Jmat.Real.erf

Percentage Accurate: 79.3% → 99.6%

Time: 19.0s

Alternatives: 11

Speedup: 279.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}\\ t_1 := \mathsf{fma}\left(0.3275911, x_m, 1\right) \cdot e^{{x_m}^{2}}\\ \mathbf{if}\;\left|x_m\right| \leq 10^{-9}:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\frac{0.254829592 + \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}, t_0, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{t_1}\right)}^{2}}{1 + \frac{0.254829592 + \frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}\right)} + -1, t_0, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{t_1}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fma 0.3275911 x_m 1.0)))
        (t_1 (* (fma 0.3275911 x_m 1.0) (exp (pow x_m 2.0)))))
   (if (<= (fabs x_m) 1e-9)
     (+
      1e-9
      (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* x_m 1.128386358070218)))
     (/
      (-
       1.0
       (pow
        (/
         (+
          0.254829592
          (/
           (fma
            (+
             1.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
              (fma 0.3275911 x_m 1.0)))
            t_0
            -0.284496736)
           (fma 0.3275911 x_m 1.0)))
         t_1)
        2.0))
      (+
       1.0
       (/
        (+
         0.254829592
         (/
          (fma
           (+
            (exp
             (log1p
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
                (fma x_m 0.3275911 1.0)))))
            -1.0)
           t_0
           -0.284496736)
          (fma 0.3275911 x_m 1.0)))
        t_1))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / fma(0.3275911, x_m, 1.0);
	double t_1 = fma(0.3275911, x_m, 1.0) * exp(pow(x_m, 2.0));
	double tmp;
	if (fabs(x_m) <= 1e-9) {
		tmp = 1e-9 + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (x_m * 1.128386358070218));
	} else {
		tmp = (1.0 - pow(((0.254829592 + (fma((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))), t_0, -0.284496736) / fma(0.3275911, x_m, 1.0))) / t_1), 2.0)) / (1.0 + ((0.254829592 + (fma((exp(log1p((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))))) + -1.0), t_0, -0.284496736) / fma(0.3275911, x_m, 1.0))) / t_1));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / fma(0.3275911, x_m, 1.0))
	t_1 = Float64(fma(0.3275911, x_m, 1.0) * exp((x_m ^ 2.0)))
	tmp = 0.0
	if (abs(x_m) <= 1e-9)
		tmp = Float64(1e-9 + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + Float64(x_m * 1.128386358070218)));
	else
		tmp = Float64(Float64(1.0 - (Float64(Float64(0.254829592 + Float64(fma(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))), t_0, -0.284496736) / fma(0.3275911, x_m, 1.0))) / t_1) ^ 2.0)) / Float64(1.0 + Float64(Float64(0.254829592 + Float64(fma(Float64(exp(log1p(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))))) + -1.0), t_0, -0.284496736) / fma(0.3275911, x_m, 1.0))) / t_1)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-9], N[(1e-9 + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[N[(N[(0.254829592 + N[(N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + -0.284496736), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.254829592 + N[(N[(N[(N[Exp[N[Log[1 + N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] * t$95$0 + -0.284496736), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000006e-9

   1. Initial program 57.6%

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

   3. Applied egg-rr 54.6%

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

   4. Taylor expanded in x around 0 97.8%

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

   if 1.00000000000000006e-9 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. div-inv 99.8%

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

      3. fma-udef 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.9%

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

      7. fma-udef 99.9%

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

      8. *-commutative 99.9%

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

      9. fma-def 99.9%

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

      10. fma-udef 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. rem-log-exp 99.9%

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

      2. flip-- 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \frac{1 - {\left(\frac{0.254829592 + \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}}{1 + \frac{0.254829592 + \frac{\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}} \]

      2. expm1-udef 99.9%

         \[\leadsto \frac{1 - {\left(\frac{0.254829592 + \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}}{1 + \frac{0.254829592 + \frac{\mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)} - 1}, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}} \]

      3. fma-udef 99.9%

         \[\leadsto \frac{1 - {\left(\frac{0.254829592 + \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}}{1 + \frac{0.254829592 + \frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{0.3275911 \cdot x + 1}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)} - 1, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}} \]

