Jmat.Real.erf

Percentage Accurate: 79.1% → 99.7%

Time: 25.2s

Alternatives: 13

Speedup: 279.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.1% 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, 0.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := e^{-{x}^{2}}\\ t_1 := \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\\ t_2 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_3 := 0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{t_2}}{t_2}\\ t_4 := \frac{t_3}{\frac{t_2}{t_0}}\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-8}:\\ \;\;\;\;10^{-9} + {\left(e^{{t_1}^{2}}\right)}^{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t_4}^{3}}{{t_4}^{2} + \mathsf{fma}\left(t_0, \frac{t_3}{t_2}, 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- (pow x 2.0))))
        (t_1 (cbrt (log (* x 1.128386358070218))))
        (t_2 (fma 0.3275911 (fabs x) 1.0))
        (t_3
         (+
          0.254829592
          (/
           (+
            -0.284496736
            (/
             (+
              1.421413741
              (/
               (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x 1.0)))
               (fma 0.3275911 x 1.0)))
             t_2))
           t_2)))
        (t_4 (/ t_3 (/ t_2 t_0))))
   (if (<= (fabs x) 2e-8)
     (+ 1e-9 (pow (exp (pow t_1 2.0)) t_1))
     (/ (- 1.0 (pow t_4 3.0)) (+ (pow t_4 2.0) (fma t_0 (/ t_3 t_2) 1.0))))))

C

x = abs(x);
double code(double x) {
	double t_0 = exp(-pow(x, 2.0));
	double t_1 = cbrt(log((x * 1.128386358070218)));
	double t_2 = fma(0.3275911, fabs(x), 1.0);
	double t_3 = 0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / t_2)) / t_2);
	double t_4 = t_3 / (t_2 / t_0);
	double tmp;
	if (fabs(x) <= 2e-8) {
		tmp = 1e-9 + pow(exp(pow(t_1, 2.0)), t_1);
	} else {
		tmp = (1.0 - pow(t_4, 3.0)) / (pow(t_4, 2.0) + fma(t_0, (t_3 / t_2), 1.0));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = exp(Float64(-(x ^ 2.0)))
	t_1 = cbrt(log(Float64(x * 1.128386358070218)))
	t_2 = fma(0.3275911, abs(x), 1.0)
	t_3 = Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / t_2)) / t_2))
	t_4 = Float64(t_3 / Float64(t_2 / t_0))
	tmp = 0.0
	if (abs(x) <= 2e-8)
		tmp = Float64(1e-9 + (exp((t_1 ^ 2.0)) ^ t_1));
	else
		tmp = Float64(Float64(1.0 - (t_4 ^ 3.0)) / Float64((t_4 ^ 2.0) + fma(t_0, Float64(t_3 / t_2), 1.0)));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[Exp[(-N[Power[x, 2.0], $MachinePrecision])], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Log[N[(x * 1.128386358070218), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2e-8], N[(1e-9 + N[Power[N[Exp[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision], t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$4, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$4, 2.0], $MachinePrecision] + N[(t$95$0 * N[(t$95$3 / t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   5. Applied egg-rr 57.6%

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

      1. *-commutative 57.6%

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

      2. distribute-neg-frac 57.6%

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

      3. distribute-neg-in 57.6%

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

      4. unsub-neg 57.6%

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

      5. metadata-eval 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

   10. Simplified 99.6%

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

       1. add-exp-log 51.0%

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

   12. Applied egg-rr 51.0%

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

       1. add-cube-cbrt 51.0%

          \[\leadsto 10^{-9} + e^{\color{blue}{\left(\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)} \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\right) \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}}} \]

       2. exp-prod 51.0%

          \[\leadsto 10^{-9} + \color{blue}{{\left(e^{\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)} \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}}\right)}^{\left(\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\right)}} \]

       3. pow2 51.0%

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

   14. Applied egg-rr 51.0%

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

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

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-udef 99.9%

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

      3. add-cube-cbrt 99.9%

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

      4. pow3 99.9%

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

      5. add-sqr-sqrt 52.6%

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

      6. fabs-sqr 52.6%

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

      7. add-sqr-sqrt 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 99.3%

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

      3. log1p-udef 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-udef 99.3%

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

      6. add-exp-log 99.3%

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

      7. add-sqr-sqrt 52.6%

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

      8. fabs-sqr 52.6%

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

      9. add-sqr-sqrt 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. fma-udef 99.3%

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

      2. associate--l+ 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-rgt-identity 99.3%

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

   9. Simplified 99.3%

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

   11. Applied egg-rr 99.3%

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

   13. Simplified 99.3%

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

4. Final simplification 75.4%

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

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := 1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{t_0}}{t_0}}{\frac{t_0}{e^{-{x}^{2}}}}\\ t_2 := \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-8}:\\ \;\;\;\;10^{-9} + {\left(e^{{t_2}^{2}}\right)}^{t_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{t_1} \cdot \sqrt[3]{{t_1}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1
         (-
          1.0
          (/
           (+
            0.254829592
            (/
             (+
              -0.284496736
              (/
               (+
                1.421413741
                (/
                 (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x 1.0)))
                 (fma 0.3275911 x 1.0)))
               t_0))
             t_0))
           (/ t_0 (exp (- (pow x 2.0)))))))
        (t_2 (cbrt (log (* x 1.128386358070218)))))
   (if (<= (fabs x) 2e-8)
     (+ 1e-9 (pow (exp (pow t_2 2.0)) t_2))
     (* (cbrt t_1) (cbrt (pow t_1 2.0))))))

