t_2 * (-1.453152027 - (1.061405429 / (-1.0 - (x_m * 0.3275911)))))) * t_1) - -0.284496736)))))

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	t_1 = Float64(1.0 / Float64(-1.0 - t_0))
	t_2 = Float64(1.0 / Float64(1.0 + t_0))
	return Float64(1.0 - Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(t_2 * Float64(0.254829592 + Float64(t_1 * Float64(Float64(Float64(1.421413741 + Float64(t_2 * Float64(-1.453152027 - Float64(1.061405429 / Float64(-1.0 - Float64(x_m * 0.3275911)))))) * t_1) - -0.284496736))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	t_0 = abs(x_m) * 0.3275911;
	t_1 = 1.0 / (-1.0 - t_0);
	t_2 = 1.0 / (1.0 + t_0);
	tmp = 1.0 - (exp((x_m * -x_m)) * (t_2 * (0.254829592 + (t_1 * (((1.421413741 + (t_2 * (-1.453152027 - (1.061405429 / (-1.0 - (x_m * 0.3275911)))))) * t_1) - -0.284496736)))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * N[(0.254829592 + N[(t$95$1 * N[(N[(N[(1.421413741 + N[(t$95$2 * N[(-1.453152027 - N[(1.061405429 / N[(-1.0 - N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 78.9%

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

3. Simplified 78.9%

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

   1. expm1-log1p-u 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   2. log1p-define 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   3. +-commutative 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   4. fma-undefine 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   5. expm1-undefine 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   6. add-exp-log 78.3%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   7. add-sqr-sqrt 36.8%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   8. fabs-sqr 36.8%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   9. add-sqr-sqrt 78.2%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

6. Applied egg-rr 78.3%

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

   1. fma-undefine 78.2%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   2. associate--l+ 78.2%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   3. metadata-eval 78.2%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   4. +-rgt-identity 78.2%