      4. *-commutative 99.9%

         \[\leadsto \frac{1 - {\left(\frac{0.254829592 + \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}}{1 + \frac{0.254829592 + \frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{x \cdot 0.3275911} + 1}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)} - 1, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}} \]

      5. fma-def 99.9%

         \[\leadsto \frac{1 - {\left(\frac{0.254829592 + \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}}{1 + \frac{0.254829592 + \frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)} - 1, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}} \]

      6. fma-udef 99.9%

         \[\leadsto \frac{1 - {\left(\frac{0.254829592 + \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}}{1 + \frac{0.254829592 + \frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{0.3275911 \cdot x + 1}}\right)} - 1, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}} \]

      7. *-commutative 99.9%

         \[\leadsto \frac{1 - {\left(\frac{0.254829592 + \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}}{1 + \frac{0.254829592 + \frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{x \cdot 0.3275911} + 1}\right)} - 1, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}} \]

      8. fma-def 99.9%

         \[\leadsto \frac{1 - {\left(\frac{0.254829592 + \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}}{1 + \frac{0.254829592 + \frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\color{blue}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}\right)} - 1, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}} \]

   9. Applied egg-rr 99.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-9}:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\frac{0.254829592 + \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}}{1 + \frac{0.254829592 + \frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\right)} + -1, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (/
          (+
           0.254829592
           (/
            (fma
             (+
              1.421413741
              (/
               (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
               (fma 0.3275911 x_m 1.0)))
             (/ 1.0 (fma 0.3275911 x_m 1.0))
             -0.284496736)
            (fma 0.3275911 x_m 1.0)))
          (* (fma 0.3275911 x_m 1.0) (exp (pow x_m 2.0))))))
   (if (<= (fabs x_m) 1e-9)
     (+
      1e-9
      (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* x_m 1.128386358070218)))
     (/ (- 1.0 (pow t_0 2.0)) (+ 1.0 t_0)))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.254829592 + N[(N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-9], N[(1e-9 + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000006e-9

   1. Initial program 57.6%

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

   3. Applied egg-rr 54.6%

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

   4. Taylor expanded in x around 0 97.8%

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

   if 1.00000000000000006e-9 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. div-inv 99.8%

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

      3. fma-udef 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.9%

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

      7. fma-udef 99.9%

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

      8. *-commutative 99.9%

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

      9. fma-def 99.9%

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

      10. fma-udef 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. rem-log-exp 99.9%

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

      2. flip-- 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 98.9%

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

Alternative 3: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 10^{-9}:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(1 - \frac{0.254829592 + \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}, \frac{1}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right) \cdot e^{{x_m}^{2}}}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 1e-9)
   (+
    1e-9
    (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* x_m 1.128386358070218)))
   (pow
    (pow
     (-
      1.0
      (/
       (+
        0.254829592
        (/
         (fma
          (+
           1.421413741
           (/
            (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
            (fma x_m 0.3275911 1.0)))
          (/ 1.0 (fma x_m 0.3275911 1.0))
          -0.284496736)
         (fma 0.3275911 x_m 1.0)))
       (* (fma 0.3275911 x_m 1.0) (exp (pow x_m 2.0)))))
     3.0)
    0.3333333333333333)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 1e-9) {
		tmp = 1e-9 + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (x_m * 1.128386358070218));
	} else {
		tmp = pow(pow((1.0 - ((0.254829592 + (fma((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))), (1.0 / fma(x_m, 0.3275911, 1.0)), -0.284496736) / fma(0.3275911, x_m, 1.0))) / (fma(0.3275911, x_m, 1.0) * exp(pow(x_m, 2.0))))), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 1e-9)
		tmp = Float64(1e-9 + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + Float64(x_m * 1.128386358070218)));
	else
		tmp = (Float64(1.0 - Float64(Float64(0.254829592 + Float64(fma(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))), Float64(1.0 / fma(x_m, 0.3275911, 1.0)), -0.284496736) / fma(0.3275911, x_m, 1.0))) / Float64(fma(0.3275911, x_m, 1.0) * exp((x_m ^ 2.0))))) ^ 3.0) ^ 0.3333333333333333;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-9], N[(1e-9 + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(1.0 - N[(N[(0.254829592 + N[(N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000006e-9