C

x = abs(x);
double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = 1.0 - ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / t_0)) / t_0)) / (t_0 / exp(-pow(x, 2.0))));
	double t_2 = cbrt(log((x * 1.128386358070218)));
	double tmp;
	if (fabs(x) <= 2e-8) {
		tmp = 1e-9 + pow(exp(pow(t_2, 2.0)), t_2);
	} else {
		tmp = cbrt(t_1) * cbrt(pow(t_1, 2.0));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = Float64(1.0 - Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / t_0)) / t_0)) / Float64(t_0 / exp(Float64(-(x ^ 2.0))))))
	t_2 = cbrt(log(Float64(x * 1.128386358070218)))
	tmp = 0.0
	if (abs(x) <= 2e-8)
		tmp = Float64(1e-9 + (exp((t_2 ^ 2.0)) ^ t_2));
	else
		tmp = Float64(cbrt(t_1) * cbrt((t_1 ^ 2.0)));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 / N[Exp[(-N[Power[x, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Log[N[(x * 1.128386358070218), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2e-8], N[(1e-9 + N[Power[N[Exp[N[Power[t$95$2, 2.0], $MachinePrecision]], $MachinePrecision], t$95$2], $MachinePrecision]), $MachinePrecision], N[(N[Power[t$95$1, 1/3], $MachinePrecision] * N[Power[N[Power[t$95$1, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   5. Applied egg-rr 57.6%

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

      1. *-commutative 57.6%

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

      2. distribute-neg-frac 57.6%

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

      3. distribute-neg-in 57.6%

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

      4. unsub-neg 57.6%

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

      5. metadata-eval 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

   10. Simplified 99.6%

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

       1. add-exp-log 51.0%

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

   12. Applied egg-rr 51.0%

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

       1. add-cube-cbrt 51.0%

          \[\leadsto 10^{-9} + e^{\color{blue}{\left(\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)} \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\right) \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}}} \]

       2. exp-prod 51.0%

          \[\leadsto 10^{-9} + \color{blue}{{\left(e^{\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)} \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}}\right)}^{\left(\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\right)}} \]

       3. pow2 51.0%

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

   14. Applied egg-rr 51.0%

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

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

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-udef 99.9%

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

      3. add-cube-cbrt 99.9%

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

      4. pow3 99.9%

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

      5. add-sqr-sqrt 52.6%

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

      6. fabs-sqr 52.6%

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

      7. add-sqr-sqrt 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 99.3%

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

      3. log1p-udef 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-udef 99.3%

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

      6. add-exp-log 99.3%

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

      7. add-sqr-sqrt 52.6%

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

      8. fabs-sqr 52.6%

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

      9. add-sqr-sqrt 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. fma-udef 99.3%

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

      2. associate--l+ 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-rgt-identity 99.3%

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

   9. Simplified 99.3%

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

   11. Applied egg-rr 99.3%

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

   13. Simplified 99.3%

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

4. Final simplification 75.3%

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

Alternative 3: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-8}:\\ \;\;\;\;10^{-9} + {\left(e^{{t_1}^{2}}\right)}^{t_1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{t_0}}{t_0}}{\frac{t_0}{e^{-{x}^{2}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1 (cbrt (log (* x 1.128386358070218)))))
   (if (<= (fabs x) 2e-8)
     (+ 1e-9 (pow (exp (pow t_1 2.0)) t_1))
     (-
      1.0
      (/
       (+
        0.254829592
        (/
         (+
          -0.284496736
          (/
           (+
            1.421413741
            (/
             (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x 1.0)))
             (fma 0.3275911 x 1.0)))
           t_0))
         t_0))
       (/ t_0 (exp (- (pow x 2.0)))))))))

C

x = abs(x);
double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = cbrt(log((x * 1.128386358070218)));
	double tmp;
	if (fabs(x) <= 2e-8) {
		tmp = 1e-9 + pow(exp(pow(t_1, 2.0)), t_1);
	} else {
		tmp = 1.0 - ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / t_0)) / t_0)) / (t_0 / exp(-pow(x, 2.0))));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = cbrt(log(Float64(x * 1.128386358070218)))
	tmp = 0.0
	if (abs(x) <= 2e-8)
		tmp = Float64(1e-9 + (exp((t_1 ^ 2.0)) ^ t_1));
	else
		tmp = Float64(1.0 - Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))) / t_0)) / t_0)) / Float64(t_0 / exp(Float64(-(x ^ 2.0))))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Log[N[(x * 1.128386358070218), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2e-8], N[(1e-9 + N[Power[N[Exp[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision], t$95$1], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 / N[Exp[(-N[Power[x, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   5. Applied egg-rr 57.6%