      \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

8. Simplified 78.3%

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

9. Final simplification 78.3%

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

Alternative 5: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ t_1 := 1 + t\_0\\ t_2 := \frac{1}{t\_1}\\ \mathbf{if}\;\left|x\_m\right| \leq 0.02:\\ \;\;\;\;\left(1 + x\_m \cdot \left(x\_m \cdot \left(-0.3754899882585643 \cdot \frac{x\_m}{t\_1} + t\_2 \cdot 0.36953108532122814\right) + t\_2 \cdot 0.8007952583978091\right)\right) + 0.999999999 \cdot \frac{1}{-1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(t\_2 \cdot \left(t\_2 \cdot \left(t\_2 \cdot \left(1.453152027 \cdot \frac{-1}{-1 - x\_m \cdot 0.3275911} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)) (t_1 (+ 1.0 t_0)) (t_2 (/ 1.0 t_1)))
   (if (<= (fabs x_m) 0.02)
     (+
      (+
       1.0
       (*
        x_m
        (+
         (*
          x_m
          (+ (* -0.3754899882585643 (/ x_m t_1)) (* t_2 0.36953108532122814)))
         (* t_2 0.8007952583978091))))
      (* 0.999999999 (/ 1.0 (- -1.0 t_0))))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        t_2
        (-
         (*
          t_2
          (-
           (*
            t_2
            (-
             (* 1.453152027 (/ -1.0 (- -1.0 (* x_m 0.3275911))))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double t_1 = 1.0 + t_0;
	double t_2 = 1.0 / t_1;
	double tmp;
	if (fabs(x_m) <= 0.02) {
		tmp = (1.0 + (x_m * ((x_m * ((-0.3754899882585643 * (x_m / t_1)) + (t_2 * 0.36953108532122814))) + (t_2 * 0.8007952583978091)))) + (0.999999999 * (1.0 / (-1.0 - t_0)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * (t_2 * ((t_2 * ((t_2 * ((1.453152027 * (-1.0 / (-1.0 - (x_m * 0.3275911)))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	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) :: tmp
    t_0 = abs(x_m) * 0.3275911d0
    t_1 = 1.0d0 + t_0
    t_2 = 1.0d0 / t_1
    if (abs(x_m) <= 0.02d0) then
        tmp = (1.0d0 + (x_m * ((x_m * (((-0.3754899882585643d0) * (x_m / t_1)) + (t_2 * 0.36953108532122814d0))) + (t_2 * 0.8007952583978091d0)))) + (0.999999999d0 * (1.0d0 / ((-1.0d0) - t_0)))
    else
        tmp = 1.0d0 + (exp((x_m * -x_m)) * (t_2 * ((t_2 * ((t_2 * ((1.453152027d0 * ((-1.0d0) / ((-1.0d0) - (x_m * 0.3275911d0)))) - 1.421413741d0)) - (-0.284496736d0))) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.abs(x_m) * 0.3275911;
	double t_1 = 1.0 + t_0;
	double t_2 = 1.0 / t_1;
	double tmp;
	if (Math.abs(x_m) <= 0.02) {
		tmp = (1.0 + (x_m * ((x_m * ((-0.3754899882585643 * (x_m / t_1)) + (t_2 * 0.36953108532122814))) + (t_2 * 0.8007952583978091)))) + (0.999999999 * (1.0 / (-1.0 - t_0)));
	} else {
		tmp = 1.0 + (Math.exp((x_m * -x_m)) * (t_2 * ((t_2 * ((t_2 * ((1.453152027 * (-1.0 / (-1.0 - (x_m * 0.3275911)))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.fabs(x_m) * 0.3275911
	t_1 = 1.0 + t_0
	t_2 = 1.0 / t_1
	tmp = 0
	if math.fabs(x_m) <= 0.02:
		tmp = (1.0 + (x_m * ((x_m * ((-0.3754899882585643 * (x_m / t_1)) + (t_2 * 0.36953108532122814))) + (t_2 * 0.8007952583978091)))) + (0.999999999 * (1.0 / (-1.0 - t_0)))
	else:
		tmp = 1.0 + (math.exp((x_m * -x_m)) * (t_2 * ((t_2 * ((t_2 * ((1.453152027 * (-1.0 / (-1.0 - (x_m * 0.3275911)))) - 1.421413741)) - -0.284496736)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(1.0 / t_1)
	tmp = 0.0
	if (abs(x_m) <= 0.02)
		tmp = Float64(Float64(1.0 + Float64(x_m * Float64(Float64(x_m * Float64(Float64(-0.3754899882585643 * Float64(x_m / t_1)) + Float64(t_2 * 0.36953108532122814))) + Float64(t_2 * 0.8007952583978091)))) + Float64(0.999999999 * Float64(1.0 / Float64(-1.0 - t_0))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(t_2 * Float64(Float64(t_2 * Float64(Float64(t_2 * Float64(Float64(1.453152027 * Float64(-1.0 / Float64(-1.0 - Float64(x_m * 0.3275911)))) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = abs(x_m) * 0.3275911;
	t_1 = 1.0 + t_0;
	t_2 = 1.0 / t_1;
	tmp = 0.0;
	if (abs(x_m) <= 0.02)
		tmp = (1.0 + (x_m * ((x_m * ((-0.3754899882585643 * (x_m / t_1)) + (t_2 * 0.36953108532122814))) + (t_2 * 0.8007952583978091)))) + (0.999999999 * (1.0 / (-1.0 - t_0)));
	else
		tmp = 1.0 + (exp((x_m * -x_m)) * (t_2 * ((t_2 * ((t_2 * ((1.453152027 * (-1.0 / (-1.0 - (x_m * 0.3275911)))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.02], N[(N[(1.0 + N[(x$95$m * N[(N[(x$95$m * N[(N[(-0.3754899882585643 * N[(x$95$m / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * 0.36953108532122814), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * 0.8007952583978091), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.999999999 * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * N[(N[(t$95$2 * N[(N[(t$95$2 * N[(N[(1.453152027 * N[(-1.0 / N[(-1.0 - N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0200000000000000004

   1. Initial program 58.2%

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

   3. Simplified 58.2%

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

      1. expm1-log1p-u 58.2%

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

      2. expm1-undefine 58.2%

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

   6. Applied egg-rr 56.5%

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

      1. expm1-define 56.5%

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

   8. Simplified 56.5%

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

   9. Taylor expanded in x around 0 56.3%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.3754899882585643 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|} + 0.36953108532122814 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + 0.8007952583978091 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]

   if 0.0200000000000000004 < (fabs.f64 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 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. log1p-define 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. +-commutative 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. fma-undefine 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. expm1-undefine 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      6. add-exp-log 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      7. add-sqr-sqrt 45.7%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      8. fabs-sqr 45.7%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      9. add-sqr-sqrt 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   6. Applied egg-rr 100.0%

