Jmat.Real.erf

Percentage Accurate: 79.3% → 99.6%

Time: 33.7s

Alternatives: 10

Speedup: 142.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right)\\ t_1 := \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{t\_0}}{t\_0}}{t\_0}\right) \cdot e^{-{x\_m}^{2}}\\ t_2 := \frac{t\_1}{t\_0}\\ \mathbf{if}\;\left|x\_m\right| \leq 1.5 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{{t\_1}^{3}}{{t\_0}^{3}}}{1 + \left(t\_2 + {t\_2}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x_m) 1.0))
        (t_1
         (*
          (+
           0.254829592
           (/
            (+
             -0.284496736
             (/
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
                t_0))
              t_0))
            t_0))
          (exp (- (pow x_m 2.0)))))
        (t_2 (/ t_1 t_0)))
   (if (<= (fabs x_m) 1.5e-10)
     (+ 1e-9 (* x_m 1.128386358070218))
     (/
      (- 1.0 (/ (pow t_1 3.0) (pow t_0 3.0)))
      (+ 1.0 (+ t_2 (pow t_2 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(0.3275911, fabs(x_m), 1.0);
	double t_1 = (0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / t_0)) / t_0)) / t_0)) * exp(-pow(x_m, 2.0));
	double t_2 = t_1 / t_0;
	double tmp;
	if (fabs(x_m) <= 1.5e-10) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = (1.0 - (pow(t_1, 3.0) / pow(t_0, 3.0))) / (1.0 + (t_2 + pow(t_2, 2.0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(0.3275911, abs(x_m), 1.0)
	t_1 = Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / t_0)) / t_0)) / t_0)) * exp(Float64(-(x_m ^ 2.0))))
	t_2 = Float64(t_1 / t_0)
	tmp = 0.0
	if (abs(x_m) <= 1.5e-10)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = Float64(Float64(1.0 - Float64((t_1 ^ 3.0) / (t_0 ^ 3.0))) / Float64(1.0 + Float64(t_2 + (t_2 ^ 2.0))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1.5e-10], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Power[t$95$1, 3.0], $MachinePrecision] / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$2 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.5e-10

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   6. Applied egg-rr 57.4%

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

      1. +-commutative 57.4%

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

      2. neg-mul-1 57.4%

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

   8. Simplified 57.4%

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

   9. Taylor expanded in x around 0 99.4%

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

       1. *-commutative 99.4%

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

   11. Simplified 99.4%

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

   if 1.5e-10 < (fabs.f64 x)