   1. Initial program 57.6%

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

   3. Applied egg-rr 54.6%

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

   4. Taylor expanded in x around 0 97.8%

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

   if 1.00000000000000006e-9 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{3}\right)}^{0.3333333333333333}} \]
   4. Step-by-step derivation

      1. +-commutative 99.8%

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

      2. div-inv 99.8%

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

      3. fma-udef 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.9%

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

      7. fma-udef 99.9%

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

      8. *-commutative 99.9%

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

      9. fma-def 99.9%

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

      10. fma-udef 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

      \[\leadsto {\left({\left(1 - \frac{0.254829592 + \frac{\color{blue}{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{1}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, -0.284496736\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{3}\right)}^{0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-9}:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(1 - \frac{0.254829592 + \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{1}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 10^{-9}:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left(-0.254829592\right) - \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right) \cdot e^{{x_m}^{2}}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 1e-9)
   (+
    1e-9
    (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* x_m 1.128386358070218)))
   (exp
    (log1p
     (/
      (-
       (- 0.254829592)
       (/
        (fma
         (+
          1.421413741
          (/
           (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
           (fma 0.3275911 x_m 1.0)))
         (/ 1.0 (fma 0.3275911 x_m 1.0))
         -0.284496736)
        (fma 0.3275911 x_m 1.0)))
      (* (fma 0.3275911 x_m 1.0) (exp (pow x_m 2.0))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 1e-9) {
		tmp = 1e-9 + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (x_m * 1.128386358070218));
	} else {
		tmp = exp(log1p(((-0.254829592 - (fma((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))), (1.0 / fma(0.3275911, x_m, 1.0)), -0.284496736) / fma(0.3275911, x_m, 1.0))) / (fma(0.3275911, x_m, 1.0) * exp(pow(x_m, 2.0))))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 1e-9)
		tmp = Float64(1e-9 + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + Float64(x_m * 1.128386358070218)));
	else
		tmp = exp(log1p(Float64(Float64(Float64(-0.254829592) - Float64(fma(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))), Float64(1.0 / fma(0.3275911, x_m, 1.0)), -0.284496736) / fma(0.3275911, x_m, 1.0))) / Float64(fma(0.3275911, x_m, 1.0) * exp((x_m ^ 2.0))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-9], N[(1e-9 + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[1 + N[(N[((-0.254829592) - N[(N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -0.284496736), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.3275911 * x$95$m + 1.0), $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) < 1.00000000000000006e-9

   1. Initial program 57.6%

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

   3. Applied egg-rr 54.6%

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

   4. Taylor expanded in x around 0 97.8%

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

   if 1.00000000000000006e-9 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. div-inv 99.8%

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

      3. fma-udef 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.9%

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

      7. fma-udef 99.9%

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

      8. *-commutative 99.9%

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

      9. fma-def 99.9%

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

      10. fma-udef 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-def 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. rem-log-exp 99.9%

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

      2. add-exp-log 99.9%

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

      3. sub-neg 99.9%

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

      4. log1p-def 99.9%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(-\frac{0.254829592 + \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, \frac{1}{\mathsf{fma}\left(x, 0.3275911, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}} \]