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

      1. *-commutative 57.6%

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

      2. distribute-neg-frac 57.6%

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

      3. distribute-neg-in 57.6%

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

      4. unsub-neg 57.6%

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

      5. metadata-eval 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

   10. Simplified 99.6%

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

       1. add-exp-log 51.0%

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

   12. Applied egg-rr 51.0%

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

       1. add-cube-cbrt 51.0%

          \[\leadsto 10^{-9} + e^{\color{blue}{\left(\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)} \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\right) \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}}} \]

       2. exp-prod 51.0%

          \[\leadsto 10^{-9} + \color{blue}{{\left(e^{\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)} \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}}\right)}^{\left(\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\right)}} \]

       3. pow2 51.0%

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

   14. Applied egg-rr 51.0%

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

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

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-udef 99.9%

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

      3. add-cube-cbrt 99.9%

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

      4. pow3 99.9%

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

      5. add-sqr-sqrt 52.6%

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

      6. fabs-sqr 52.6%

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

      7. add-sqr-sqrt 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 99.3%

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

      3. log1p-udef 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-udef 99.3%

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

      6. add-exp-log 99.3%

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

      7. add-sqr-sqrt 52.6%

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

      8. fabs-sqr 52.6%

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

      9. add-sqr-sqrt 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. fma-udef 99.3%

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

      2. associate--l+ 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-rgt-identity 99.3%

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

   9. Simplified 99.3%

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

   11. Applied egg-rr 99.3%

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

       1. cancel-sign-sub-inv 99.3%

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

       2. associate-*l/ 99.3%

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

       3. fma-udef 99.3%

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

       4. +-commutative 99.3%

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

       5. associate-/l* 99.3%

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

       6. +-commutative 99.3%

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

       7. fma-udef 99.3%

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

   13. Simplified 99.3%

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

4. Final simplification 75.3%

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

Alternative 4: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ t_2 := \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-8}:\\ \;\;\;\;10^{-9} + {\left(e^{{t_2}^{2}}\right)}^{t_2}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(t_1 \cdot \left(\left(-0.284496736 + t_1 \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot {\left(\sqrt[3]{{\left(\sqrt[3]{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}^{3}}\right)}^{3}\right)\right) \cdot \frac{-1}{t_0} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_1 (/ 1.0 t_0))
        (t_2 (cbrt (log (* x 1.128386358070218)))))
   (if (<= (fabs x) 2e-8)
     (+ 1e-9 (pow (exp (pow t_2 2.0)) t_2))
     (+
      1.0
      (*
       (exp (* x (- x)))
       (*
        t_1
        (-
         (*
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (*
              (/ 1.0 (+ 1.0 (* x 0.3275911)))
              (pow
               (cbrt
                (pow
                 (cbrt (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x 1.0))))
                 3.0))
               3.0)))))
          (/ -1.0 t_0))
         0.254829592)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = cbrt(log((x * 1.128386358070218)));
	double tmp;
	if (fabs(x) <= 2e-8) {
		tmp = 1e-9 + pow(exp(pow(t_2, 2.0)), t_2);
	} else {
		tmp = 1.0 + (exp((x * -x)) * (t_1 * (((-0.284496736 + (t_1 * (1.421413741 + ((1.0 / (1.0 + (x * 0.3275911))) * pow(cbrt(pow(cbrt((-1.453152027 + (1.061405429 / fma(0.3275911, x, 1.0)))), 3.0)), 3.0))))) * (-1.0 / t_0)) - 0.254829592)));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	t_2 = cbrt(log(Float64(x * 1.128386358070218)))
	tmp = 0.0
	if (abs(x) <= 2e-8)
		tmp = Float64(1e-9 + (exp((t_2 ^ 2.0)) ^ t_2));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x * Float64(-x))) * Float64(t_1 * Float64(Float64(Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(Float64(1.0 / Float64(1.0 + Float64(x * 0.3275911))) * (cbrt((cbrt(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x, 1.0)))) ^ 3.0)) ^ 3.0))))) * Float64(-1.0 / t_0)) - 0.254829592))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Log[N[(x * 1.128386358070218), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2e-8], N[(1e-9 + N[Power[N[Exp[N[Power[t$95$2, 2.0], $MachinePrecision]], $MachinePrecision], t$95$2], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(N[(N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(N[(1.0 / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Power[N[Power[N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   5. Applied egg-rr 57.6%

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

      1. *-commutative 57.6%

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

      2. distribute-neg-frac 57.6%

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

      3. distribute-neg-in 57.6%

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

      4. unsub-neg 57.6%

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

      5. metadata-eval 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

   10. Simplified 99.6%

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

       1. add-exp-log 51.0%

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

   12. Applied egg-rr 51.0%

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

       1. add-cube-cbrt 51.0%

          \[\leadsto 10^{-9} + e^{\color{blue}{\left(\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)} \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\right) \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}}} \]

       2. exp-prod 51.0%

          \[\leadsto 10^{-9} + \color{blue}{{\left(e^{\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)} \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}}\right)}^{\left(\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\right)}} \]

