Jmat.Real.erf

Percentage Accurate: 79.2% → 99.7%

Time: 17.9s

Alternatives: 8

Speedup: 142.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ t_1 := -1 - t\_0\\ t_2 := 1 + t\_0\\ \mathbf{if}\;\left|x\_m\right| \leq 5 \cdot 10^{-9}:\\ \;\;\;\;10^{-9} - x\_m \cdot \mathsf{fma}\left(x\_m, 0.00011824294398844343, -1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{e^{-{x\_m}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{t\_2}^{4}} + 1.421413741 \cdot \frac{1}{{t\_2}^{2}}\right)\right) + \left(0.284496736 \cdot \frac{1}{t\_1} + 1.453152027 \cdot \frac{-1}{{t\_2}^{3}}\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)) (t_1 (- -1.0 t_0)) (t_2 (+ 1.0 t_0)))
   (if (<= (fabs x_m) 5e-9)
     (- 1e-9 (* x_m (fma x_m 0.00011824294398844343 -1.128386358070218)))
     (+
      1.0
      (/
       (*
        (exp (- (pow x_m 2.0)))
        (+
         (+
          0.254829592
          (+
           (* 1.061405429 (/ 1.0 (pow t_2 4.0)))
           (* 1.421413741 (/ 1.0 (pow t_2 2.0)))))
         (+
          (* 0.284496736 (/ 1.0 t_1))
          (* 1.453152027 (/ -1.0 (pow t_2 3.0))))))
       t_1)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double t_1 = -1.0 - t_0;
	double t_2 = 1.0 + t_0;
	double tmp;
	if (fabs(x_m) <= 5e-9) {
		tmp = 1e-9 - (x_m * fma(x_m, 0.00011824294398844343, -1.128386358070218));
	} else {
		tmp = 1.0 + ((exp(-pow(x_m, 2.0)) * ((0.254829592 + ((1.061405429 * (1.0 / pow(t_2, 4.0))) + (1.421413741 * (1.0 / pow(t_2, 2.0))))) + ((0.284496736 * (1.0 / t_1)) + (1.453152027 * (-1.0 / pow(t_2, 3.0)))))) / t_1);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	t_1 = Float64(-1.0 - t_0)
	t_2 = Float64(1.0 + t_0)
	tmp = 0.0
	if (abs(x_m) <= 5e-9)
		tmp = Float64(1e-9 - Float64(x_m * fma(x_m, 0.00011824294398844343, -1.128386358070218)));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(-(x_m ^ 2.0))) * Float64(Float64(0.254829592 + Float64(Float64(1.061405429 * Float64(1.0 / (t_2 ^ 4.0))) + Float64(1.421413741 * Float64(1.0 / (t_2 ^ 2.0))))) + Float64(Float64(0.284496736 * Float64(1.0 / t_1)) + Float64(1.453152027 * Float64(-1.0 / (t_2 ^ 3.0)))))) / t_1));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 5e-9], N[(1e-9 - N[(x$95$m * N[(x$95$m * 0.00011824294398844343 + -1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision] * N[(N[(0.254829592 + N[(N[(1.061405429 * N[(1.0 / N[Power[t$95$2, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.421413741 * N[(1.0 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.284496736 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.453152027 * N[(-1.0 / N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.0000000000000001e-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. Simplified 57.6%

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

   6. Applied egg-rr 56.7%

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

   7. Taylor expanded in x around 0 56.8%

      \[\leadsto {\left({\left(1 - \color{blue}{\left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)}\right)}^{3}\right)}^{0.3333333333333333} \]
   8. Step-by-step derivation

      1. pow1 56.8%

         \[\leadsto {\left({\color{blue}{\left({\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{1}\right)}}^{3}\right)}^{0.3333333333333333} \]

      2. metadata-eval 56.8%

         \[\leadsto {\left({\left({\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{\color{blue}{\left(3 \cdot 0.3333333333333333\right)}}\right)}^{3}\right)}^{0.3333333333333333} \]

      3. pow-pow 56.8%

         \[\leadsto {\left({\color{blue}{\left({\left({\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \]

      4. add-sqr-sqrt 56.8%

         \[\leadsto {\left({\left({\color{blue}{\left(\sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}} \cdot \sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}}\right)}}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \]

      5. unpow-prod-down 56.8%

         \[\leadsto {\left({\color{blue}{\left({\left(\sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 91.9%

      \[\leadsto {\left({\color{blue}{\left({\left({\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \]
   10. Step-by-step derivation

       1. unpow1/3 92.8%

          \[\leadsto {\left({\left(\color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}} \cdot {\left({\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \]

