Jmat.Real.erf

Percentage Accurate: 78.9% → 99.2%

Time: 23.8s

Alternatives: 8

Speedup: 7.4×

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: 78.9% 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: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{1 + t\_0}\\ \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;10^{-9} + \left(x\_m \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot t\_1\right) + \left({x\_m}^{2} \cdot \left(t\_1 \cdot 0.36953108532122814 - 0.10731592869189407\right) + {x\_m}^{3} \cdot \left(0.03515574312769914 + 0.3754899882585643 \cdot \frac{1}{-1 - t\_0}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-0.7778892405807117}{e^{{x\_m}^{2}}}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) 0.3275911)) (t_1 (/ 1.0 (+ 1.0 t_0))))
   (if (<= (fabs x_m) 2e-5)
     (+
      1e-9
      (+
       (* x_m (+ 0.3275910996724089 (* 0.8007952583978091 t_1)))
       (+
        (* (pow x_m 2.0) (- (* t_1 0.36953108532122814) 0.10731592869189407))
        (*
         (pow x_m 3.0)
         (+
          0.03515574312769914
          (* 0.3754899882585643 (/ 1.0 (- -1.0 t_0))))))))
     (+ 1.0 (/ (/ -0.7778892405807117 (exp (pow x_m 2.0))) x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * 0.3275911;
	double t_1 = 1.0 / (1.0 + t_0);
	double tmp;
	if (fabs(x_m) <= 2e-5) {
		tmp = 1e-9 + ((x_m * (0.3275910996724089 + (0.8007952583978091 * t_1))) + ((pow(x_m, 2.0) * ((t_1 * 0.36953108532122814) - 0.10731592869189407)) + (pow(x_m, 3.0) * (0.03515574312769914 + (0.3754899882585643 * (1.0 / (-1.0 - t_0)))))));
	} else {
		tmp = 1.0 + ((-0.7778892405807117 / exp(pow(x_m, 2.0))) / x_m);
	}
	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 (abs(x_m) <= 2d-5) then
        tmp = 1d-9 + ((x_m * (0.3275910996724089d0 + (0.8007952583978091d0 * t_1))) + (((x_m ** 2.0d0) * ((t_1 * 0.36953108532122814d0) - 0.10731592869189407d0)) + ((x_m ** 3.0d0) * (0.03515574312769914d0 + (0.3754899882585643d0 * (1.0d0 / ((-1.0d0) - t_0)))))))
    else
        tmp = 1.0d0 + (((-0.7778892405807117d0) / exp((x_m ** 2.0d0))) / x_m)
    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 (Math.abs(x_m) <= 2e-5) {
		tmp = 1e-9 + ((x_m * (0.3275910996724089 + (0.8007952583978091 * t_1))) + ((Math.pow(x_m, 2.0) * ((t_1 * 0.36953108532122814) - 0.10731592869189407)) + (Math.pow(x_m, 3.0) * (0.03515574312769914 + (0.3754899882585643 * (1.0 / (-1.0 - t_0)))))));
	} else {
		tmp = 1.0 + ((-0.7778892405807117 / Math.exp(Math.pow(x_m, 2.0))) / x_m);
	}
	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 math.fabs(x_m) <= 2e-5:
		tmp = 1e-9 + ((x_m * (0.3275910996724089 + (0.8007952583978091 * t_1))) + ((math.pow(x_m, 2.0) * ((t_1 * 0.36953108532122814) - 0.10731592869189407)) + (math.pow(x_m, 3.0) * (0.03515574312769914 + (0.3754899882585643 * (1.0 / (-1.0 - t_0)))))))
	else:
		tmp = 1.0 + ((-0.7778892405807117 / math.exp(math.pow(x_m, 2.0))) / x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * 0.3275911)
	t_1 = Float64(1.0 / Float64(1.0 + t_0))
	tmp = 0.0
	if (abs(x_m) <= 2e-5)
		tmp = Float64(1e-9 + Float64(Float64(x_m * Float64(0.3275910996724089 + Float64(0.8007952583978091 * t_1))) + Float64(Float64((x_m ^ 2.0) * Float64(Float64(t_1 * 0.36953108532122814) - 0.10731592869189407)) + Float64((x_m ^ 3.0) * Float64(0.03515574312769914 + Float64(0.3754899882585643 * Float64(1.0 / Float64(-1.0 - t_0))))))));
	else
		tmp = Float64(1.0 + Float64(Float64(-0.7778892405807117 / exp((x_m ^ 2.0))) / x_m));
	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 (abs(x_m) <= 2e-5)
		tmp = 1e-9 + ((x_m * (0.3275910996724089 + (0.8007952583978091 * t_1))) + (((x_m ^ 2.0) * ((t_1 * 0.36953108532122814) - 0.10731592869189407)) + ((x_m ^ 3.0) * (0.03515574312769914 + (0.3754899882585643 * (1.0 / (-1.0 - t_0)))))));
	else
		tmp = 1.0 + ((-0.7778892405807117 / exp((x_m ^ 2.0))) / x_m);
	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[N[Abs[x$95$m], $MachinePrecision], 2e-5], N[(1e-9 + N[(N[(x$95$m * N[(0.3275910996724089 + N[(0.8007952583978091 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$1 * 0.36953108532122814), $MachinePrecision] - 0.10731592869189407), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * N[(0.03515574312769914 + N[(0.3754899882585643 * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-0.7778892405807117 / N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000016e-5