   7. Applied egg-rr 99.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-9}:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left(-0.254829592\right) - \frac{\mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* x_m 0.3275911))))
        (t_1 (+ 1.0 (* (fabs x_m) 0.3275911)))
        (t_2 (/ 1.0 t_1)))
   (if (<= (fabs x_m) 1e-9)
     (+
      1e-9
      (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* x_m 1.128386358070218)))
     (+
      1.0
      (*
       t_0
       (*
        (exp (* x_m (- x_m)))
        (-
         (*
          t_0
          (-
           (*
            t_2
            (-
             (* t_2 (+ 1.453152027 (* 1.061405429 (/ -1.0 t_1))))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (x_m * 0.3275911));
	double t_1 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_2 = 1.0 / t_1;
	double tmp;
	if (fabs(x_m) <= 1e-9) {
		tmp = 1e-9 + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0 + (t_0 * (exp((x_m * -x_m)) * ((t_0 * ((t_2 * ((t_2 * (1.453152027 + (1.061405429 * (-1.0 / t_1)))) - 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 = 1.0d0 / (1.0d0 + (x_m * 0.3275911d0))
    t_1 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    t_2 = 1.0d0 / t_1
    if (abs(x_m) <= 1d-9) then
        tmp = 1d-9 + (((-0.00011824294398844343d0) * (x_m ** 2.0d0)) + (x_m * 1.128386358070218d0))
    else
        tmp = 1.0d0 + (t_0 * (exp((x_m * -x_m)) * ((t_0 * ((t_2 * ((t_2 * (1.453152027d0 + (1.061405429d0 * ((-1.0d0) / t_1)))) - 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 = 1.0 / (1.0 + (x_m * 0.3275911));
	double t_1 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double t_2 = 1.0 / t_1;
	double tmp;
	if (Math.abs(x_m) <= 1e-9) {
		tmp = 1e-9 + ((-0.00011824294398844343 * Math.pow(x_m, 2.0)) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0 + (t_0 * (Math.exp((x_m * -x_m)) * ((t_0 * ((t_2 * ((t_2 * (1.453152027 + (1.061405429 * (-1.0 / t_1)))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 / (1.0 + (x_m * 0.3275911))
	t_1 = 1.0 + (math.fabs(x_m) * 0.3275911)
	t_2 = 1.0 / t_1
	tmp = 0
	if math.fabs(x_m) <= 1e-9:
		tmp = 1e-9 + ((-0.00011824294398844343 * math.pow(x_m, 2.0)) + (x_m * 1.128386358070218))
	else:
		tmp = 1.0 + (t_0 * (math.exp((x_m * -x_m)) * ((t_0 * ((t_2 * ((t_2 * (1.453152027 + (1.061405429 * (-1.0 / t_1)))) - 1.421413741)) - -0.284496736)) - 0.254829592)))
	return tmp

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 / (1.0 + (x_m * 0.3275911));
	t_1 = 1.0 + (abs(x_m) * 0.3275911);
	t_2 = 1.0 / t_1;
	tmp = 0.0;
	if (abs(x_m) <= 1e-9)
		tmp = 1e-9 + ((-0.00011824294398844343 * (x_m ^ 2.0)) + (x_m * 1.128386358070218));
	else
		tmp = 1.0 + (t_0 * (exp((x_m * -x_m)) * ((t_0 * ((t_2 * ((t_2 * (1.453152027 + (1.061405429 * (-1.0 / t_1)))) - 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[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-9], N[(1e-9 + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$0 * N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 * N[(N[(t$95$2 * N[(N[(t$95$2 * N[(1.453152027 + N[(1.061405429 * N[(-1.0 / t$95$1), $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) < 1.00000000000000006e-9

   1. Initial program 57.6%

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

   3. Applied egg-rr 54.6%

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

   4. Taylor expanded in x around 0 97.8%

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

   if 1.00000000000000006e-9 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. pow1 99.9%

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

      2. add-sqr-sqrt 48.0%

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

      3. fabs-sqr 48.0%

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

      4. add-sqr-sqrt 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. unpow1 99.9%

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

      2. *-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 99.9%

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

      1. pow1 99.9%

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

      2. add-sqr-sqrt 48.0%

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

      3. fabs-sqr 48.0%

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

      4. add-sqr-sqrt 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. unpow1 99.9%