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

      1. fma-undefine 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. associate--l+ 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. metadata-eval 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. +-rgt-identity 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   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. Taylor expanded in x around inf 99.5%

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

       1. expm1-log1p-u 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. log1p-define 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. +-commutative 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. fma-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       5. expm1-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       6. add-exp-log 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       7. add-sqr-sqrt 45.7%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       8. fabs-sqr 45.7%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       9. add-sqr-sqrt 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   11. Applied egg-rr 99.5%

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

       1. fma-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. associate--l+ 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. +-rgt-identity 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   13. Simplified 99.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;\left(1 + x \cdot \left(x \cdot \left(-0.3754899882585643 \cdot \frac{x}{1 + \left|x\right| \cdot 0.3275911} + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.36953108532122814\right) + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.8007952583978091\right)\right) + 0.999999999 \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.453152027 \cdot \frac{-1}{-1 - x \cdot 0.3275911} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ t_1 := 1 + t\_0\\ t_2 := \frac{1}{t\_1}\\ \mathbf{if}\;\left|x\_m\right| \leq 0.02:\\ \;\;\;\;\left(1 + x\_m \cdot \left(x\_m \cdot \left(-0.3754899882585643 \cdot \frac{x\_m}{t\_1} + t\_2 \cdot 0.36953108532122814\right) + t\_2 \cdot 0.8007952583978091\right)\right) + 0.999999999 \cdot \frac{1}{-1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 - 0.254829592 \cdot \frac{1}{\left(1 + x\_m \cdot 0.3275911\right) \cdot e^{{x\_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)) (t_1 (+ 1.0 t_0)) (t_2 (/ 1.0 t_1)))
   (if (<= (fabs x_m) 0.02)
     (+
      (+
       1.0
       (*
        x_m
        (+
         (*
          x_m
          (+ (* -0.3754899882585643 (/ x_m t_1)) (* t_2 0.36953108532122814)))
         (* t_2 0.8007952583978091))))
      (* 0.999999999 (/ 1.0 (- -1.0 t_0))))
     (-
      1.0
      (*
       0.254829592
       (/ 1.0 (* (+ 1.0 (* x_m 0.3275911)) (exp (pow x_m 2.0)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double t_1 = 1.0 + t_0;
	double t_2 = 1.0 / t_1;
	double tmp;
	if (fabs(x_m) <= 0.02) {
		tmp = (1.0 + (x_m * ((x_m * ((-0.3754899882585643 * (x_m / t_1)) + (t_2 * 0.36953108532122814))) + (t_2 * 0.8007952583978091)))) + (0.999999999 * (1.0 / (-1.0 - t_0)));
	} else {
		tmp = 1.0 - (0.254829592 * (1.0 / ((1.0 + (x_m * 0.3275911)) * exp(pow(x_m, 2.0)))));
	}
	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) :: tmp
    t_0 = abs(x_m) * 0.3275911d0
    t_1 = 1.0d0 + t_0
    t_2 = 1.0d0 / t_1
    if (abs(x_m) <= 0.02d0) then
        tmp = (1.0d0 + (x_m * ((x_m * (((-0.3754899882585643d0) * (x_m / t_1)) + (t_2 * 0.36953108532122814d0))) + (t_2 * 0.8007952583978091d0)))) + (0.999999999d0 * (1.0d0 / ((-1.0d0) - t_0)))
    else
        tmp = 1.0d0 - (0.254829592d0 * (1.0d0 / ((1.0d0 + (x_m * 0.3275911d0)) * exp((x_m ** 2.0d0)))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.abs(x_m) * 0.3275911;
	double t_1 = 1.0 + t_0;
	double t_2 = 1.0 / t_1;
	double tmp;
	if (Math.abs(x_m) <= 0.02) {
		tmp = (1.0 + (x_m * ((x_m * ((-0.3754899882585643 * (x_m / t_1)) + (t_2 * 0.36953108532122814))) + (t_2 * 0.8007952583978091)))) + (0.999999999 * (1.0 / (-1.0 - t_0)));
	} else {
		tmp = 1.0 - (0.254829592 * (1.0 / ((1.0 + (x_m * 0.3275911)) * Math.exp(Math.pow(x_m, 2.0)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.fabs(x_m) * 0.3275911
	t_1 = 1.0 + t_0
	t_2 = 1.0 / t_1
	tmp = 0
	if math.fabs(x_m) <= 0.02:
		tmp = (1.0 + (x_m * ((x_m * ((-0.3754899882585643 * (x_m / t_1)) + (t_2 * 0.36953108532122814))) + (t_2 * 0.8007952583978091)))) + (0.999999999 * (1.0 / (-1.0 - t_0)))
	else:
		tmp = 1.0 - (0.254829592 * (1.0 / ((1.0 + (x_m * 0.3275911)) * math.exp(math.pow(x_m, 2.0)))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(1.0 / t_1)
	tmp = 0.0
	if (abs(x_m) <= 0.02)
		tmp = Float64(Float64(1.0 + Float64(x_m * Float64(Float64(x_m * Float64(Float64(-0.3754899882585643 * Float64(x_m / t_1)) + Float64(t_2 * 0.36953108532122814))) + Float64(t_2 * 0.8007952583978091)))) + Float64(0.999999999 * Float64(1.0 / Float64(-1.0 - t_0))));
	else
		tmp = Float64(1.0 - Float64(0.254829592 * Float64(1.0 / Float64(Float64(1.0 + Float64(x_m * 0.3275911)) * exp((x_m ^ 2.0))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = abs(x_m) * 0.3275911;
	t_1 = 1.0 + t_0;
	t_2 = 1.0 / t_1;
	tmp = 0.0;
	if (abs(x_m) <= 0.02)
		tmp = (1.0 + (x_m * ((x_m * ((-0.3754899882585643 * (x_m / t_1)) + (t_2 * 0.36953108532122814))) + (t_2 * 0.8007952583978091)))) + (0.999999999 * (1.0 / (-1.0 - t_0)));
	else
		tmp = 1.0 - (0.254829592 * (1.0 / ((1.0 + (x_m * 0.3275911)) * exp((x_m ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.02], N[(N[(1.0 + N[(x$95$m * N[(N[(x$95$m * N[(N[(-0.3754899882585643 * N[(x$95$m / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * 0.36953108532122814), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * 0.8007952583978091), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.999999999 * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.254829592 * N[(1.0 / N[(N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0200000000000000004

