Jmat.Real.erf

Percentage Accurate: 79.4% → 99.7%

Time: 11.0s

Alternatives: 6

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: 79.4% 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, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x\_m\right| \cdot 0.3275911}\\ \mathbf{if}\;\left|x\_m\right| \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x\_m \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(t\_0 \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x\_m}}\right)}^{2}\right) + x\_m \cdot 0.3333333333333333\right)} \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + \frac{1}{1 + x\_m \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right)\right)\right)\right)\right) \cdot e^{x\_m \cdot \left(-x\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x_m) 0.3275911)))))
   (if (<= (fabs x_m) 4e-8)
     (+ (* x_m 1.128386358070218) 1e-9)
     (-
      1.0
      (*
       (*
        t_0
        (+
         0.254829592
         (*
          (/
           1.0
           (+
            1.0
            (*
             0.3275911
             (+ (log (pow (cbrt (exp x_m)) 2.0)) (* x_m 0.3333333333333333)))))
          (+
           -0.284496736
           (*
            t_0
            (+
             1.421413741
             (*
              (/ 1.0 (+ 1.0 (* x_m 0.3275911)))
              (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0))))))))))
       (exp (* x_m (- x_m))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (fabs(x_m) * 0.3275911));
	double tmp;
	if (fabs(x_m) <= 4e-8) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0 - ((t_0 * (0.254829592 + ((1.0 / (1.0 + (0.3275911 * (log(pow(cbrt(exp(x_m)), 2.0)) + (x_m * 0.3333333333333333))))) * (-0.284496736 + (t_0 * (1.421413741 + ((1.0 / (1.0 + (x_m * 0.3275911))) * (-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0)))))))))) * exp((x_m * -x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))
	tmp = 0.0
	if (abs(x_m) <= 4e-8)
		tmp = Float64(Float64(x_m * 1.128386358070218) + 1e-9);
	else
		tmp = Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * Float64(log((cbrt(exp(x_m)) ^ 2.0)) + Float64(x_m * 0.3333333333333333))))) * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911))) * Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0)))))))))) * exp(Float64(x_m * Float64(-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[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 4e-8], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[(N[Log[N[Power[N[Power[N[Exp[x$95$m], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[(x$95$m * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $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) < 4.0000000000000001e-8

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 56.9%

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

   7. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

   9. Simplified 98.3%

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

       1. unpow2 98.3%

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

       2. add-sqr-sqrt 98.4%

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

       3. metadata-eval 98.4%

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

       4. metadata-eval 98.4%

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

       5. sqrt-pow2 98.4%

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

       6. +-commutative 98.4%

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

       7. sqrt-pow2 98.4%

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

       8. metadata-eval 98.4%

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

       9. metadata-eval 98.4%

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

   11. Applied egg-rr 98.4%

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

   if 4.0000000000000001e-8 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 52.8%

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

      2. fabs-sqr 52.8%

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

      3. add-sqr-sqrt 99.2%

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

      4. add-log-exp 99.2%

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

      5. add-cube-cbrt 99.2%

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

      6. log-prod 99.2%

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

      7. pow2 99.2%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in x around 0 99.2%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. fma-neg 99.2%

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

      2. +-commutative 99.2%

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

      3. fma-define 99.2%

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

      4. rem-square-sqrt 52.8%

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

      5. fabs-sqr 52.8%

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

      6. rem-square-sqrt 99.1%

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

      7. metadata-eval 99.1%

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

      8. fma-undefine 99.2%

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

      9. associate-*r/ 99.2%

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

      10. metadata-eval 99.2%

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

      11. +-commutative 99.2%

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

   10. Simplified 99.2%

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

       1. expm1-log1p-u 99.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + 0.3333333333333333 \cdot x\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       2. log1p-define 99.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + 0.3333333333333333 \cdot x\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       3. +-commutative 99.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + 0.3333333333333333 \cdot x\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. fma-undefine 99.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + 0.3333333333333333 \cdot x\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       5. expm1-undefine 99.2%

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

       6. add-exp-log 99.2%

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

       7. add-sqr-sqrt 52.9%

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

       8. fabs-sqr 52.9%

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

       9. add-sqr-sqrt 99.2%

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

   12. Applied egg-rr 99.2%

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

       1. fma-undefine 99.2%

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

       2. associate--l+ 99.2%

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

       3. metadata-eval 99.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + 0.3333333333333333 \cdot x\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       4. metadata-eval 99.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + 0.3333333333333333 \cdot x\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0.3275911 \cdot 0}\right)} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       5. distribute-lft-in 99.2%

          \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + 0.3333333333333333 \cdot x\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{0.3275911 \cdot \left(x + 0\right)}} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