       3. pow2 51.0%

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

   14. Applied egg-rr 51.0%

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

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

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-udef 99.9%

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

      3. add-cube-cbrt 99.9%

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

      4. pow3 99.9%

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

      5. add-sqr-sqrt 52.6%

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

      6. fabs-sqr 52.6%

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

      7. add-sqr-sqrt 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 99.3%

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

      3. log1p-udef 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-udef 99.3%

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

      6. add-exp-log 99.3%

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

      7. add-sqr-sqrt 52.6%

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

      8. fabs-sqr 52.6%

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

      9. add-sqr-sqrt 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. fma-udef 99.3%

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

      2. associate--l+ 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-rgt-identity 99.3%

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

   9. Simplified 99.3%

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

       1. rem-cube-cbrt 99.3%

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

   11. Applied egg-rr 99.3%

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

4. Final simplification 75.3%

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

Alternative 5: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := \frac{1}{1 + \left|x\right| \cdot 0.3275911}\\ t_2 := \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-8}:\\ \;\;\;\;10^{-9} + {\left(e^{{t_2}^{2}}\right)}^{t_2}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(t_1 \cdot \left(t_1 \cdot \left(\frac{1}{t_0} \cdot \left({\left(\sqrt[3]{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}^{3} \cdot \frac{-1}{t_0} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x 0.3275911)))
        (t_1 (/ 1.0 (+ 1.0 (* (fabs x) 0.3275911))))
        (t_2 (cbrt (log (* x 1.128386358070218)))))
   (if (<= (fabs x) 2e-8)
     (+ 1e-9 (pow (exp (pow t_2 2.0)) t_2))
     (+
      1.0
      (*
       (exp (* x (- x)))
       (*
        t_1
        (-
         (*
          t_1
          (-
           (*
            (/ 1.0 t_0)
            (-
             (*
              (pow
               (cbrt (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x 1.0))))
               3.0)
              (/ -1.0 t_0))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = 1.0 / (1.0 + (fabs(x) * 0.3275911));
	double t_2 = cbrt(log((x * 1.128386358070218)));
	double tmp;
	if (fabs(x) <= 2e-8) {
		tmp = 1e-9 + pow(exp(pow(t_2, 2.0)), t_2);
	} else {
		tmp = 1.0 + (exp((x * -x)) * (t_1 * ((t_1 * (((1.0 / t_0) * ((pow(cbrt((-1.453152027 + (1.061405429 / fma(0.3275911, x, 1.0)))), 3.0) * (-1.0 / t_0)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 + Float64(x * 0.3275911))
	t_1 = Float64(1.0 / Float64(1.0 + Float64(abs(x) * 0.3275911)))
	t_2 = cbrt(log(Float64(x * 1.128386358070218)))
	tmp = 0.0
	if (abs(x) <= 2e-8)
		tmp = Float64(1e-9 + (exp((t_2 ^ 2.0)) ^ t_2));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x * Float64(-x))) * Float64(t_1 * Float64(Float64(t_1 * Float64(Float64(Float64(1.0 / t_0) * Float64(Float64((cbrt(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x, 1.0)))) ^ 3.0) * Float64(-1.0 / t_0)) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Log[N[(x * 1.128386358070218), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2e-8], N[(1e-9 + N[Power[N[Exp[N[Power[t$95$2, 2.0], $MachinePrecision]], $MachinePrecision], t$95$2], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(N[(t$95$1 * N[(N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(N[Power[N[Power[N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[(-1.0 / t$95$0), $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) < 2e-8

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   5. Applied egg-rr 57.6%

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

      1. *-commutative 57.6%

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

      2. distribute-neg-frac 57.6%

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

      3. distribute-neg-in 57.6%

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

      4. unsub-neg 57.6%

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

      5. metadata-eval 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

   10. Simplified 99.6%

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

       1. add-exp-log 51.0%

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

   12. Applied egg-rr 51.0%

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

       1. add-cube-cbrt 51.0%

          \[\leadsto 10^{-9} + e^{\color{blue}{\left(\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)} \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\right) \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}}} \]

       2. exp-prod 51.0%

          \[\leadsto 10^{-9} + \color{blue}{{\left(e^{\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)} \cdot \sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}}\right)}^{\left(\sqrt[3]{\log \left(x \cdot 1.128386358070218\right)}\right)}} \]

       3. pow2 51.0%

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

   14. Applied egg-rr 51.0%

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

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

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-udef 99.9%

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

      3. add-cube-cbrt 99.9%

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

      4. pow3 99.9%

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

      5. add-sqr-sqrt 52.6%

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

      6. fabs-sqr 52.6%

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

      7. add-sqr-sqrt 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 99.3%

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

      3. log1p-udef 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-udef 99.3%

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

      6. add-exp-log 99.3%

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

      7. add-sqr-sqrt 52.6%

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

      8. fabs-sqr 52.6%

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

      9. add-sqr-sqrt 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. fma-udef 99.3%

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

      2. associate--l+ 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-rgt-identity 99.3%