   1. Initial program 58.2%

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

   3. Simplified 58.2%

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

      1. expm1-log1p-u 58.2%

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

      2. expm1-undefine 58.2%

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

   6. Applied egg-rr 56.5%

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

      1. expm1-define 56.5%

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

   8. Simplified 56.5%

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

   9. Taylor expanded in x around 0 56.3%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.3754899882585643 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|} + 0.36953108532122814 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + 0.8007952583978091 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right)\right) - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]

   if 0.0200000000000000004 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 99.5%

      \[\leadsto \color{blue}{1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. log1p-define 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. +-commutative 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. fma-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       5. expm1-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       6. add-exp-log 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       7. add-sqr-sqrt 45.7%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       8. fabs-sqr 45.7%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       9. add-sqr-sqrt 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   11. Applied egg-rr 99.5%

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

       1. fma-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. associate--l+ 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. +-rgt-identity 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   13. Simplified 99.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;\left(1 + x \cdot \left(x \cdot \left(-0.3754899882585643 \cdot \frac{x}{1 + \left|x\right| \cdot 0.3275911} + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.36953108532122814\right) + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.8007952583978091\right)\right) + 0.999999999 \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911}\\ \mathbf{else}:\\ \;\;\;\;1 - 0.254829592 \cdot \frac{1}{\left(1 + x \cdot 0.3275911\right) \cdot e^{{x}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\left|x\_m\right| \leq 0.02:\\ \;\;\;\;\left(1 + x\_m \cdot \left(\frac{1}{t\_1} \cdot 0.8007952583978091 + \frac{x\_m}{t\_1} \cdot 0.36953108532122814\right)\right) + 0.999999999 \cdot \frac{1}{-1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 - 0.254829592 \cdot \frac{1}{\left(1 + x\_m \cdot 0.3275911\right) \cdot e^{{x\_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)) (t_1 (+ 1.0 t_0)))
   (if (<= (fabs x_m) 0.02)
     (+
      (+
       1.0
       (*
        x_m
        (+
         (* (/ 1.0 t_1) 0.8007952583978091)
         (* (/ x_m t_1) 0.36953108532122814))))
      (* 0.999999999 (/ 1.0 (- -1.0 t_0))))
     (-
      1.0
      (*
       0.254829592
       (/ 1.0 (* (+ 1.0 (* x_m 0.3275911)) (exp (pow x_m 2.0)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double t_1 = 1.0 + t_0;
	double tmp;
	if (fabs(x_m) <= 0.02) {
		tmp = (1.0 + (x_m * (((1.0 / t_1) * 0.8007952583978091) + ((x_m / t_1) * 0.36953108532122814)))) + (0.999999999 * (1.0 / (-1.0 - t_0)));
	} else {
		tmp = 1.0 - (0.254829592 * (1.0 / ((1.0 + (x_m * 0.3275911)) * exp(pow(x_m, 2.0)))));
	}
	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 = abs(x_m) * 0.3275911d0
    t_1 = 1.0d0 + t_0
    if (abs(x_m) <= 0.02d0) then
        tmp = (1.0d0 + (x_m * (((1.0d0 / t_1) * 0.8007952583978091d0) + ((x_m / t_1) * 0.36953108532122814d0)))) + (0.999999999d0 * (1.0d0 / ((-1.0d0) - t_0)))
    else
        tmp = 1.0d0 - (0.254829592d0 * (1.0d0 / ((1.0d0 + (x_m * 0.3275911d0)) * exp((x_m ** 2.0d0)))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.abs(x_m) * 0.3275911;
	double t_1 = 1.0 + t_0;
	double tmp;
	if (Math.abs(x_m) <= 0.02) {
		tmp = (1.0 + (x_m * (((1.0 / t_1) * 0.8007952583978091) + ((x_m / t_1) * 0.36953108532122814)))) + (0.999999999 * (1.0 / (-1.0 - t_0)));
	} else {
		tmp = 1.0 - (0.254829592 * (1.0 / ((1.0 + (x_m * 0.3275911)) * Math.exp(Math.pow(x_m, 2.0)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.fabs(x_m) * 0.3275911
	t_1 = 1.0 + t_0
	tmp = 0
	if math.fabs(x_m) <= 0.02:
		tmp = (1.0 + (x_m * (((1.0 / t_1) * 0.8007952583978091) + ((x_m / t_1) * 0.36953108532122814)))) + (0.999999999 * (1.0 / (-1.0 - t_0)))
	else:
		tmp = 1.0 - (0.254829592 * (1.0 / ((1.0 + (x_m * 0.3275911)) * math.exp(math.pow(x_m, 2.0)))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	t_1 = Float64(1.0 + t_0)
	tmp = 0.0
	if (abs(x_m) <= 0.02)
		tmp = Float64(Float64(1.0 + Float64(x_m * Float64(Float64(Float64(1.0 / t_1) * 0.8007952583978091) + Float64(Float64(x_m / t_1) * 0.36953108532122814)))) + Float64(0.999999999 * Float64(1.0 / Float64(-1.0 - t_0))));
	else
		tmp = Float64(1.0 - Float64(0.254829592 * Float64(1.0 / Float64(Float64(1.0 + Float64(x_m * 0.3275911)) * exp((x_m ^ 2.0))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = abs(x_m) * 0.3275911;
	t_1 = 1.0 + t_0;
	tmp = 0.0;
	if (abs(x_m) <= 0.02)
		tmp = (1.0 + (x_m * (((1.0 / t_1) * 0.8007952583978091) + ((x_m / t_1) * 0.36953108532122814)))) + (0.999999999 * (1.0 / (-1.0 - t_0)));
	else
		tmp = 1.0 - (0.254829592 * (1.0 / ((1.0 + (x_m * 0.3275911)) * exp((x_m ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.02], N[(N[(1.0 + N[(x$95$m * N[(N[(N[(1.0 / t$95$1), $MachinePrecision] * 0.8007952583978091), $MachinePrecision] + N[(N[(x$95$m / t$95$1), $MachinePrecision] * 0.36953108532122814), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.999999999 * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.254829592 * N[(1.0 / N[(N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0200000000000000004