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 56.8%

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

      1. associate-*l/ 56.8%

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

      2. metadata-eval 56.8%

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

   8. Simplified 56.8%

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

   9. Taylor expanded in x around 0 56.8%

      \[\leadsto \color{blue}{\left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 56.8%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

       2. log1p-define 56.8%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

       3. +-commutative 56.8%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

       4. fma-undefine 56.8%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

       5. expm1-undefine 56.8%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

       6. add-exp-log 56.8%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

       7. add-sqr-sqrt 28.4%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

       8. fabs-sqr 28.4%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

       9. add-sqr-sqrt 56.8%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   11. Applied egg-rr 56.8%

       \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   12. Step-by-step derivation

       1. fma-undefine 56.8%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 56.8%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 56.8%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 56.8%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Simplified 56.8%

       \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 97.9%

       \[\leadsto \color{blue}{10^{-9} + \left(x \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + \left({x}^{2} \cdot \left(0.36953108532122814 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 0.10731592869189407\right) + {x}^{3} \cdot \left(0.03515574312769914 - 0.3754899882585643 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)} \]

   if 2.00000000000000016e-5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. expm1-log1p-u 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

       2. log1p-define 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

       3. +-commutative 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

       4. fma-undefine 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

       5. expm1-undefine 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

       6. add-exp-log 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

       7. add-sqr-sqrt 0.3%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

       8. fabs-sqr 0.3%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

       9. add-sqr-sqrt 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   11. Applied egg-rr 100.0%

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

       1. fma-undefine 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Simplified 100.0%

       \[\leadsto 1 + -0.254829592 \cdot \frac{e^{-1 \cdot {x}^{2}}}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around inf 100.0%

       \[\leadsto 1 + \color{blue}{-0.7778892405807117 \cdot \frac{e^{-1 \cdot {x}^{2}}}{x}} \]
   15. Step-by-step derivation

       1. associate-*r/ 100.0%

          \[\leadsto 1 + \color{blue}{\frac{-0.7778892405807117 \cdot e^{-1 \cdot {x}^{2}}}{x}} \]

       2. neg-mul-1 100.0%

          \[\leadsto 1 + \frac{-0.7778892405807117 \cdot e^{\color{blue}{-{x}^{2}}}}{x} \]

       3. exp-neg 100.0%

          \[\leadsto 1 + \frac{-0.7778892405807117 \cdot \color{blue}{\frac{1}{e^{{x}^{2}}}}}{x} \]

       4. associate-*r/ 100.0%

          \[\leadsto 1 + \frac{\color{blue}{\frac{-0.7778892405807117 \cdot 1}{e^{{x}^{2}}}}}{x} \]

       5. metadata-eval 100.0%

          \[\leadsto 1 + \frac{\frac{\color{blue}{-0.7778892405807117}}{e^{{x}^{2}}}}{x} \]