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

   9. Simplified 99.3%

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

       1. expm1-log1p-u 99.3%

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

       2. expm1-udef 99.3%

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

       3. log1p-udef 99.3%

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

       4. +-commutative 99.3%

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

       5. fma-udef 99.3%

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

       6. add-exp-log 99.3%

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

       7. add-sqr-sqrt 52.6%

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

       8. fabs-sqr 52.6%

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

       9. add-sqr-sqrt 99.3%

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

   11. Applied egg-rr 99.2%

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

       1. fma-udef 99.3%

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

       2. associate--l+ 99.3%

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

       3. metadata-eval 99.3%

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

       4. +-rgt-identity 99.3%

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

   13. Simplified 99.2%

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

4. Final simplification 75.3%

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

Alternative 6: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x\right| \cdot 0.3275911}\\ t_1 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;x \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + {\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(t_0 \cdot \left(t_0 \cdot \left(\frac{1}{t_1} \cdot \left({\left(\sqrt[3]{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}^{3} \cdot \frac{-1}{t_1} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x) 0.3275911))))
        (t_1 (+ 1.0 (* x 0.3275911))))
   (if (<= x 1.7e-6)
     (+ 1e-9 (pow (cbrt (* x 1.128386358070218)) 3.0))
     (+
      1.0
      (*
       (exp (* x (- x)))
       (*
        t_0
        (-
         (*
          t_0
          (-
           (*
            (/ 1.0 t_1)
            (-
             (*
              (pow
               (cbrt (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x 1.0))))
               3.0)
              (/ -1.0 t_1))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 / (1.0 + (fabs(x) * 0.3275911));
	double t_1 = 1.0 + (x * 0.3275911);
	double tmp;
	if (x <= 1.7e-6) {
		tmp = 1e-9 + pow(cbrt((x * 1.128386358070218)), 3.0);
	} else {
		tmp = 1.0 + (exp((x * -x)) * (t_0 * ((t_0 * (((1.0 / t_1) * ((pow(cbrt((-1.453152027 + (1.061405429 / fma(0.3275911, x, 1.0)))), 3.0) * (-1.0 / t_1)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x) * 0.3275911)))
	t_1 = Float64(1.0 + Float64(x * 0.3275911))
	tmp = 0.0
	if (x <= 1.7e-6)
		tmp = Float64(1e-9 + (cbrt(Float64(x * 1.128386358070218)) ^ 3.0));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x * Float64(-x))) * Float64(t_0 * Float64(Float64(t_0 * Float64(Float64(Float64(1.0 / t_1) * Float64(Float64((cbrt(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x, 1.0)))) ^ 3.0) * Float64(-1.0 / t_1)) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.7e-6], N[(1e-9 + N[Power[N[Power[N[(x * 1.128386358070218), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(N[(t$95$0 * N[(N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(N[(N[Power[N[Power[N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.70000000000000003e-6

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   5. Applied egg-rr 39.9%

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

      1. *-commutative 39.9%

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

      2. distribute-neg-frac 39.9%

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

      3. distribute-neg-in 39.9%

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

      4. unsub-neg 39.9%

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

      5. metadata-eval 39.9%

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

   7. Simplified 39.9%

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

   8. Taylor expanded in x around 0 67.7%

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

      1. *-commutative 67.7%

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

   10. Simplified 67.7%

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

       1. add-cube-cbrt 67.7%

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

       2. pow3 67.7%

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

   12. Applied egg-rr 67.7%

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

   if 1.70000000000000003e-6 < x

   1. Initial program 99.7%

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

   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. Step-by-step derivation

      1. +-commutative 99.7%

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

      2. fma-udef 99.7%

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

      3. add-cube-cbrt 99.8%

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

      4. pow3 99.8%

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

      5. add-sqr-sqrt 99.8%

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

      6. fabs-sqr 99.8%

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

      7. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. expm1-log1p-u 99.8%

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

      2. expm1-udef 99.8%

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

      3. log1p-udef 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-udef 99.8%

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

      6. add-exp-log 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. associate--l+ 99.8%

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

      3. metadata-eval 99.8%

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

      4. +-rgt-identity 99.8%

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

   9. Simplified 99.8%

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

       1. expm1-log1p-u 99.8%

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

       2. expm1-udef 99.8%

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

       3. log1p-udef 99.8%

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

       4. +-commutative 99.8%

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

       5. fma-udef 99.8%

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

       6. add-exp-log 99.8%

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

       7. add-sqr-sqrt 99.8%

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

       8. fabs-sqr 99.8%

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

       9. add-sqr-sqrt 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

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

       4. +-rgt-identity 99.8%

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

   13. Simplified 99.8%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + {\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left({\left(\sqrt[3]{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}^{3} \cdot \frac{-1}{1 + x \cdot 0.3275911} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]

Alternative 7: 99.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;10^{-9} + {\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= x 9.5e-7)
     (+ 1e-9 (pow (cbrt (* x 1.128386358070218)) 3.0))
     (+
      1.0
      (*
       (exp (* x (- x)))
       (*
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+ 1.421413741 (* t_1 (+ -1.453152027 (/ 1.061405429 t_0))))))))
        (/ -1.0 (+ 1.0 (* (fabs x) 0.3275911)))))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x <= 9.5e-7) {
		tmp = 1e-9 + pow(cbrt((x * 1.128386358070218)), 3.0);
	} else {
		tmp = 1.0 + (exp((x * -x)) * ((0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0)))))))) * (-1.0 / (1.0 + (fabs(x) * 0.3275911)))));
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x <= 9.5e-7) {
		tmp = 1e-9 + Math.pow(Math.cbrt((x * 1.128386358070218)), 3.0);
	} else {
		tmp = 1.0 + (Math.exp((x * -x)) * ((0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0)))))))) * (-1.0 / (1.0 + (Math.abs(x) * 0.3275911)))));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 + Float64(x * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x <= 9.5e-7)
		tmp = Float64(1e-9 + (cbrt(Float64(x * 1.128386358070218)) ^ 3.0));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x * Float64(-x))) * Float64(Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(t_1 * Float64(-1.453152027 + Float64(1.061405429 / t_0)))))))) * Float64(-1.0 / Float64(1.0 + Float64(abs(x) * 0.3275911))))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x, 9.5e-7], N[(1e-9 + N[Power[N[Power[N[(x * 1.128386358070218), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000001e-7