   1. Initial program 58.2%

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

   3. Simplified 58.2%

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

      1. expm1-log1p-u 58.2%

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

      2. expm1-undefine 58.2%

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

   6. Applied egg-rr 56.5%

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

      1. expm1-define 56.5%

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

   8. Simplified 56.5%

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

   9. Taylor expanded in x around 0 56.2%

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

   if 0.0200000000000000004 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 99.5%

      \[\leadsto \color{blue}{1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. log1p-define 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. +-commutative 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. fma-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       5. expm1-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       6. add-exp-log 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       7. add-sqr-sqrt 45.7%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       8. fabs-sqr 45.7%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       9. add-sqr-sqrt 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   11. Applied egg-rr 99.5%

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

       1. fma-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. associate--l+ 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. +-rgt-identity 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   13. Simplified 99.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.8007952583978091 + \frac{x}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.36953108532122814\right)\right) + 0.999999999 \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911}\\ \mathbf{else}:\\ \;\;\;\;1 - 0.254829592 \cdot \frac{1}{\left(1 + x \cdot 0.3275911\right) \cdot e^{{x}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ \mathbf{if}\;\left|x\_m\right| \leq 0.02:\\ \;\;\;\;\left(1 + \frac{x\_m}{1 + t\_0} \cdot 0.8007952583978091\right) + 0.999999999 \cdot \frac{1}{-1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.254829592 \cdot \frac{1}{e^{{x\_m}^{2}} \cdot \left(-1 - x\_m \cdot 0.3275911\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)))
   (if (<= (fabs x_m) 0.02)
     (+
      (+ 1.0 (* (/ x_m (+ 1.0 t_0)) 0.8007952583978091))
      (* 0.999999999 (/ 1.0 (- -1.0 t_0))))
     (+
      1.0
      (*
       0.254829592
       (/ 1.0 (* (exp (pow x_m 2.0)) (- -1.0 (* x_m 0.3275911)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double tmp;
	if (fabs(x_m) <= 0.02) {
		tmp = (1.0 + ((x_m / (1.0 + t_0)) * 0.8007952583978091)) + (0.999999999 * (1.0 / (-1.0 - t_0)));
	} else {
		tmp = 1.0 + (0.254829592 * (1.0 / (exp(pow(x_m, 2.0)) * (-1.0 - (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) :: tmp
    t_0 = abs(x_m) * 0.3275911d0
    if (abs(x_m) <= 0.02d0) then
        tmp = (1.0d0 + ((x_m / (1.0d0 + t_0)) * 0.8007952583978091d0)) + (0.999999999d0 * (1.0d0 / ((-1.0d0) - t_0)))
    else
        tmp = 1.0d0 + (0.254829592d0 * (1.0d0 / (exp((x_m ** 2.0d0)) * ((-1.0d0) - (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 = Math.abs(x_m) * 0.3275911;
	double tmp;
	if (Math.abs(x_m) <= 0.02) {
		tmp = (1.0 + ((x_m / (1.0 + t_0)) * 0.8007952583978091)) + (0.999999999 * (1.0 / (-1.0 - t_0)));
	} else {
		tmp = 1.0 + (0.254829592 * (1.0 / (Math.exp(Math.pow(x_m, 2.0)) * (-1.0 - (x_m * 0.3275911)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.fabs(x_m) * 0.3275911
	tmp = 0
	if math.fabs(x_m) <= 0.02:
		tmp = (1.0 + ((x_m / (1.0 + t_0)) * 0.8007952583978091)) + (0.999999999 * (1.0 / (-1.0 - t_0)))
	else:
		tmp = 1.0 + (0.254829592 * (1.0 / (math.exp(math.pow(x_m, 2.0)) * (-1.0 - (x_m * 0.3275911)))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	tmp = 0.0
	if (abs(x_m) <= 0.02)
		tmp = Float64(Float64(1.0 + Float64(Float64(x_m / Float64(1.0 + t_0)) * 0.8007952583978091)) + Float64(0.999999999 * Float64(1.0 / Float64(-1.0 - t_0))));
	else
		tmp = Float64(1.0 + Float64(0.254829592 * Float64(1.0 / Float64(exp((x_m ^ 2.0)) * Float64(-1.0 - Float64(x_m * 0.3275911))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = abs(x_m) * 0.3275911;
	tmp = 0.0;
	if (abs(x_m) <= 0.02)
		tmp = (1.0 + ((x_m / (1.0 + t_0)) * 0.8007952583978091)) + (0.999999999 * (1.0 / (-1.0 - t_0)));
	else
		tmp = 1.0 + (0.254829592 * (1.0 / (exp((x_m ^ 2.0)) * (-1.0 - (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[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.02], N[(N[(1.0 + N[(N[(x$95$m / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * 0.8007952583978091), $MachinePrecision]), $MachinePrecision] + N[(0.999999999 * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.254829592 * N[(1.0 / N[(N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision] * N[(-1.0 - N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0200000000000000004