   16. Simplified 100.0%

       \[\leadsto 1 + \color{blue}{\frac{\frac{-0.7778892405807117}{e^{{x}^{2}}}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;10^{-9} + \left(x \cdot \left(0.3275910996724089 + 0.8007952583978091 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right) + \left({x}^{2} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.36953108532122814 - 0.10731592869189407\right) + {x}^{3} \cdot \left(0.03515574312769914 + 0.3754899882585643 \cdot \frac{1}{-1 - \left|x\right| \cdot 0.3275911}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-0.7778892405807117}{e^{{x}^{2}}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x\_m\right| \cdot 0.3275911}\\ \mathbf{if}\;\left|x\_m\right| \leq 10^{-8}:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(x\_m \cdot -0.00011824251945160904 + 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(t\_0 \cdot \left(t\_0 \cdot \left(t\_0 \cdot \left(\frac{1}{1 + x\_m \cdot 0.3275911} \cdot \left(\frac{1.061405429}{-1 - x\_m \cdot 0.3275911} - -1.453152027\right) - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x_m) 0.3275911)))))
   (if (<= (fabs x_m) 1e-8)
     (+ 1e-9 (* x_m (+ (* x_m -0.00011824251945160904) 1.128386358070218)))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        t_0
        (-
         (*
          t_0
          (-
           (*
            t_0
            (-
             (*
              (/ 1.0 (+ 1.0 (* x_m 0.3275911)))
              (- (/ 1.061405429 (- -1.0 (* x_m 0.3275911))) -1.453152027))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (fabs(x_m) * 0.3275911));
	double tmp;
	if (fabs(x_m) <= 1e-8) {
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * (t_0 * ((t_0 * ((t_0 * (((1.0 / (1.0 + (x_m * 0.3275911))) * ((1.061405429 / (-1.0 - (x_m * 0.3275911))) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (1.0d0 + (abs(x_m) * 0.3275911d0))
    if (abs(x_m) <= 1d-8) then
        tmp = 1d-9 + (x_m * ((x_m * (-0.00011824251945160904d0)) + 1.128386358070218d0))
    else
        tmp = 1.0d0 + (exp((x_m * -x_m)) * (t_0 * ((t_0 * ((t_0 * (((1.0d0 / (1.0d0 + (x_m * 0.3275911d0))) * ((1.061405429d0 / ((-1.0d0) - (x_m * 0.3275911d0))) - (-1.453152027d0))) - 1.421413741d0)) - (-0.284496736d0))) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (Math.abs(x_m) * 0.3275911));
	double tmp;
	if (Math.abs(x_m) <= 1e-8) {
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218));
	} else {
		tmp = 1.0 + (Math.exp((x_m * -x_m)) * (t_0 * ((t_0 * ((t_0 * (((1.0 / (1.0 + (x_m * 0.3275911))) * ((1.061405429 / (-1.0 - (x_m * 0.3275911))) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 / (1.0 + (math.fabs(x_m) * 0.3275911))
	tmp = 0
	if math.fabs(x_m) <= 1e-8:
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218))
	else:
		tmp = 1.0 + (math.exp((x_m * -x_m)) * (t_0 * ((t_0 * ((t_0 * (((1.0 / (1.0 + (x_m * 0.3275911))) * ((1.061405429 / (-1.0 - (x_m * 0.3275911))) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))
	tmp = 0.0
	if (abs(x_m) <= 1e-8)
		tmp = Float64(1e-9 + Float64(x_m * Float64(Float64(x_m * -0.00011824251945160904) + 1.128386358070218)));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(t_0 * Float64(Float64(t_0 * Float64(Float64(t_0 * Float64(Float64(Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911))) * Float64(Float64(1.061405429 / Float64(-1.0 - Float64(x_m * 0.3275911))) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 / (1.0 + (abs(x_m) * 0.3275911));
	tmp = 0.0;
	if (abs(x_m) <= 1e-8)
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218));
	else
		tmp = 1.0 + (exp((x_m * -x_m)) * (t_0 * ((t_0 * ((t_0 * (((1.0 / (1.0 + (x_m * 0.3275911))) * ((1.061405429 / (-1.0 - (x_m * 0.3275911))) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-8], N[(1e-9 + N[(x$95$m * N[(N[(x$95$m * -0.00011824251945160904), $MachinePrecision] + 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(N[(t$95$0 * N[(N[(t$95$0 * N[(N[(N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.061405429 / N[(-1.0 - N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.453152027), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1e-8

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

   6. Applied egg-rr 57.3%

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

      1. +-commutative 57.3%

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

      2. associate-+l+ 57.3%

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

   8. Simplified 57.3%

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

   9. Taylor expanded in x around 0 98.7%

      \[\leadsto \frac{\color{blue}{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}}{{\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2} + \left(1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}\right)} \]

   10. Taylor expanded in x around 0 97.3%

       \[\leadsto \frac{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}{\color{blue}{2.999999997 + -3.3851590719538813 \cdot x}} \]
   11. Step-by-step derivation

       1. *-commutative 97.3%

          \[\leadsto \frac{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}{2.999999997 + \color{blue}{x \cdot -3.3851590719538813}} \]

   12. Simplified 97.3%

       \[\leadsto \frac{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}{\color{blue}{2.999999997 + x \cdot -3.3851590719538813}} \]