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

       2. *-commutative 99.9%

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

   12. Simplified 99.9%

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

4. Final simplification 98.9%

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

Alternative 6: 99.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + x_m \cdot 0.3275911}\\ t_1 := 1 + \left|x_m\right| \cdot 0.3275911\\ \mathbf{if}\;x_m \leq 8.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 + t_0 \cdot \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(t_0 \cdot \left(t_0 \cdot \left(\left(-1.453152027 + \frac{1.061405429}{t_1}\right) \cdot \frac{-1}{t_1} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* x_m 0.3275911))))
        (t_1 (+ 1.0 (* (fabs x_m) 0.3275911))))
   (if (<= x_m 8.5e-6)
     (+
      1e-9
      (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* x_m 1.128386358070218)))
     (+
      1.0
      (*
       t_0
       (*
        (exp (* x_m (- x_m)))
        (-
         (*
          t_0
          (-
           (*
            t_0
            (-
             (* (+ -1.453152027 (/ 1.061405429 t_1)) (/ -1.0 t_1))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (x_m * 0.3275911));
	double t_1 = 1.0 + (fabs(x_m) * 0.3275911);
	double tmp;
	if (x_m <= 8.5e-6) {
		tmp = 1e-9 + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0 + (t_0 * (exp((x_m * -x_m)) * ((t_0 * ((t_0 * (((-1.453152027 + (1.061405429 / t_1)) * (-1.0 / t_1)) - 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) :: tmp
    t_0 = 1.0d0 / (1.0d0 + (x_m * 0.3275911d0))
    t_1 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    if (x_m <= 8.5d-6) then
        tmp = 1d-9 + (((-0.00011824294398844343d0) * (x_m ** 2.0d0)) + (x_m * 1.128386358070218d0))
    else
        tmp = 1.0d0 + (t_0 * (exp((x_m * -x_m)) * ((t_0 * ((t_0 * ((((-1.453152027d0) + (1.061405429d0 / t_1)) * ((-1.0d0) / t_1)) - 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 = 1.0 / (1.0 + (x_m * 0.3275911));
	double t_1 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double tmp;
	if (x_m <= 8.5e-6) {
		tmp = 1e-9 + ((-0.00011824294398844343 * Math.pow(x_m, 2.0)) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0 + (t_0 * (Math.exp((x_m * -x_m)) * ((t_0 * ((t_0 * (((-1.453152027 + (1.061405429 / t_1)) * (-1.0 / t_1)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 / (1.0 + (x_m * 0.3275911))
	t_1 = 1.0 + (math.fabs(x_m) * 0.3275911)
	tmp = 0
	if x_m <= 8.5e-6:
		tmp = 1e-9 + ((-0.00011824294398844343 * math.pow(x_m, 2.0)) + (x_m * 1.128386358070218))
	else:
		tmp = 1.0 + (t_0 * (math.exp((x_m * -x_m)) * ((t_0 * ((t_0 * (((-1.453152027 + (1.061405429 / t_1)) * (-1.0 / t_1)) - 1.421413741)) - -0.284496736)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911)))
	t_1 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	tmp = 0.0
	if (x_m <= 8.5e-6)
		tmp = Float64(1e-9 + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + Float64(x_m * 1.128386358070218)));
	else
		tmp = Float64(1.0 + Float64(t_0 * Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(t_0 * Float64(Float64(t_0 * Float64(Float64(Float64(-1.453152027 + Float64(1.061405429 / t_1)) * Float64(-1.0 / t_1)) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 / (1.0 + (x_m * 0.3275911));
	t_1 = 1.0 + (abs(x_m) * 0.3275911);
	tmp = 0.0;
	if (x_m <= 8.5e-6)
		tmp = 1e-9 + ((-0.00011824294398844343 * (x_m ^ 2.0)) + (x_m * 1.128386358070218));
	else
		tmp = 1.0 + (t_0 * (exp((x_m * -x_m)) * ((t_0 * ((t_0 * (((-1.453152027 + (1.061405429 / t_1)) * (-1.0 / t_1)) - 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[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 8.5e-6], N[(1e-9 + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$0 * N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 * N[(N[(t$95$0 * N[(N[(N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.4999999999999999e-6

   1. Initial program 73.2%

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

   3. Applied egg-rr 71.3%

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

   4. Taylor expanded in x around 0 62.3%

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

   if 8.4999999999999999e-6 < x

   1. Initial program 99.8%

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

   3. Simplified 99.7%

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

      1. pow1 99.8%

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

      2. add-sqr-sqrt 99.8%

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

      3. fabs-sqr 99.8%

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

      4. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.7%

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

      1. unpow1 99.8%

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

      2. *-commutative 99.8%

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

   7. Simplified 99.7%

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

      1. pow1 99.8%

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

      2. add-sqr-sqrt 99.8%

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

      3. fabs-sqr 99.8%

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

      4. add-sqr-sqrt 99.8%

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

   9. Applied egg-rr 99.7%

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

       1. unpow1 99.8%

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

       2. *-commutative 99.8%

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

   11. Simplified 99.7%

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

       1. pow1 99.8%

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

       2. add-sqr-sqrt 99.8%

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

       3. fabs-sqr 99.8%

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

       4. add-sqr-sqrt 99.8%

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

   13. Applied egg-rr 99.7%

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

       1. unpow1 99.8%

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

       2. *-commutative 99.8%

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

   15. Simplified 99.7%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(\left(-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]