   1. Initial program 58.2%

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

   3. Simplified 58.2%

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

      1. expm1-log1p-u 58.2%

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

      2. expm1-undefine 58.2%

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

   6. Applied egg-rr 56.5%

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

      1. expm1-define 56.5%

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

   8. Simplified 56.5%

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

   9. Taylor expanded in x around 0 56.1%

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

   if 0.0200000000000000004 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 99.5%

      \[\leadsto \color{blue}{1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. log1p-define 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. +-commutative 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. fma-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       5. expm1-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       6. add-exp-log 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       7. add-sqr-sqrt 45.7%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       8. fabs-sqr 45.7%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       9. add-sqr-sqrt 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   11. Applied egg-rr 99.5%

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

       1. fma-undefine 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. associate--l+ 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. +-rgt-identity 100.0%

          \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   13. Simplified 99.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;\left(1 + \frac{x}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.8007952583978091\right) + 0.999999999 \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(-1 - x \cdot 0.3275911\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.4% accurate, 77.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 + 0.999999999 \cdot \frac{1}{-1 - x\_m \cdot 0.3275911} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (+ 1.0 (* 0.999999999 (/ 1.0 (- -1.0 (* x_m 0.3275911))))))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 + (0.999999999 * (1.0 / (-1.0 - (x_m * 0.3275911))));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1.0d0 + (0.999999999d0 * (1.0d0 / ((-1.0d0) - (x_m * 0.3275911d0))))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0 + (0.999999999 * (1.0 / (-1.0 - (x_m * 0.3275911))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0 + (0.999999999 * (1.0 / (-1.0 - (x_m * 0.3275911))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 + Float64(0.999999999 * Float64(1.0 / Float64(-1.0 - Float64(x_m * 0.3275911)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0 + (0.999999999 * (1.0 / (-1.0 - (x_m * 0.3275911))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 + N[(0.999999999 * N[(1.0 / N[(-1.0 - N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

3. Simplified 78.9%

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

   1. expm1-log1p-u 79.0%

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

   2. expm1-undefine 79.0%

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

6. Applied egg-rr 78.1%

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

   1. expm1-define 78.1%

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

8. Simplified 78.1%

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

9. Taylor expanded in x around 0 76.7%

   \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 78.3%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

    2. log1p-define 78.3%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

    3. +-commutative 78.3%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

    4. fma-undefine 78.3%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

    5. expm1-undefine 78.3%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

    6. add-exp-log 78.3%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

    7. add-sqr-sqrt 36.8%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

    8. fabs-sqr 36.8%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

    9. add-sqr-sqrt 78.2%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

11. Applied egg-rr 76.6%

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

    1. fma-undefine 78.2%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

    2. associate--l+ 78.2%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

    3. metadata-eval 78.2%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

    4. +-rgt-identity 78.2%

       \[\leadsto 1 - \left(\frac{1}{1 + {\left(\sqrt[3]{0.3275911 \cdot x}\right)}^{3}} \cdot \left(0.254829592 + e^{-\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot x}}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

13. Simplified 76.6%

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

14. Final simplification 76.6%

    \[\leadsto 1 + 0.999999999 \cdot \frac{1}{-1 - x \cdot 0.3275911} \]
15. Add Preprocessing

Alternative 10: 56.4% accurate, 856.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0

Julia

x_m = abs(x)
function code(x_m)
	return 1.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1.0

Derivation

1. Initial program 78.9%

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

3. Simplified 78.9%

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

   1. add-cube-cbrt 78.9%

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

   2. pow3 78.9%

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

   3. add-sqr-sqrt 36.8%

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

   4. fabs-sqr 36.8%

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

   5. add-sqr-sqrt 78.3%

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

6. Applied egg-rr 78.3%

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

7. Taylor expanded in x around inf 55.0%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024106 
(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)))))))