   13. Taylor expanded in x around 0 98.7%

       \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824251945160904 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   14. Step-by-step derivation

       1. unpow2 98.7%

          \[\leadsto 10^{-9} + \left(-0.00011824251945160904 \cdot \color{blue}{\left(x \cdot x\right)} + 1.128386358070218 \cdot x\right) \]

       2. associate-*r* 98.7%

          \[\leadsto 10^{-9} + \left(\color{blue}{\left(-0.00011824251945160904 \cdot x\right) \cdot x} + 1.128386358070218 \cdot x\right) \]

       3. distribute-rgt-out 98.7%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot \left(-0.00011824251945160904 \cdot x + 1.128386358070218\right)} \]

   15. Simplified 98.7%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(-0.00011824251945160904 \cdot x + 1.128386358070218\right)} \]

   if 1e-8 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

      2. log1p-define 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

      3. +-commutative 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

      4. fma-undefine 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

      5. expm1-undefine 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

      6. add-exp-log 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

      7. add-sqr-sqrt 0.3%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

      8. fabs-sqr 0.3%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   6. Applied egg-rr 99.4%

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

      1. fma-undefine 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

      2. associate--l+ 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

      3. metadata-eval 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

      4. +-rgt-identity 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   8. Simplified 99.4%

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

      1. expm1-log1p-u 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

      2. log1p-define 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

      3. +-commutative 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

      4. fma-undefine 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

      5. expm1-undefine 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

      6. add-exp-log 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

      7. add-sqr-sqrt 0.3%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

      8. fabs-sqr 0.3%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 2.5%

         \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 99.4%

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

       1. fma-undefine 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 99.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-8}:\\ \;\;\;\;10^{-9} + x \cdot \left(x \cdot -0.00011824251945160904 + 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(\frac{1.061405429}{-1 - x \cdot 0.3275911} - -1.453152027\right) - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(x\_m \cdot -0.00011824251945160904 + 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-0.7778892405807117}{e^{{x\_m}^{2}}}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 2e-5)
   (+ 1e-9 (* x_m (+ (* x_m -0.00011824251945160904) 1.128386358070218)))
   (+ 1.0 (/ (/ -0.7778892405807117 (exp (pow x_m 2.0))) x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 2e-5) {
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218));
	} else {
		tmp = 1.0 + ((-0.7778892405807117 / exp(pow(x_m, 2.0))) / x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (abs(x_m) <= 2d-5) then
        tmp = 1d-9 + (x_m * ((x_m * (-0.00011824251945160904d0)) + 1.128386358070218d0))
    else
        tmp = 1.0d0 + (((-0.7778892405807117d0) / exp((x_m ** 2.0d0))) / x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 2e-5) {
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218));
	} else {
		tmp = 1.0 + ((-0.7778892405807117 / Math.exp(Math.pow(x_m, 2.0))) / x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 2e-5:
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218))
	else:
		tmp = 1.0 + ((-0.7778892405807117 / math.exp(math.pow(x_m, 2.0))) / x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 2e-5)
		tmp = Float64(1e-9 + Float64(x_m * Float64(Float64(x_m * -0.00011824251945160904) + 1.128386358070218)));
	else
		tmp = Float64(1.0 + Float64(Float64(-0.7778892405807117 / exp((x_m ^ 2.0))) / x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 2e-5)
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218));
	else
		tmp = 1.0 + ((-0.7778892405807117 / exp((x_m ^ 2.0))) / x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-5], N[(1e-9 + N[(x$95$m * N[(N[(x$95$m * -0.00011824251945160904), $MachinePrecision] + 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-0.7778892405807117 / N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000016e-5

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 56.8%

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

      1. +-commutative 56.8%

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

      2. associate-+l+ 56.8%

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

   8. Simplified 56.8%

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

   9. Taylor expanded in x around 0 97.9%

      \[\leadsto \frac{\color{blue}{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}}{{\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2} + \left(1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}\right)} \]

   10. Taylor expanded in x around 0 96.5%

       \[\leadsto \frac{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}{\color{blue}{2.999999997 + -3.3851590719538813 \cdot x}} \]
   11. Step-by-step derivation

       1. *-commutative 96.5%

          \[\leadsto \frac{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}{2.999999997 + \color{blue}{x \cdot -3.3851590719538813}} \]

   12. Simplified 96.5%

       \[\leadsto \frac{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}{\color{blue}{2.999999997 + x \cdot -3.3851590719538813}} \]