Alternative 7: 99.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.1:\\ \;\;\;\;10^{-9} + \left(\left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right) + -0.37545125292247583 \cdot {x_m}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (+
    1e-9
    (+
     (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* x_m 1.128386358070218))
     (* -0.37545125292247583 (pow x_m 3.0))))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = 1e-9 + (((-0.00011824294398844343 * pow(x_m, 2.0)) + (x_m * 1.128386358070218)) + (-0.37545125292247583 * pow(x_m, 3.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.1d0) then
        tmp = 1d-9 + ((((-0.00011824294398844343d0) * (x_m ** 2.0d0)) + (x_m * 1.128386358070218d0)) + ((-0.37545125292247583d0) * (x_m ** 3.0d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = 1e-9 + (((-0.00011824294398844343 * Math.pow(x_m, 2.0)) + (x_m * 1.128386358070218)) + (-0.37545125292247583 * Math.pow(x_m, 3.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.1:
		tmp = 1e-9 + (((-0.00011824294398844343 * math.pow(x_m, 2.0)) + (x_m * 1.128386358070218)) + (-0.37545125292247583 * math.pow(x_m, 3.0)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(1e-9 + Float64(Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + Float64(x_m * 1.128386358070218)) + Float64(-0.37545125292247583 * (x_m ^ 3.0))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.1)
		tmp = 1e-9 + (((-0.00011824294398844343 * (x_m ^ 2.0)) + (x_m * 1.128386358070218)) + (-0.37545125292247583 * (x_m ^ 3.0)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(1e-9 + N[(N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision] + N[(-0.37545125292247583 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 73.5%

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

   3. Applied egg-rr 71.6%

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

   4. Taylor expanded in x around 0 62.8%

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

   if 1.1000000000000001 < x

   1. Initial program 100.0%

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

      1. pow1 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. fabs-sqr 100.0%

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

      4. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;10^{-9} + \left(\left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right) + -0.37545125292247583 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 99.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.88:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x_m}^{2} + x_m \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88)
   (+
    1e-9
    (+ (* -0.00011824294398844343 (pow x_m 2.0)) (* x_m 1.128386358070218)))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = 1e-9 + ((-0.00011824294398844343 * pow(x_m, 2.0)) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.88d0) then
        tmp = 1d-9 + (((-0.00011824294398844343d0) * (x_m ** 2.0d0)) + (x_m * 1.128386358070218d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = 1e-9 + ((-0.00011824294398844343 * Math.pow(x_m, 2.0)) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.88:
		tmp = 1e-9 + ((-0.00011824294398844343 * math.pow(x_m, 2.0)) + (x_m * 1.128386358070218))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.88)
		tmp = Float64(1e-9 + Float64(Float64(-0.00011824294398844343 * (x_m ^ 2.0)) + Float64(x_m * 1.128386358070218)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.88)
		tmp = 1e-9 + ((-0.00011824294398844343 * (x_m ^ 2.0)) + (x_m * 1.128386358070218));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.88], N[(1e-9 + N[(N[(-0.00011824294398844343 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 73.5%

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

   3. Applied egg-rr 71.6%

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

   4. Taylor expanded in x around 0 61.9%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

      1. pow1 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. fabs-sqr 100.0%

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

      4. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 99.3% accurate, 121.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 73.5%

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

   3. Applied egg-rr 71.6%

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

   4. Taylor expanded in x around 0 62.0%

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

      1. *-commutative 62.0%

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

   6. Simplified 62.0%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

      1. pow1 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. fabs-sqr 100.0%

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

      4. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 71.2%

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

Alternative 10: 97.6% accurate, 279.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 73.2%

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

   3. Applied egg-rr 71.3%

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

   4. Taylor expanded in x around 0 65.4%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 99.8%

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

      1. pow1 99.8%

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

      2. add-sqr-sqrt 99.8%

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

      3. fabs-sqr 99.8%

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

      4. add-sqr-sqrt 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around inf 96.1%

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

4. Final simplification 73.2%

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

Alternative 11: 52.9% accurate, 856.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.9%

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

3. Applied egg-rr 78.4%

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

4. Taylor expanded in x around 0 51.6%

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

5. Final simplification 51.6%

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

Reproduce

herbie shell --seed 2023333 
(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)))))))