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

      1. expm1-log1p-u 99.0%

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

      2. log1p-define 99.0%

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

      3. +-commutative 99.0%

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

      4. fma-undefine 99.0%

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

      5. expm1-undefine 99.0%

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

      6. add-exp-log 99.0%

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

      7. add-sqr-sqrt 45.6%

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

      8. fabs-sqr 45.6%

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

      9. add-sqr-sqrt 98.1%

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

   6. Applied egg-rr 98.1%

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

      1. fma-undefine 98.1%

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

      2. associate--l+ 98.1%

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

      3. metadata-eval 98.1%

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

      4. +-rgt-identity 98.1%

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

   8. Simplified 98.1%

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

   10. Applied egg-rr 98.1%

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

   12. Simplified 98.1%

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

       1. cube-div 98.1%

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

       2. *-commutative 98.1%

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

   14. Applied egg-rr 98.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.5 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{{\left(\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) \cdot e^{-{x}^{2}}\right)}^{3}}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}}}{1 + \left(\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) \cdot e^{-{x}^{2}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + {\left(\frac{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) \cdot e^{-{x}^{2}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right)\\ \mathbf{if}\;\left|x\_m\right| \leq 1.5 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(1 - \frac{e^{-{x\_m}^{2}} \cdot \left(\left(0.254829592 + \left(\frac{1.421413741}{{t\_0}^{2}} + \frac{1.061405429}{{t\_0}^{4}}\right)\right) - \left(\frac{1.453152027}{{t\_0}^{3}} + \frac{0.284496736}{t\_0}\right)\right)}{t\_0}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x_m) 1.0)))
   (if (<= (fabs x_m) 1.5e-10)
     (+ 1e-9 (* x_m 1.128386358070218))
     (pow
      (pow
       (-
        1.0
        (/
         (*
          (exp (- (pow x_m 2.0)))
          (-
           (+
            0.254829592
            (+ (/ 1.421413741 (pow t_0 2.0)) (/ 1.061405429 (pow t_0 4.0))))
           (+ (/ 1.453152027 (pow t_0 3.0)) (/ 0.284496736 t_0))))
         t_0))
       3.0)
      0.3333333333333333))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(0.3275911, fabs(x_m), 1.0);
	double tmp;
	if (fabs(x_m) <= 1.5e-10) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = pow(pow((1.0 - ((exp(-pow(x_m, 2.0)) * ((0.254829592 + ((1.421413741 / pow(t_0, 2.0)) + (1.061405429 / pow(t_0, 4.0)))) - ((1.453152027 / pow(t_0, 3.0)) + (0.284496736 / t_0)))) / t_0)), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(0.3275911, abs(x_m), 1.0)
	tmp = 0.0
	if (abs(x_m) <= 1.5e-10)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = (Float64(1.0 - Float64(Float64(exp(Float64(-(x_m ^ 2.0))) * Float64(Float64(0.254829592 + Float64(Float64(1.421413741 / (t_0 ^ 2.0)) + Float64(1.061405429 / (t_0 ^ 4.0)))) - Float64(Float64(1.453152027 / (t_0 ^ 3.0)) + Float64(0.284496736 / t_0)))) / t_0)) ^ 3.0) ^ 0.3333333333333333;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1.5e-10], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(1.0 - N[(N[(N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision] * N[(N[(0.254829592 + N[(N[(1.421413741 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + N[(1.061405429 / N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1.453152027 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.284496736 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.5e-10

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   6. Applied egg-rr 57.4%

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

      1. +-commutative 57.4%

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

      2. neg-mul-1 57.4%

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

   8. Simplified 57.4%

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

   9. Taylor expanded in x around 0 99.4%

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

       1. *-commutative 99.4%

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

   11. Simplified 99.4%

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

   if 1.5e-10 < (fabs.f64 x)

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around inf 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in x around inf 99.1%

      \[\leadsto {\left({\color{blue}{\left(1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}\right)}}^{3}\right)}^{0.3333333333333333} \]