       2. unpow1/3 94.1%

          \[\leadsto {\left({\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}}\right)}^{3}\right)}^{0.3333333333333333} \]

   11. Simplified 94.1%

       \[\leadsto {\left({\color{blue}{\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)}}^{3}\right)}^{0.3333333333333333} \]
   12. Step-by-step derivation

       1. unpow1/3 94.7%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)}^{3}}} \]

       2. unpow3 94.6%

          \[\leadsto \sqrt[3]{\color{blue}{\left(\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right) \cdot \left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)\right) \cdot \left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)}} \]

       3. add-cbrt-cube 95.2%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}} \]

       4. cbrt-unprod 96.1%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5} \cdot {\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}} \]

       5. pow-prod-up 96.1%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{\left(1.5 + 1.5\right)}}} \]

       6. metadata-eval 96.1%

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

       7. pow3 96.1%

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

       8. add-cbrt-cube 97.6%

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

       9. sub-neg 97.6%

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

   13. Applied egg-rr 97.6%

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

       1. sub-neg 97.6%

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

   15. Simplified 97.6%

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

   if 5.0000000000000001e-9 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.9%

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

4. Final simplification 98.7%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x\_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;\left|x\_m\right| \leq 5 \cdot 10^{-9}:\\ \;\;\;\;10^{-9} - x\_m \cdot \mathsf{fma}\left(x\_m, 0.00011824294398844343, -1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{1}{1 + 0.3275911 \cdot \log \left(e^{x\_m}\right)} \cdot \left(0.254829592 + t\_1 \cdot \left(-0.284496736 + t\_1 \cdot \left(1.421413741 + t\_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t\_0}\right)\right)\right)\right)\right) \cdot e^{-x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x_m) 5e-9)
     (- 1e-9 (* x_m (fma x_m 0.00011824294398844343 -1.128386358070218)))
     (-
      1.0
      (*
       (*
        (/ 1.0 (+ 1.0 (* 0.3275911 (log (exp x_m)))))
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+ 1.421413741 (* t_1 (+ -1.453152027 (/ 1.061405429 t_0)))))))))
       (exp (- (* x_m x_m))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x_m) <= 5e-9) {
		tmp = 1e-9 - (x_m * fma(x_m, 0.00011824294398844343, -1.128386358070218));
	} else {
		tmp = 1.0 - (((1.0 / (1.0 + (0.3275911 * log(exp(x_m))))) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))) * exp(-(x_m * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x_m) <= 5e-9)
		tmp = Float64(1e-9 - Float64(x_m * fma(x_m, 0.00011824294398844343, -1.128386358070218)));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * log(exp(x_m))))) * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(t_1 * Float64(-1.453152027 + Float64(1.061405429 / t_0))))))))) * exp(Float64(-Float64(x_m * x_m)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 5e-9], N[(1e-9 - N[(x$95$m * N[(x$95$m * 0.00011824294398844343 + -1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Log[N[Exp[x$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(x$95$m * x$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.0000000000000001e-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. Simplified 57.6%

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

   6. Applied egg-rr 56.7%

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

   7. Taylor expanded in x around 0 56.8%

      \[\leadsto {\left({\left(1 - \color{blue}{\left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)}\right)}^{3}\right)}^{0.3333333333333333} \]
   8. Step-by-step derivation

      1. pow1 56.8%

         \[\leadsto {\left({\color{blue}{\left({\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{1}\right)}}^{3}\right)}^{0.3333333333333333} \]

      2. metadata-eval 56.8%

         \[\leadsto {\left({\left({\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{\color{blue}{\left(3 \cdot 0.3333333333333333\right)}}\right)}^{3}\right)}^{0.3333333333333333} \]

      3. pow-pow 56.8%

         \[\leadsto {\left({\color{blue}{\left({\left({\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \]

      4. add-sqr-sqrt 56.8%

         \[\leadsto {\left({\left({\color{blue}{\left(\sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}} \cdot \sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}}\right)}}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \]

      5. unpow-prod-down 56.8%

         \[\leadsto {\left({\color{blue}{\left({\left(\sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 91.9%

      \[\leadsto {\left({\color{blue}{\left({\left({\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \]
   10. Step-by-step derivation

       1. unpow1/3 92.8%

          \[\leadsto {\left({\left(\color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}} \cdot {\left({\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \]

       2. unpow1/3 94.1%

          \[\leadsto {\left({\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}}\right)}^{3}\right)}^{0.3333333333333333} \]