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   5. Applied egg-rr 39.9%

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

      1. *-commutative 39.9%

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

      2. distribute-neg-frac 39.9%

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

      3. distribute-neg-in 39.9%

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

      4. unsub-neg 39.9%

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

      5. metadata-eval 39.9%

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

   7. Simplified 39.9%

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

   8. Taylor expanded in x around 0 67.7%

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

      1. *-commutative 67.7%

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

   10. Simplified 67.7%

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

       1. add-cube-cbrt 67.7%

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

       2. pow3 67.7%

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

   12. Applied egg-rr 67.7%

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

   if 9.5000000000000001e-7 < x

   1. Initial program 99.7%

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

   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. Step-by-step derivation

      1. expm1-log1p-u 99.8%

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

      2. expm1-udef 99.8%

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

      3. log1p-udef 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-udef 99.8%

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

      6. add-exp-log 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.7%

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

      1. fma-udef 99.8%

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

      2. associate--l+ 99.8%

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

      3. metadata-eval 99.8%

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

      4. +-rgt-identity 99.8%

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

   7. Simplified 99.7%

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

      1. expm1-log1p-u 99.8%

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

      2. expm1-udef 99.8%

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

      3. log1p-udef 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-udef 99.8%

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

      6. add-exp-log 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   9. Applied egg-rr 99.7%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

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

       4. +-rgt-identity 99.8%

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

   11. Simplified 99.7%

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

       1. expm1-log1p-u 99.8%

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

       2. expm1-udef 99.8%

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

       3. log1p-udef 99.8%

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

       4. +-commutative 99.8%

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

       5. fma-udef 99.8%

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

       6. add-exp-log 99.8%

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

       7. add-sqr-sqrt 99.8%

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

       8. fabs-sqr 99.8%

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

       9. add-sqr-sqrt 99.8%

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

   13. Applied egg-rr 99.7%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

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

       4. +-rgt-identity 99.8%

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

   15. Simplified 99.7%

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

       1. expm1-log1p-u 99.8%

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

       2. expm1-udef 99.8%

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

       3. log1p-udef 99.8%

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

       4. +-commutative 99.8%

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

       5. fma-udef 99.8%

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

       6. add-exp-log 99.8%

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

       7. add-sqr-sqrt 99.8%

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

       8. fabs-sqr 99.8%

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

       9. add-sqr-sqrt 99.8%

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

   17. Applied egg-rr 99.7%

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

       1. fma-udef 99.8%

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

       2. associate--l+ 99.8%

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

       3. metadata-eval 99.8%

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

       4. +-rgt-identity 99.8%

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

   19. Simplified 99.7%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;10^{-9} + {\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\left(0.254829592 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\right)\\ \end{array} \]

Alternative 8: 99.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;x \leq 0.3:\\ \;\;\;\;10^{-9} + {\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{t_0} \cdot \left(\left(x \cdot -0.2193742730720041 + 1.029667143\right) \cdot \frac{-1}{t_0} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x 0.3275911))))
   (if (<= x 0.3)
     (+ 1e-9 (pow (cbrt (* x 1.128386358070218)) 3.0))
     (+
      1.0
      (*
       (exp (* x (- x)))
       (*
        (/ 1.0 (+ 1.0 (* (fabs x) 0.3275911)))
        (-
         (*
          (/ 1.0 t_0)
          (-
           (* (+ (* x -0.2193742730720041) 1.029667143) (/ -1.0 t_0))
           -0.284496736))
         0.254829592)))))))

C

x = abs(x);
double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double tmp;
	if (x <= 0.3) {
		tmp = 1e-9 + pow(cbrt((x * 1.128386358070218)), 3.0);
	} else {
		tmp = 1.0 + (exp((x * -x)) * ((1.0 / (1.0 + (fabs(x) * 0.3275911))) * (((1.0 / t_0) * ((((x * -0.2193742730720041) + 1.029667143) * (-1.0 / t_0)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = 1.0 + (x * 0.3275911);
	double tmp;
	if (x <= 0.3) {
		tmp = 1e-9 + Math.pow(Math.cbrt((x * 1.128386358070218)), 3.0);
	} else {
		tmp = 1.0 + (Math.exp((x * -x)) * ((1.0 / (1.0 + (Math.abs(x) * 0.3275911))) * (((1.0 / t_0) * ((((x * -0.2193742730720041) + 1.029667143) * (-1.0 / t_0)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = Float64(1.0 + Float64(x * 0.3275911))
	tmp = 0.0
	if (x <= 0.3)
		tmp = Float64(1e-9 + (cbrt(Float64(x * 1.128386358070218)) ^ 3.0));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x * Float64(-x))) * Float64(Float64(1.0 / Float64(1.0 + Float64(abs(x) * 0.3275911))) * Float64(Float64(Float64(1.0 / t_0) * Float64(Float64(Float64(Float64(x * -0.2193742730720041) + 1.029667143) * Float64(-1.0 / t_0)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.3], N[(1e-9 + N[Power[N[Power[N[(x * 1.128386358070218), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(N[(N[(x * -0.2193742730720041), $MachinePrecision] + 1.029667143), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.299999999999999989