   10. Simplified 99.1%

       \[\leadsto {\left({\color{blue}{\left(1 - \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(\frac{1.421413741}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{2}} + \frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{4}}\right)\right) - \left(\frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{3}} + \frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x\_m\right| \cdot 0.3275911\\ \mathbf{if}\;\left|x\_m\right| \leq 1.5 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{e^{-{x\_m}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{t\_0}^{4}} + 1.421413741 \cdot \frac{1}{{t\_0}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{t\_0} + 1.453152027 \cdot \frac{1}{{t\_0}^{3}}\right)\right)}{-1 - x\_m \cdot 0.3275911}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911))))
   (if (<= (fabs x_m) 1.5e-10)
     (+ 1e-9 (* x_m 1.128386358070218))
     (+
      1.0
      (/
       (*
        (exp (- (pow x_m 2.0)))
        (-
         (+
          0.254829592
          (+
           (* 1.061405429 (/ 1.0 (pow t_0 4.0)))
           (* 1.421413741 (/ 1.0 (pow t_0 2.0)))))
         (+
          (* 0.284496736 (/ 1.0 t_0))
          (* 1.453152027 (/ 1.0 (pow t_0 3.0))))))
       (- -1.0 (* x_m 0.3275911)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double tmp;
	if (fabs(x_m) <= 1.5e-10) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + ((exp(-pow(x_m, 2.0)) * ((0.254829592 + ((1.061405429 * (1.0 / pow(t_0, 4.0))) + (1.421413741 * (1.0 / pow(t_0, 2.0))))) - ((0.284496736 * (1.0 / t_0)) + (1.453152027 * (1.0 / pow(t_0, 3.0)))))) / (-1.0 - (x_m * 0.3275911)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    if (abs(x_m) <= 1.5d-10) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 1.0d0 + ((exp(-(x_m ** 2.0d0)) * ((0.254829592d0 + ((1.061405429d0 * (1.0d0 / (t_0 ** 4.0d0))) + (1.421413741d0 * (1.0d0 / (t_0 ** 2.0d0))))) - ((0.284496736d0 * (1.0d0 / t_0)) + (1.453152027d0 * (1.0d0 / (t_0 ** 3.0d0)))))) / ((-1.0d0) - (x_m * 0.3275911d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double tmp;
	if (Math.abs(x_m) <= 1.5e-10) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + ((Math.exp(-Math.pow(x_m, 2.0)) * ((0.254829592 + ((1.061405429 * (1.0 / Math.pow(t_0, 4.0))) + (1.421413741 * (1.0 / Math.pow(t_0, 2.0))))) - ((0.284496736 * (1.0 / t_0)) + (1.453152027 * (1.0 / Math.pow(t_0, 3.0)))))) / (-1.0 - (x_m * 0.3275911)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (math.fabs(x_m) * 0.3275911)
	tmp = 0
	if math.fabs(x_m) <= 1.5e-10:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 1.0 + ((math.exp(-math.pow(x_m, 2.0)) * ((0.254829592 + ((1.061405429 * (1.0 / math.pow(t_0, 4.0))) + (1.421413741 * (1.0 / math.pow(t_0, 2.0))))) - ((0.284496736 * (1.0 / t_0)) + (1.453152027 * (1.0 / math.pow(t_0, 3.0)))))) / (-1.0 - (x_m * 0.3275911)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	tmp = 0.0
	if (abs(x_m) <= 1.5e-10)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(-(x_m ^ 2.0))) * Float64(Float64(0.254829592 + Float64(Float64(1.061405429 * Float64(1.0 / (t_0 ^ 4.0))) + Float64(1.421413741 * Float64(1.0 / (t_0 ^ 2.0))))) - Float64(Float64(0.284496736 * Float64(1.0 / t_0)) + Float64(1.453152027 * Float64(1.0 / (t_0 ^ 3.0)))))) / Float64(-1.0 - Float64(x_m * 0.3275911))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (abs(x_m) * 0.3275911);
	tmp = 0.0;
	if (abs(x_m) <= 1.5e-10)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 1.0 + ((exp(-(x_m ^ 2.0)) * ((0.254829592 + ((1.061405429 * (1.0 / (t_0 ^ 4.0))) + (1.421413741 * (1.0 / (t_0 ^ 2.0))))) - ((0.284496736 * (1.0 / t_0)) + (1.453152027 * (1.0 / (t_0 ^ 3.0)))))) / (-1.0 - (x_m * 0.3275911)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1.5e-10], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision] * N[(N[(0.254829592 + N[(N[(1.061405429 * N[(1.0 / N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.421413741 * N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.284496736 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.453152027 * N[(1.0 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.5e-10

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   6. Applied egg-rr 57.4%

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

      1. +-commutative 57.4%

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

      2. neg-mul-1 57.4%

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

   8. Simplified 57.4%

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

   9. Taylor expanded in x around 0 99.4%

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

       1. *-commutative 99.4%

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

   11. Simplified 99.4%

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

   if 1.5e-10 < (fabs.f64 x)