   11. Simplified 94.1%

       \[\leadsto {\left({\color{blue}{\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)}}^{3}\right)}^{0.3333333333333333} \]
   12. Step-by-step derivation

       1. unpow1/3 94.7%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)}^{3}}} \]

       2. unpow3 94.6%

          \[\leadsto \sqrt[3]{\color{blue}{\left(\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right) \cdot \left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)\right) \cdot \left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)}} \]

       3. add-cbrt-cube 95.2%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}} \]

       4. cbrt-unprod 96.1%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5} \cdot {\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}} \]

       5. pow-prod-up 96.1%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{\left(1.5 + 1.5\right)}}} \]

       6. metadata-eval 96.1%

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

       7. pow3 96.1%

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

       8. add-cbrt-cube 97.6%

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

       9. sub-neg 97.6%

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

   13. Applied egg-rr 97.6%

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

       1. sub-neg 97.6%

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

   15. Simplified 97.6%

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

   if 5.0000000000000001e-9 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 47.5%

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

      2. fabs-sqr 47.5%

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

      3. add-sqr-sqrt 99.2%

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

      4. add-log-exp 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 98.4%

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

Alternative 3: 99.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ t_1 := 1 + x\_m \cdot 0.3275911\\ t_2 := \frac{1}{t\_1}\\ \mathbf{if}\;x\_m \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} - x\_m \cdot \mathsf{fma}\left(x\_m, 0.00011824294398844343, -1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 + e^{-x\_m \cdot x\_m} \cdot \left(\frac{1}{1 + t\_0} \cdot \left(\left(-0.284496736 + t\_2 \cdot \left(1.421413741 + t\_2 \cdot \left(-1.453152027 + \frac{1.061405429}{t\_1}\right)\right)\right) \cdot \frac{1}{-1 - t\_0} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911))
        (t_1 (+ 1.0 (* x_m 0.3275911)))
        (t_2 (/ 1.0 t_1)))
   (if (<= x_m 4.5e-6)
     (- 1e-9 (* x_m (fma x_m 0.00011824294398844343 -1.128386358070218)))
     (+
      1.0
      (*
       (exp (- (* x_m x_m)))
       (*
        (/ 1.0 (+ 1.0 t_0))
        (-
         (*
          (+
           -0.284496736
           (*
            t_2
            (+ 1.421413741 (* t_2 (+ -1.453152027 (/ 1.061405429 t_1))))))
          (/ 1.0 (- -1.0 t_0)))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double t_1 = 1.0 + (x_m * 0.3275911);
	double t_2 = 1.0 / t_1;
	double tmp;
	if (x_m <= 4.5e-6) {
		tmp = 1e-9 - (x_m * fma(x_m, 0.00011824294398844343, -1.128386358070218));
	} else {
		tmp = 1.0 + (exp(-(x_m * x_m)) * ((1.0 / (1.0 + t_0)) * (((-0.284496736 + (t_2 * (1.421413741 + (t_2 * (-1.453152027 + (1.061405429 / t_1)))))) * (1.0 / (-1.0 - t_0))) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	t_1 = Float64(1.0 + Float64(x_m * 0.3275911))
	t_2 = Float64(1.0 / t_1)
	tmp = 0.0
	if (x_m <= 4.5e-6)
		tmp = Float64(1e-9 - Float64(x_m * fma(x_m, 0.00011824294398844343, -1.128386358070218)));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(-Float64(x_m * x_m))) * Float64(Float64(1.0 / Float64(1.0 + t_0)) * Float64(Float64(Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(t_2 * Float64(-1.453152027 + Float64(1.061405429 / t_1)))))) * Float64(1.0 / Float64(-1.0 - t_0))) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, If[LessEqual[x$95$m, 4.5e-6], N[(1e-9 - N[(x$95$m * N[(x$95$m * 0.00011824294398844343 + -1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[(-N[(x$95$m * x$95$m), $MachinePrecision])], $MachinePrecision] * N[(N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(t$95$2 * N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.50000000000000011e-6

   1. Initial program 72.5%

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

   3. Simplified 72.5%

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

   6. Applied egg-rr 71.4%

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

   7. Taylor expanded in x around 0 37.8%

      \[\leadsto {\left({\left(1 - \color{blue}{\left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)}\right)}^{3}\right)}^{0.3333333333333333} \]
   8. Step-by-step derivation

      1. pow1 37.8%

         \[\leadsto {\left({\color{blue}{\left({\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{1}\right)}}^{3}\right)}^{0.3333333333333333} \]

      2. metadata-eval 37.8%

         \[\leadsto {\left({\left({\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{\color{blue}{\left(3 \cdot 0.3333333333333333\right)}}\right)}^{3}\right)}^{0.3333333333333333} \]