       6. +-rgt-identity 99.2%

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

   14. Simplified 99.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + x \cdot 0.3333333333333333\right)} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{1 + t\_0}\\ \mathbf{if}\;\left|x\_m\right| \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x\_m \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(t\_1 \cdot \left(t\_1 \cdot \left(t\_1 \cdot \left(\mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}, -1.453152027\right) \cdot \frac{1}{-1 - t\_0} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)) (t_1 (/ 1.0 (+ 1.0 t_0))))
   (if (<= (fabs x_m) 4e-8)
     (+ (* x_m 1.128386358070218) 1e-9)
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        t_1
        (-
         (*
          t_1
          (-
           (*
            t_1
            (-
             (*
              (fma 1.061405429 (/ 1.0 (fma 0.3275911 x_m 1.0)) -1.453152027)
              (/ 1.0 (- -1.0 t_0)))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double t_1 = 1.0 / (1.0 + t_0);
	double tmp;
	if (fabs(x_m) <= 4e-8) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * (t_1 * ((t_1 * ((t_1 * ((fma(1.061405429, (1.0 / fma(0.3275911, x_m, 1.0)), -1.453152027) * (1.0 / (-1.0 - t_0))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	t_1 = Float64(1.0 / Float64(1.0 + t_0))
	tmp = 0.0
	if (abs(x_m) <= 4e-8)
		tmp = Float64(Float64(x_m * 1.128386358070218) + 1e-9);
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(t_1 * Float64(Float64(t_1 * Float64(Float64(t_1 * Float64(Float64(fma(1.061405429, Float64(1.0 / fma(0.3275911, x_m, 1.0)), -1.453152027) * Float64(1.0 / Float64(-1.0 - t_0))) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 4e-8], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(N[(t$95$1 * N[(N[(t$95$1 * N[(N[(N[(1.061405429 * N[(1.0 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.0000000000000001e-8

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 56.9%

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

   7. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

   9. Simplified 98.3%

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

       1. unpow2 98.3%

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

       2. add-sqr-sqrt 98.4%

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

       3. metadata-eval 98.4%

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

       4. metadata-eval 98.4%

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

       5. sqrt-pow2 98.4%

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

       6. +-commutative 98.4%

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

       7. sqrt-pow2 98.4%

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

       8. metadata-eval 98.4%

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

       9. metadata-eval 98.4%

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

   11. Applied egg-rr 98.4%

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

   if 4.0000000000000001e-8 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. +-commutative 99.8%

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

      2. div-inv 99.8%

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

      3. fma-define 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-undefine 99.8%

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

      6. add-sqr-sqrt 52.8%

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

      7. fabs-sqr 52.8%

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

      8. add-sqr-sqrt 99.2%

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

   6. Applied egg-rr 99.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\mathsf{fma}\left(1.061405429, \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027\right) \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x\_m\right| \cdot 0.3275911}\\ \mathbf{if}\;x\_m \leq 1.25 \cdot 10^{-6}:\\ \;\;\;\;x\_m \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(t\_0 \cdot \left(t\_0 \cdot \left(t\_0 \cdot \left(t\_0 \cdot \left(\frac{1.061405429}{-1 - x\_m \cdot 0.3275911} - -1.453152027\right) - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x_m) 0.3275911)))))
   (if (<= x_m 1.25e-6)
     (+ (* x_m 1.128386358070218) 1e-9)
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        t_0
        (-
         (*
          t_0
          (-
           (*
            t_0
            (-
             (*
              t_0
              (- (/ 1.061405429 (- -1.0 (* x_m 0.3275911))) -1.453152027))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (fabs(x_m) * 0.3275911));
	double tmp;
	if (x_m <= 1.25e-6) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * (t_0 * ((t_0 * ((t_0 * ((t_0 * ((1.061405429 / (-1.0 - (x_m * 0.3275911))) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (1.0d0 + (abs(x_m) * 0.3275911d0))
    if (x_m <= 1.25d-6) then
        tmp = (x_m * 1.128386358070218d0) + 1d-9
    else
        tmp = 1.0d0 + (exp((x_m * -x_m)) * (t_0 * ((t_0 * ((t_0 * ((t_0 * ((1.061405429d0 / ((-1.0d0) - (x_m * 0.3275911d0))) - (-1.453152027d0))) - 1.421413741d0)) - (-0.284496736d0))) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (Math.abs(x_m) * 0.3275911));
	double tmp;
	if (x_m <= 1.25e-6) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0 + (Math.exp((x_m * -x_m)) * (t_0 * ((t_0 * ((t_0 * ((t_0 * ((1.061405429 / (-1.0 - (x_m * 0.3275911))) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 / (1.0 + (math.fabs(x_m) * 0.3275911))
	tmp = 0
	if x_m <= 1.25e-6:
		tmp = (x_m * 1.128386358070218) + 1e-9
	else:
		tmp = 1.0 + (math.exp((x_m * -x_m)) * (t_0 * ((t_0 * ((t_0 * ((t_0 * ((1.061405429 / (-1.0 - (x_m * 0.3275911))) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))
	tmp = 0.0
	if (x_m <= 1.25e-6)
		tmp = Float64(Float64(x_m * 1.128386358070218) + 1e-9);
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(t_0 * Float64(Float64(t_0 * Float64(Float64(t_0 * Float64(Float64(t_0 * Float64(Float64(1.061405429 / Float64(-1.0 - Float64(x_m * 0.3275911))) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 / (1.0 + (abs(x_m) * 0.3275911));
	tmp = 0.0;
	if (x_m <= 1.25e-6)
		tmp = (x_m * 1.128386358070218) + 1e-9;
	else
		tmp = 1.0 + (exp((x_m * -x_m)) * (t_0 * ((t_0 * ((t_0 * ((t_0 * ((1.061405429 / (-1.0 - (x_m * 0.3275911))) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.25e-6], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(N[(t$95$0 * N[(N[(t$95$0 * N[(N[(t$95$0 * N[(N[(1.061405429 / N[(-1.0 - N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.453152027), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2500000000000001e-6