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. expm1-log1p-u 99.0%

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

      2. log1p-define 99.0%

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

      3. +-commutative 99.0%

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

      4. fma-undefine 99.0%

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

      5. expm1-undefine 99.0%

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

      6. add-exp-log 99.0%

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

      7. add-sqr-sqrt 45.6%

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

      8. fabs-sqr 45.6%

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

      9. add-sqr-sqrt 98.1%

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

   7. Applied egg-rr 98.1%

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

      1. fma-undefine 98.1%

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

      2. associate--l+ 98.1%

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

      3. metadata-eval 98.1%

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

      4. +-rgt-identity 98.1%

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

   9. Simplified 98.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.5 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{e^{-{x}^{2}} \cdot \left(\left(0.254829592 + \left(1.061405429 \cdot \frac{1}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{4}} + 1.421413741 \cdot \frac{1}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{2}}\right)\right) - \left(0.284496736 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911} + 1.453152027 \cdot \frac{1}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{3}}\right)\right)}{-1 - x \cdot 0.3275911}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x\_m\right| \cdot 0.3275911}\\ \mathbf{if}\;\left|x\_m\right| \leq 1.5 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot {\left(\sqrt{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}\right)}^{2}\right)\right)\right) \cdot e^{x\_m \cdot \left(-x\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x_m) 0.3275911)))))
   (if (<= (fabs x_m) 1.5e-10)
     (+ 1e-9 (* x_m 1.128386358070218))
     (-
      1.0
      (*
       (*
        t_0
        (+
         0.254829592
         (*
          t_0
          (+
           -0.284496736
           (*
            t_0
            (pow
             (sqrt
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
                (fma 0.3275911 x_m 1.0))))
             2.0))))))
       (exp (* x_m (- x_m))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (fabs(x_m) * 0.3275911));
	double tmp;
	if (fabs(x_m) <= 1.5e-10) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * pow(sqrt((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)))), 2.0)))))) * exp((x_m * -x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))
	tmp = 0.0
	if (abs(x_m) <= 1.5e-10)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * (sqrt(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)))) ^ 2.0)))))) * exp(Float64(x_m * Float64(-x_m)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1.5e-10], N[(1e-9 + N[(x$95$m * 1.128386358070218), $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[Power[N[Sqrt[N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.5e-10

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   6. Applied egg-rr 57.4%

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

      1. +-commutative 57.4%

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

      2. neg-mul-1 57.4%

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

   8. Simplified 57.4%

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

   9. Taylor expanded in x around 0 99.4%

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

       1. *-commutative 99.4%

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

   11. Simplified 99.4%

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

   if 1.5e-10 < (fabs.f64 x)

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

      1. associate-*l/ 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. +-commutative 99.0%

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

      4. fma-undefine 99.0%

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

      5. +-commutative 99.0%

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

      6. fma-undefine 99.0%

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

      7. add-sqr-sqrt 99.1%

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

      8. pow2 99.1%

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

   6. Applied egg-rr 98.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.5 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot {\left(\sqrt{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}^{2}\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x\_m\right| \cdot 0.3275911}\\ \mathbf{if}\;x\_m \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + \frac{1}{1 + x\_m \cdot 0.3275911} \cdot \left(1.421413741 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x\_m, 1\right)\right)}^{2}} + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x_m) 0.3275911)))))
   (if (<= x_m 1.6e-10)
     (+ 1e-9 (* x_m 1.128386358070218))
     (-
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        t_0
        (+
         0.254829592
         (*
          t_0
          (+
           -0.284496736
           (*
            (/ 1.0 (+ 1.0 (* x_m 0.3275911)))
            (+
             1.421413741
             (+
              (/ 1.061405429 (pow (fma 0.3275911 x_m 1.0) 2.0))
              (/ -1.453152027 (fma 0.3275911 x_m 1.0))))))))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (fabs(x_m) * 0.3275911));
	double tmp;
	if (x_m <= 1.6e-10) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 - (exp((x_m * -x_m)) * (t_0 * (0.254829592 + (t_0 * (-0.284496736 + ((1.0 / (1.0 + (x_m * 0.3275911))) * (1.421413741 + ((1.061405429 / pow(fma(0.3275911, x_m, 1.0), 2.0)) + (-1.453152027 / fma(0.3275911, x_m, 1.0))))))))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))
	tmp = 0.0
	if (x_m <= 1.6e-10)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = Float64(1.0 - Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911))) * Float64(1.421413741 + Float64(Float64(1.061405429 / (fma(0.3275911, x_m, 1.0) ^ 2.0)) + Float64(-1.453152027 / fma(0.3275911, x_m, 1.0)))))))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.6e-10], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.421413741 + N[(N[(1.061405429 / N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.453152027 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5999999999999999e-10