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   5. Applied egg-rr 39.9%

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

      1. *-commutative 39.9%

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

      2. distribute-neg-frac 39.9%

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

      3. distribute-neg-in 39.9%

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

      4. unsub-neg 39.9%

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

      5. metadata-eval 39.9%

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

   7. Simplified 39.9%

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

   8. Taylor expanded in x around 0 67.6%

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

      1. *-commutative 67.6%

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

   10. Simplified 67.6%

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

       1. add-cube-cbrt 67.6%

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

       2. pow3 67.6%

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

   12. Applied egg-rr 67.6%

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

   if 0.299999999999999989 < x

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-def 100.0%

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

      7. unpow1 100.0%

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

      8. sqr-pow 100.0%

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

      9. fabs-sqr 100.0%

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

      10. sqr-pow 100.0%

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

      11. unpow1 100.0%

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

      12. associate-*r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-neg-frac 100.0%

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

      15. metadata-eval 100.0%

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

      16. +-commutative 100.0%

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

      17. fma-def 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(1.029667143 + -0.2193742730720041 \cdot x\right)}\right)\right)\right) \cdot e^{-x \cdot x} \]
   8. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(-0.2193742730720041 \cdot x + 1.029667143\right)}\right)\right)\right) \cdot e^{-x \cdot x} \]

      2. *-commutative 100.0%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\color{blue}{x \cdot -0.2193742730720041} + 1.029667143\right)\right)\right)\right) \cdot e^{-x \cdot x} \]

   9. Simplified 100.0%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(x \cdot -0.2193742730720041 + 1.029667143\right)}\right)\right)\right) \cdot e^{-x \cdot x} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 100.0%

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

       2. expm1-udef 100.0%

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

       3. log1p-udef 100.0%

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

       4. +-commutative 100.0%

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

       5. fma-udef 100.0%

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

       6. add-exp-log 100.0%

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

       7. add-sqr-sqrt 100.0%

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

       8. fabs-sqr 100.0%

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

       9. add-sqr-sqrt 100.0%

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

   11. Applied egg-rr 100.0%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \cdot \left(x \cdot -0.2193742730720041 + 1.029667143\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   12. Step-by-step derivation

       1. fma-udef 100.0%

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

       2. associate--l+ 100.0%

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

       3. metadata-eval 100.0%

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

       4. +-rgt-identity 100.0%

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

   13. Simplified 100.0%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(x \cdot -0.2193742730720041 + 1.029667143\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   14. Step-by-step derivation

       1. expm1-log1p-u 100.0%

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

       2. expm1-udef 100.0%

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

       3. log1p-udef 100.0%

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

       4. +-commutative 100.0%

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

       5. fma-udef 100.0%

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

       6. add-exp-log 100.0%

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

       7. add-sqr-sqrt 100.0%

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

       8. fabs-sqr 100.0%

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

       9. add-sqr-sqrt 100.0%

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

   15. Applied egg-rr 100.0%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(x \cdot -0.2193742730720041 + 1.029667143\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
   16. Step-by-step derivation

       1. fma-udef 100.0%

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

       2. associate--l+ 100.0%

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

       3. metadata-eval 100.0%

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

       4. +-rgt-identity 100.0%

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

   17. Simplified 100.0%

       \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot x} \cdot \left(x \cdot -0.2193742730720041 + 1.029667143\right)\right)\right)\right) \cdot e^{-x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.3:\\ \;\;\;\;10^{-9} + {\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(\left(x \cdot -0.2193742730720041 + 1.029667143\right) \cdot \frac{-1}{1 + x \cdot 0.3275911} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]

Alternative 9: 99.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;10^{-9} + {\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.89) (+ 1e-9 (pow (cbrt (* x 1.128386358070218)) 3.0)) 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.89) {
		tmp = 1e-9 + pow(cbrt((x * 1.128386358070218)), 3.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.89) {
		tmp = 1e-9 + Math.pow(Math.cbrt((x * 1.128386358070218)), 3.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.89)
		tmp = Float64(1e-9 + (cbrt(Float64(x * 1.128386358070218)) ^ 3.0));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.89], N[(1e-9 + N[Power[N[Power[N[(x * 1.128386358070218), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.890000000000000013

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   5. Applied egg-rr 39.9%

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

      1. *-commutative 39.9%

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

      2. distribute-neg-frac 39.9%

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

      3. distribute-neg-in 39.9%

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

      4. unsub-neg 39.9%

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

      5. metadata-eval 39.9%

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

   7. Simplified 39.9%

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

   8. Taylor expanded in x around 0 67.6%

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

      1. *-commutative 67.6%

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

   10. Simplified 67.6%

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

       1. add-cube-cbrt 67.6%

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

       2. pow3 67.6%

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

   12. Applied egg-rr 67.6%

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

   if 0.890000000000000013 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 0.0%