      3. pow-pow 37.8%

         \[\leadsto {\left({\color{blue}{\left({\left({\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \]

      4. add-sqr-sqrt 36.9%

         \[\leadsto {\left({\left({\color{blue}{\left(\sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}} \cdot \sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}}\right)}}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \]

      5. unpow-prod-down 36.9%

         \[\leadsto {\left({\color{blue}{\left({\left(\sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 60.2%

      \[\leadsto {\left({\color{blue}{\left({\left({\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \]
   10. Step-by-step derivation

       1. unpow1/3 60.8%

          \[\leadsto {\left({\left(\color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}} \cdot {\left({\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \]

       2. unpow1/3 61.6%

          \[\leadsto {\left({\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}}\right)}^{3}\right)}^{0.3333333333333333} \]

   11. Simplified 61.6%

       \[\leadsto {\left({\color{blue}{\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)}}^{3}\right)}^{0.3333333333333333} \]
   12. Step-by-step derivation

       1. unpow1/3 62.0%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)}^{3}}} \]

       2. unpow3 62.0%

          \[\leadsto \sqrt[3]{\color{blue}{\left(\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right) \cdot \left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)\right) \cdot \left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)}} \]

       3. add-cbrt-cube 62.4%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}} \]

       4. cbrt-unprod 62.9%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5} \cdot {\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}} \]

       5. pow-prod-up 62.7%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{\left(1.5 + 1.5\right)}}} \]

       6. metadata-eval 62.7%

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

       7. pow3 62.7%

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

       8. add-cbrt-cube 63.7%

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

       9. sub-neg 63.7%

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

   13. Applied egg-rr 63.7%

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

       1. sub-neg 63.7%

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

   15. Simplified 63.7%

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

   if 4.50000000000000011e-6 < x

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 99.7%

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

      2. log1p-define 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-undefine 99.7%

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

      5. expm1-undefine 99.7%

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

      6. add-exp-log 99.7%

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

      7. add-sqr-sqrt 99.7%

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

      8. fabs-sqr 99.7%

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

      9. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. fma-undefine 99.7%

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

      2. associate--l+ 99.7%

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

      3. metadata-eval 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. +-commutative 99.7%

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

      7. +-lft-identity 99.7%

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

   8. Simplified 99.7%

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

      1. expm1-log1p-u 99.7%

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

      2. log1p-define 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-undefine 99.7%

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

      5. expm1-undefine 99.7%

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

      6. add-exp-log 99.7%

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

      7. add-sqr-sqrt 99.7%

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

      8. fabs-sqr 99.7%

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

      9. add-sqr-sqrt 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. fma-undefine 99.7%

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

       2. associate--l+ 99.7%

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

       3. metadata-eval 99.7%

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

       4. metadata-eval 99.7%

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

       5. distribute-lft-in 99.7%

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

       6. +-commutative 99.7%

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

       7. +-lft-identity 99.7%

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

   12. Simplified 99.7%

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

       1. expm1-log1p-u 99.7%

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

       2. log1p-define 99.7%

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

       3. +-commutative 99.7%

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

       4. fma-undefine 99.7%

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

       5. expm1-undefine 99.7%

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

       6. add-exp-log 99.7%

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

       7. add-sqr-sqrt 99.7%

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

       8. fabs-sqr 99.7%

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

       9. add-sqr-sqrt 99.7%

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

   14. Applied egg-rr 99.7%

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

       1. fma-undefine 99.7%

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

       2. associate--l+ 99.7%

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

       3. metadata-eval 99.7%

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

       4. metadata-eval 99.7%

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

       5. distribute-lft-in 99.7%

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

       6. +-commutative 99.7%

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

       7. +-lft-identity 99.7%

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

   16. Simplified 99.7%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 + e^{-x \cdot x} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right) \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 7.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88)
   (- 1e-9 (* x_m (fma x_m 0.00011824294398844343 -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 * fma(x_m, 0.00011824294398844343, -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 * fma(x_m, 0.00011824294398844343, -1.128386358070218)));
	else
		tmp = 1.0;
	end
	return 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 * N[(x$95$m * 0.00011824294398844343 + -1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 72.5%

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

   3. Simplified 72.5%

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

   6. Applied egg-rr 71.4%

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

   7. Taylor expanded in x around 0 37.9%

      \[\leadsto {\left({\left(1 - \color{blue}{\left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)}\right)}^{3}\right)}^{0.3333333333333333} \]
   8. Step-by-step derivation