   1. Initial program 73.5%

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

   3. Simplified 73.5%

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

   6. Applied egg-rr 36.9%

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

      1. +-commutative 36.9%

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

      2. neg-mul-1 36.9%

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

   8. Simplified 36.9%

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

   9. Taylor expanded in x around 0 62.2%

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

       1. *-commutative 62.2%

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

   11. Simplified 62.2%

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

   if 1.5999999999999999e-10 < x

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around 0 98.3%

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

      1. associate--l+ 98.5%

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

      2. sub-neg 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

      5. +-commutative 98.5%

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

      6. metadata-eval 98.5%

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

      7. fabs-mul 98.5%

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

      8. rem-square-sqrt 98.5%

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

      9. fabs-sqr 98.5%

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

      10. rem-square-sqrt 98.5%

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

      11. fma-undefine 98.5%

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

      12. associate-*r/ 98.4%

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

      13. metadata-eval 98.4%

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

      14. distribute-neg-frac 98.4%

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

      15. metadata-eval 98.4%

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

      16. +-commutative 98.4%

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

   7. Simplified 98.4%

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

      1. expm1-log1p-u 98.4%

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

      2. log1p-define 98.4%

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

      3. +-commutative 98.4%

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

      4. fma-undefine 98.4%

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

      5. expm1-undefine 98.4%

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

      6. add-exp-log 98.4%

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

      7. add-sqr-sqrt 98.4%

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

      8. fabs-sqr 98.4%

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

      9. add-sqr-sqrt 98.4%

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

   9. Applied egg-rr 98.4%

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

       1. fma-undefine 98.4%

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

       2. associate--l+ 98.4%

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

       3. metadata-eval 98.4%

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

       4. +-rgt-identity 98.4%

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