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

      1. *-commutative 0.0%

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

      2. distribute-neg-frac 0.0%

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

      3. distribute-neg-in 0.0%

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

      4. unsub-neg 0.0%

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

      5. metadata-eval 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 100.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;10^{-9} + {\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 99.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;10^{-9} + e^{\log \left(x \cdot 1.128386358070218\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.89) (+ 1e-9 (exp (log (* x 1.128386358070218)))) 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.89) {
		tmp = 1e-9 + exp(log((x * 1.128386358070218)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.89d0) then
        tmp = 1d-9 + exp(log((x * 1.128386358070218d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.89) {
		tmp = 1e-9 + Math.exp(Math.log((x * 1.128386358070218)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.89:
		tmp = 1e-9 + math.exp(math.log((x * 1.128386358070218)))
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.89)
		tmp = Float64(1e-9 + exp(log(Float64(x * 1.128386358070218))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.89)
		tmp = 1e-9 + exp(log((x * 1.128386358070218)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.89], N[(1e-9 + N[Exp[N[Log[N[(x * 1.128386358070218), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.890000000000000013

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   5. Applied egg-rr 39.9%

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

      1. *-commutative 39.9%

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

      2. distribute-neg-frac 39.9%

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

      3. distribute-neg-in 39.9%

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

      4. unsub-neg 39.9%

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

      5. metadata-eval 39.9%

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

   7. Simplified 39.9%

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

   8. Taylor expanded in x around 0 67.6%

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

      1. *-commutative 67.6%

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

   10. Simplified 67.6%

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

       1. add-exp-log 34.6%

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

   12. Applied egg-rr 34.6%

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

   if 0.890000000000000013 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 0.0%

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

      1. *-commutative 0.0%

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

      2. distribute-neg-frac 0.0%

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

      3. distribute-neg-in 0.0%

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

      4. unsub-neg 0.0%

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

      5. metadata-eval 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 100.0%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;10^{-9} + e^{\log \left(x \cdot 1.128386358070218\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 99.3% accurate, 121.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.89) (+ 1e-9 (* x 1.128386358070218)) 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.89) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.89d0) then
        tmp = 1d-9 + (x * 1.128386358070218d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.89) {
		tmp = 1e-9 + (x * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.89:
		tmp = 1e-9 + (x * 1.128386358070218)
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.89)
		tmp = Float64(1e-9 + Float64(x * 1.128386358070218));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.89)
		tmp = 1e-9 + (x * 1.128386358070218);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.89], N[(1e-9 + N[(x * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.890000000000000013

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   5. Applied egg-rr 39.9%

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

      1. *-commutative 39.9%

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

      2. distribute-neg-frac 39.9%

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

      3. distribute-neg-in 39.9%

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

      4. unsub-neg 39.9%

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

      5. metadata-eval 39.9%

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

   7. Simplified 39.9%

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

   8. Taylor expanded in x around 0 67.6%

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

      1. *-commutative 67.6%

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

   10. Simplified 67.6%

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

   if 0.890000000000000013 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 0.0%

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

      1. *-commutative 0.0%

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

      2. distribute-neg-frac 0.0%

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

      3. distribute-neg-in 0.0%

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

      4. unsub-neg 0.0%

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

      5. metadata-eval 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 100.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 97.7% accurate, 279.5× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 2.8e-5) 1e-9 1.0))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.8d-5) then
        tmp = 1d-9
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.8e-5], 1e-9, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   5. Applied egg-rr 39.9%

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

      1. *-commutative 39.9%

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

      2. distribute-neg-frac 39.9%

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

      3. distribute-neg-in 39.9%

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

      4. unsub-neg 39.9%

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

      5. metadata-eval 39.9%

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

   7. Simplified 39.9%

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

   8. Taylor expanded in x around 0 69.3%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Applied egg-rr 0.5%

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

      1. *-commutative 0.5%

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

      2. distribute-neg-frac 0.5%

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

      3. distribute-neg-in 0.5%

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

      4. unsub-neg 0.5%

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

      5. metadata-eval 0.5%

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

   7. Simplified 0.5%

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

   8. Taylor expanded in x around inf 98.7%

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

4. Final simplification 77.1%

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

Alternative 13: 53.4% accurate, 856.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 1e-9)

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 1d-9
end function

Java

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

Python

x = abs(x)
def code(x):
	return 1e-9

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 1e-9

Derivation

1. Initial program 78.9%

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

3. Simplified 78.9%

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

5. Applied egg-rr 29.5%

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

   1. *-commutative 29.5%

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

   2. distribute-neg-frac 29.5%

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

   3. distribute-neg-in 29.5%

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

   4. unsub-neg 29.5%

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

   5. metadata-eval 29.5%

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

7. Simplified 29.5%

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

8. Taylor expanded in x around 0 53.8%

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

9. Final simplification 53.8%

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

Reproduce

herbie shell --seed 2024024 
(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)))))))