   1. Initial program 69.7%

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

   3. Simplified 69.7%

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

   6. Applied egg-rr 41.6%

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

   7. Taylor expanded in x around 0 70.5%

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

      1. *-commutative 70.5%

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

   9. Simplified 70.5%

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

       1. unpow2 70.5%

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

       2. add-sqr-sqrt 70.8%

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

       3. metadata-eval 70.8%

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

       4. metadata-eval 70.8%

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

       5. sqrt-pow2 70.8%

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

       6. +-commutative 70.8%

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

       7. sqrt-pow2 70.8%

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

       8. metadata-eval 70.8%

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

       9. metadata-eval 70.8%

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

   11. Applied egg-rr 70.8%

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

   if 1.2500000000000001e-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 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + 0.3333333333333333 \cdot x\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. log1p-define 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + 0.3333333333333333 \cdot x\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      3. +-commutative 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + 0.3333333333333333 \cdot x\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. fma-undefine 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + 0.3333333333333333 \cdot x\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. expm1-undefine 99.8%

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

      6. add-exp-log 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   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 99.8%

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

      2. associate--l+ 99.8%

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

      3. metadata-eval 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + 0.3333333333333333 \cdot x\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      4. metadata-eval 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + 0.3333333333333333 \cdot x\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0.3275911 \cdot 0}\right)} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      5. distribute-lft-in 99.8%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left(\log \left({\left(\sqrt[3]{e^{x}}\right)}^{2}\right) + 0.3333333333333333 \cdot x\right)} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{0.3275911 \cdot \left(x + 0\right)}} \cdot \left(-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

      6. +-rgt-identity 99.8%

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

   8. Simplified 99.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-6}:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1.061405429}{-1 - x \cdot 0.3275911} - -1.453152027\right) - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 85.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88) (+ (* x_m 1.128386358070218) 1e-9) 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;
	} 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
    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;
	} 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
	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(Float64(x_m * 1.128386358070218) + 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 <= 0.88)
		tmp = (x_m * 1.128386358070218) + 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, 0.88], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 69.8%

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

   3. Simplified 69.8%

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

   6. Applied egg-rr 41.6%

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

   7. Taylor expanded in x around 0 70.4%

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

      1. *-commutative 70.4%

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

   9. Simplified 70.4%

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

       1. unpow2 70.4%

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

       2. add-sqr-sqrt 70.7%

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

       3. metadata-eval 70.7%

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

       4. metadata-eval 70.7%

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

       5. sqrt-pow2 70.7%

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

       6. +-commutative 70.7%

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

       7. sqrt-pow2 70.7%

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

       8. metadata-eval 70.7%

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

       9. metadata-eval 70.7%

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

   11. Applied egg-rr 70.7%

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

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around inf 100.0%

      \[\leadsto {\left(\sqrt{\color{blue}{1}}\right)}^{2} \]
   8. Step-by-step derivation

      1. metadata-eval 100.0%

         \[\leadsto {\color{blue}{1}}^{2} \]

      2. metadata-eval 100.0%

         \[\leadsto \color{blue}{1} \]

   9. Applied egg-rr 100.0%

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

Alternative 5: 97.6% 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 69.7%

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

   3. Simplified 69.7%

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

   6. Applied egg-rr 41.6%

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

   7. Taylor expanded in x around 0 70.5%

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

      1. *-commutative 70.5%

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

   9. Simplified 70.5%

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

   10. Taylor expanded in x around 0 72.9%

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

   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

   6. Applied egg-rr 0.6%

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

   7. Taylor expanded in x around inf 98.6%

      \[\leadsto {\left(\sqrt{\color{blue}{1}}\right)}^{2} \]
   8. Step-by-step derivation

      1. metadata-eval 98.6%

         \[\leadsto {\color{blue}{1}}^{2} \]

      2. metadata-eval 98.6%

         \[\leadsto \color{blue}{1} \]

   9. Applied egg-rr 98.6%

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

Alternative 6: 52.6% accurate, 856.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

3. Simplified 77.0%

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

6. Applied egg-rr 31.7%

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

7. Taylor expanded in x around 0 54.9%

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

   1. *-commutative 54.9%

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

9. Simplified 54.9%

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

10. Taylor expanded in x around 0 57.9%

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

Reproduce

herbie shell --seed 2024111 
(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)))))))