      1. pow1 37.9%

         \[\leadsto {\left({\color{blue}{\left({\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{1}\right)}}^{3}\right)}^{0.3333333333333333} \]

      2. metadata-eval 37.9%

         \[\leadsto {\left({\left({\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{\color{blue}{\left(3 \cdot 0.3333333333333333\right)}}\right)}^{3}\right)}^{0.3333333333333333} \]

      3. pow-pow 37.9%

         \[\leadsto {\left({\color{blue}{\left({\left({\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \]

      4. add-sqr-sqrt 37.0%

         \[\leadsto {\left({\left({\color{blue}{\left(\sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}} \cdot \sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}}\right)}}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \]

      5. unpow-prod-down 37.0%

         \[\leadsto {\left({\color{blue}{\left({\left(\sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\left(1 - \left(0.999999999 + x \cdot \left(0.00011824294398844343 \cdot x - 1.128386358070218\right)\right)\right)}^{3}}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 60.2%

      \[\leadsto {\left({\color{blue}{\left({\left({\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \]
   10. Step-by-step derivation

       1. unpow1/3 60.8%

          \[\leadsto {\left({\left(\color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}} \cdot {\left({\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \]

       2. unpow1/3 61.6%

          \[\leadsto {\left({\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}}\right)}^{3}\right)}^{0.3333333333333333} \]

   11. Simplified 61.6%

       \[\leadsto {\left({\color{blue}{\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)}}^{3}\right)}^{0.3333333333333333} \]
   12. Step-by-step derivation

       1. unpow1/3 62.0%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)}^{3}}} \]

       2. unpow3 62.0%

          \[\leadsto \sqrt[3]{\color{blue}{\left(\left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right) \cdot \left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)\right) \cdot \left(\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}\right)}} \]

       3. add-cbrt-cube 62.3%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}} \]

       4. cbrt-unprod 62.9%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5} \cdot {\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{1.5}}} \]

       5. pow-prod-up 62.7%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(10^{-9} - x \cdot \mathsf{fma}\left(x, 0.00011824294398844343, -1.128386358070218\right)\right)}^{\left(1.5 + 1.5\right)}}} \]

       6. metadata-eval 62.7%

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

       7. pow3 62.7%

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

       8. add-cbrt-cube 63.6%

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

       9. sub-neg 63.6%

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

   13. Applied egg-rr 63.6%

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

       1. sub-neg 63.6%

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

   15. Simplified 63.6%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. add-log-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

Alternative 5: 99.3% accurate, 71.2× speedup

Math

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

FPCore

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

C

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

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.88:
		tmp = x_m * (1.128386358070218 + (1e-9 / x_m))
	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(x_m * Float64(1.128386358070218 + Float64(1e-9 / x_m)));
	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 = x_m * (1.128386358070218 + (1e-9 / x_m));
	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[(x$95$m * N[(1.128386358070218 + N[(1e-9 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in x around 0 68.7%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in x around 0 37.4%

      \[\leadsto 1 - \color{blue}{\left(0.999999999 + -1.128386358070218 \cdot x\right)} \]
   9. Step-by-step derivation

      1. *-commutative 37.4%

         \[\leadsto 1 - \left(0.999999999 + \color{blue}{x \cdot -1.128386358070218}\right) \]

   10. Simplified 37.4%

       \[\leadsto 1 - \color{blue}{\left(0.999999999 + x \cdot -1.128386358070218\right)} \]

   11. Taylor expanded in x around inf 63.5%

       \[\leadsto \color{blue}{x \cdot \left(1.128386358070218 + 10^{-9} \cdot \frac{1}{x}\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ 63.7%

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

       2. metadata-eval 63.7%

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

   13. Simplified 63.7%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. add-log-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

Alternative 6: 99.3% accurate, 85.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in x around 0 68.7%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in x around 0 63.7%

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

      1. *-commutative 63.7%

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

   10. Simplified 63.7%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. add-log-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

Alternative 7: 97.7% accurate, 142.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in x around 0 68.9%

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

   7. Simplified 70.5%

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

   8. Taylor expanded in x around 0 66.4%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 99.7%

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

      2. fabs-sqr 99.7%

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

      3. add-sqr-sqrt 99.7%

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

      4. add-log-exp 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 98.6%

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

Alternative 8: 53.0% 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 79.1%

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

3. Simplified 79.1%

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

5. Taylor expanded in x around 0 76.2%

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

7. Simplified 77.4%

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

8. Taylor expanded in x around 0 53.0%

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

Reproduce

herbie shell --seed 2024135 
(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)))))))