   11. Simplified 98.4%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{1 + t\_0}\\ \mathbf{if}\;x\_m \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(\left(0.254829592 + t\_1 \cdot \left(-0.284496736 + t\_1 \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(1 + x\_m \cdot 0.3275911\right)}^{2}}\right) + 1.453152027 \cdot \frac{1}{-1 - x\_m \cdot 0.3275911}\right)\right)\right) \cdot \frac{1}{-1 - t\_0}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)) (t_1 (/ 1.0 (+ 1.0 t_0))))
   (if (<= x_m 1.6e-10)
     (+ 1e-9 (* x_m 1.128386358070218))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+
             (+
              1.421413741
              (* 1.061405429 (/ 1.0 (pow (+ 1.0 (* x_m 0.3275911)) 2.0))))
             (* 1.453152027 (/ 1.0 (- -1.0 (* x_m 0.3275911)))))))))
        (/ 1.0 (- -1.0 t_0))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double t_1 = 1.0 / (1.0 + t_0);
	double tmp;
	if (x_m <= 1.6e-10) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((0.254829592 + (t_1 * (-0.284496736 + (t_1 * ((1.421413741 + (1.061405429 * (1.0 / pow((1.0 + (x_m * 0.3275911)), 2.0)))) + (1.453152027 * (1.0 / (-1.0 - (x_m * 0.3275911))))))))) * (1.0 / (-1.0 - t_0))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(x_m) * 0.3275911d0
    t_1 = 1.0d0 / (1.0d0 + t_0)
    if (x_m <= 1.6d-10) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 1.0d0 + (exp((x_m * -x_m)) * ((0.254829592d0 + (t_1 * ((-0.284496736d0) + (t_1 * ((1.421413741d0 + (1.061405429d0 * (1.0d0 / ((1.0d0 + (x_m * 0.3275911d0)) ** 2.0d0)))) + (1.453152027d0 * (1.0d0 / ((-1.0d0) - (x_m * 0.3275911d0))))))))) * (1.0d0 / ((-1.0d0) - t_0))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.abs(x_m) * 0.3275911;
	double t_1 = 1.0 / (1.0 + t_0);
	double tmp;
	if (x_m <= 1.6e-10) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + (Math.exp((x_m * -x_m)) * ((0.254829592 + (t_1 * (-0.284496736 + (t_1 * ((1.421413741 + (1.061405429 * (1.0 / Math.pow((1.0 + (x_m * 0.3275911)), 2.0)))) + (1.453152027 * (1.0 / (-1.0 - (x_m * 0.3275911))))))))) * (1.0 / (-1.0 - t_0))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.fabs(x_m) * 0.3275911
	t_1 = 1.0 / (1.0 + t_0)
	tmp = 0
	if x_m <= 1.6e-10:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 1.0 + (math.exp((x_m * -x_m)) * ((0.254829592 + (t_1 * (-0.284496736 + (t_1 * ((1.421413741 + (1.061405429 * (1.0 / math.pow((1.0 + (x_m * 0.3275911)), 2.0)))) + (1.453152027 * (1.0 / (-1.0 - (x_m * 0.3275911))))))))) * (1.0 / (-1.0 - t_0))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	t_1 = Float64(1.0 / Float64(1.0 + t_0))
	tmp = 0.0
	if (x_m <= 1.6e-10)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(Float64(1.421413741 + Float64(1.061405429 * Float64(1.0 / (Float64(1.0 + Float64(x_m * 0.3275911)) ^ 2.0)))) + Float64(1.453152027 * Float64(1.0 / Float64(-1.0 - Float64(x_m * 0.3275911))))))))) * Float64(1.0 / Float64(-1.0 - t_0)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = abs(x_m) * 0.3275911;
	t_1 = 1.0 / (1.0 + t_0);
	tmp = 0.0;
	if (x_m <= 1.6e-10)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 1.0 + (exp((x_m * -x_m)) * ((0.254829592 + (t_1 * (-0.284496736 + (t_1 * ((1.421413741 + (1.061405429 * (1.0 / ((1.0 + (x_m * 0.3275911)) ^ 2.0)))) + (1.453152027 * (1.0 / (-1.0 - (x_m * 0.3275911))))))))) * (1.0 / (-1.0 - t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.6e-10], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(N[(1.421413741 + N[(1.061405429 * N[(1.0 / N[Power[N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.453152027 * N[(1.0 / N[(-1.0 - N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5999999999999999e-10

   1. Initial program 73.5%

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

   3. Simplified 73.5%

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

   6. Applied egg-rr 36.9%

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

      1. +-commutative 36.9%

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

      2. neg-mul-1 36.9%

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

   8. Simplified 36.9%

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

   9. Taylor expanded in x around 0 62.2%

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

       1. *-commutative 62.2%

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

   11. Simplified 62.2%

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

   if 1.5999999999999999e-10 < x

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around 0 98.3%

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

      1. expm1-log1p-u 98.4%

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

      2. log1p-define 98.4%

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

      3. +-commutative 98.4%

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

      4. fma-undefine 98.4%

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

      5. expm1-undefine 98.4%

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

      6. add-exp-log 98.4%

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

      7. add-sqr-sqrt 98.4%

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

      8. fabs-sqr 98.4%

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

      9. add-sqr-sqrt 98.4%

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

   7. Applied egg-rr 98.3%

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

      1. fma-undefine 98.4%

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

      2. associate--l+ 98.4%

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

      3. metadata-eval 98.4%

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

      4. +-rgt-identity 98.4%

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

   9. Simplified 98.3%

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

       1. expm1-log1p-u 98.4%

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

       2. log1p-define 98.4%

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

       3. +-commutative 98.4%

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

       4. fma-undefine 98.4%

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

       5. expm1-undefine 98.4%

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

       6. add-exp-log 98.4%

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

       7. add-sqr-sqrt 98.4%

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

       8. fabs-sqr 98.4%

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

       9. add-sqr-sqrt 98.4%

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

   11. Applied egg-rr 98.3%

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

       1. fma-undefine 98.4%

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

       2. associate--l+ 98.4%

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

       3. metadata-eval 98.4%

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

       4. +-rgt-identity 98.4%

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

   13. Simplified 98.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(1 + x \cdot 0.3275911\right)}^{2}}\right) + 1.453152027 \cdot \frac{1}{-1 - x \cdot 0.3275911}\right)\right)\right) \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{1 + t\_0}\\ t_2 := 1 + x\_m \cdot 0.3275911\\ \mathbf{if}\;x\_m \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(t\_1 \cdot \left(t\_1 \cdot \left(\left(1.421413741 + \frac{1}{t\_2} \cdot \left(-1.453152027 + \frac{1.061405429}{t\_2}\right)\right) \cdot \frac{1}{-1 - t\_0} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911))
        (t_1 (/ 1.0 (+ 1.0 t_0)))
        (t_2 (+ 1.0 (* x_m 0.3275911))))
   (if (<= x_m 1.6e-10)
     (+ 1e-9 (* x_m 1.128386358070218))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        t_1
        (-
         (*
          t_1
          (-
           (*
            (+
             1.421413741
             (* (/ 1.0 t_2) (+ -1.453152027 (/ 1.061405429 t_2))))
            (/ 1.0 (- -1.0 t_0)))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double t_1 = 1.0 / (1.0 + t_0);
	double t_2 = 1.0 + (x_m * 0.3275911);
	double tmp;
	if (x_m <= 1.6e-10) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * (t_1 * ((t_1 * (((1.421413741 + ((1.0 / t_2) * (-1.453152027 + (1.061405429 / t_2)))) * (1.0 / (-1.0 - t_0))) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = abs(x_m) * 0.3275911d0
    t_1 = 1.0d0 / (1.0d0 + t_0)
    t_2 = 1.0d0 + (x_m * 0.3275911d0)
    if (x_m <= 1.6d-10) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 1.0d0 + (exp((x_m * -x_m)) * (t_1 * ((t_1 * (((1.421413741d0 + ((1.0d0 / t_2) * ((-1.453152027d0) + (1.061405429d0 / t_2)))) * (1.0d0 / ((-1.0d0) - t_0))) - (-0.284496736d0))) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.abs(x_m) * 0.3275911;
	double t_1 = 1.0 / (1.0 + t_0);
	double t_2 = 1.0 + (x_m * 0.3275911);
	double tmp;
	if (x_m <= 1.6e-10) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 + (Math.exp((x_m * -x_m)) * (t_1 * ((t_1 * (((1.421413741 + ((1.0 / t_2) * (-1.453152027 + (1.061405429 / t_2)))) * (1.0 / (-1.0 - t_0))) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.fabs(x_m) * 0.3275911
	t_1 = 1.0 / (1.0 + t_0)
	t_2 = 1.0 + (x_m * 0.3275911)
	tmp = 0
	if x_m <= 1.6e-10:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 1.0 + (math.exp((x_m * -x_m)) * (t_1 * ((t_1 * (((1.421413741 + ((1.0 / t_2) * (-1.453152027 + (1.061405429 / t_2)))) * (1.0 / (-1.0 - t_0))) - -0.284496736)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	t_1 = Float64(1.0 / Float64(1.0 + t_0))
	t_2 = Float64(1.0 + Float64(x_m * 0.3275911))
	tmp = 0.0
	if (x_m <= 1.6e-10)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(t_1 * Float64(Float64(t_1 * Float64(Float64(Float64(1.421413741 + Float64(Float64(1.0 / t_2) * Float64(-1.453152027 + Float64(1.061405429 / t_2)))) * Float64(1.0 / Float64(-1.0 - t_0))) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = abs(x_m) * 0.3275911;
	t_1 = 1.0 / (1.0 + t_0);
	t_2 = 1.0 + (x_m * 0.3275911);
	tmp = 0.0;
	if (x_m <= 1.6e-10)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 1.0 + (exp((x_m * -x_m)) * (t_1 * ((t_1 * (((1.421413741 + ((1.0 / t_2) * (-1.453152027 + (1.061405429 / t_2)))) * (1.0 / (-1.0 - t_0))) - -0.284496736)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.6e-10], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(N[(t$95$1 * N[(N[(N[(1.421413741 + N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(-1.453152027 + N[(1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5999999999999999e-10

   1. Initial program 73.5%

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

   3. Simplified 73.5%

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

   6. Applied egg-rr 36.9%

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

      1. +-commutative 36.9%

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

      2. neg-mul-1 36.9%

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

   8. Simplified 36.9%

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

   9. Taylor expanded in x around 0 62.2%

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

       1. *-commutative 62.2%

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

   11. Simplified 62.2%

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

   if 1.5999999999999999e-10 < x

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

      1. expm1-log1p-u 98.4%

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

      2. log1p-define 98.4%

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

      3. +-commutative 98.4%

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

      4. fma-undefine 98.4%

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

      5. expm1-undefine 98.4%

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

      6. add-exp-log 98.4%

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

      7. add-sqr-sqrt 98.4%

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

      8. fabs-sqr 98.4%

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

      9. add-sqr-sqrt 98.4%

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

   6. Applied egg-rr 98.4%

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

      1. fma-undefine 98.4%

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

      2. associate--l+ 98.4%

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

      3. metadata-eval 98.4%

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

      4. +-rgt-identity 98.4%

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

   8. Simplified 98.4%

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

      1. expm1-log1p-u 98.4%

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

      2. log1p-define 98.4%

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

      3. +-commutative 98.4%

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

      4. fma-undefine 98.4%

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

      5. expm1-undefine 98.4%

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

      6. add-exp-log 98.4%

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

      7. add-sqr-sqrt 98.4%

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

      8. fabs-sqr 98.4%

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

      9. add-sqr-sqrt 98.4%

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

   10. Applied egg-rr 98.4%

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

       1. fma-undefine 98.4%

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

       2. associate--l+ 98.4%

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

       3. metadata-eval 98.4%

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

       4. +-rgt-identity 98.4%

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

   12. Simplified 98.4%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right) \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% accurate, 85.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.6e-10) (+ 1e-9 (* x_m 1.128386358070218)) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.6e-10) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.6d-10) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.6e-10) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.6e-10:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.6e-10)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.6e-10)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.6e-10], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.5999999999999999e-10

   1. Initial program 73.5%

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

   3. Simplified 73.5%

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

   6. Applied egg-rr 36.9%

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

      1. +-commutative 36.9%

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

      2. neg-mul-1 36.9%

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

   8. Simplified 36.9%

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

   9. Taylor expanded in x around 0 62.2%

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

       1. *-commutative 62.2%

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

   11. Simplified 62.2%

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

   if 1.5999999999999999e-10 < x

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

   6. Applied egg-rr 3.3%

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

      1. +-commutative 3.3%

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

      2. neg-mul-1 3.3%

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

   8. Simplified 3.3%

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

   9. Taylor expanded in x around inf 94.5%

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

Alternative 9: 97.5% accurate, 142.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.5 \cdot 10^{-16}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.5e-16) 1e-9 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.5e-16) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.5d-16) then
        tmp = 1d-9
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.5e-16) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.5e-16:
		tmp = 1e-9
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.5e-16)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.5e-16)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.5e-16], 1e-9, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.5000000000000002e-16

   1. Initial program 73.9%

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

   3. Simplified 73.9%

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

   6. Applied egg-rr 36.5%

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

      1. +-commutative 36.5%

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

      2. neg-mul-1 36.5%

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

   8. Simplified 36.5%

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

   9. Taylor expanded in x around 0 65.3%

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

   if 2.5000000000000002e-16 < x

   1. Initial program 95.8%

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

   3. Simplified 95.8%

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

   6. Applied egg-rr 6.3%

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

      1. +-commutative 6.3%

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

      2. neg-mul-1 6.3%

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

   8. Simplified 6.3%

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

   9. Taylor expanded in x around inf 89.5%

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

Alternative 10: 52.9% accurate, 856.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

3. Simplified 79.6%

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

6. Applied egg-rr 28.6%

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

   1. +-commutative 28.6%

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

   2. neg-mul-1 28.6%

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

8. Simplified 28.6%

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

9. Taylor expanded in x around 0 51.6%

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

Reproduce

herbie shell --seed 2024103 
(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)))))))