   13. Taylor expanded in x around 0 97.9%

       \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824251945160904 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   14. Step-by-step derivation

       1. unpow2 97.9%

          \[\leadsto 10^{-9} + \left(-0.00011824251945160904 \cdot \color{blue}{\left(x \cdot x\right)} + 1.128386358070218 \cdot x\right) \]

       2. associate-*r* 97.9%

          \[\leadsto 10^{-9} + \left(\color{blue}{\left(-0.00011824251945160904 \cdot x\right) \cdot x} + 1.128386358070218 \cdot x\right) \]

       3. distribute-rgt-out 97.9%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot \left(-0.00011824251945160904 \cdot x + 1.128386358070218\right)} \]

   15. Simplified 97.9%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(-0.00011824251945160904 \cdot x + 1.128386358070218\right)} \]

   if 2.00000000000000016e-5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. expm1-log1p-u 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

       2. log1p-define 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

       3. +-commutative 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

       4. fma-undefine 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

       5. expm1-undefine 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

       6. add-exp-log 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

       7. add-sqr-sqrt 0.3%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

       8. fabs-sqr 0.3%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

       9. add-sqr-sqrt 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   11. Applied egg-rr 100.0%

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

       1. fma-undefine 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Simplified 100.0%

       \[\leadsto 1 + -0.254829592 \cdot \frac{e^{-1 \cdot {x}^{2}}}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around inf 100.0%

       \[\leadsto 1 + \color{blue}{-0.7778892405807117 \cdot \frac{e^{-1 \cdot {x}^{2}}}{x}} \]
   15. Step-by-step derivation

       1. associate-*r/ 100.0%

          \[\leadsto 1 + \color{blue}{\frac{-0.7778892405807117 \cdot e^{-1 \cdot {x}^{2}}}{x}} \]

       2. neg-mul-1 100.0%

          \[\leadsto 1 + \frac{-0.7778892405807117 \cdot e^{\color{blue}{-{x}^{2}}}}{x} \]

       3. exp-neg 100.0%

          \[\leadsto 1 + \frac{-0.7778892405807117 \cdot \color{blue}{\frac{1}{e^{{x}^{2}}}}}{x} \]

       4. associate-*r/ 100.0%

          \[\leadsto 1 + \frac{\color{blue}{\frac{-0.7778892405807117 \cdot 1}{e^{{x}^{2}}}}}{x} \]

       5. metadata-eval 100.0%

          \[\leadsto 1 + \frac{\frac{\color{blue}{-0.7778892405807117}}{e^{{x}^{2}}}}{x} \]

   16. Simplified 100.0%

       \[\leadsto 1 + \color{blue}{\frac{\frac{-0.7778892405807117}{e^{{x}^{2}}}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;10^{-9} + x \cdot \left(x \cdot -0.00011824251945160904 + 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-0.7778892405807117}{e^{{x}^{2}}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(x\_m \cdot -0.00011824251945160904 + 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{1 + x\_m \cdot 0.3275911} \cdot -0.254829592\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 2e-5)
   (+ 1e-9 (* x_m (+ (* x_m -0.00011824251945160904) 1.128386358070218)))
   (+ 1.0 (* (/ 1.0 (+ 1.0 (* x_m 0.3275911))) -0.254829592))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 2e-5) {
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218));
	} else {
		tmp = 1.0 + ((1.0 / (1.0 + (x_m * 0.3275911))) * -0.254829592);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (abs(x_m) <= 2d-5) then
        tmp = 1d-9 + (x_m * ((x_m * (-0.00011824251945160904d0)) + 1.128386358070218d0))
    else
        tmp = 1.0d0 + ((1.0d0 / (1.0d0 + (x_m * 0.3275911d0))) * (-0.254829592d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 2e-5) {
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218));
	} else {
		tmp = 1.0 + ((1.0 / (1.0 + (x_m * 0.3275911))) * -0.254829592);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 2e-5:
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218))
	else:
		tmp = 1.0 + ((1.0 / (1.0 + (x_m * 0.3275911))) * -0.254829592)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 2e-5)
		tmp = Float64(1e-9 + Float64(x_m * Float64(Float64(x_m * -0.00011824251945160904) + 1.128386358070218)));
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911))) * -0.254829592));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 2e-5)
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218));
	else
		tmp = 1.0 + ((1.0 / (1.0 + (x_m * 0.3275911))) * -0.254829592);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-5], N[(1e-9 + N[(x$95$m * N[(N[(x$95$m * -0.00011824251945160904), $MachinePrecision] + 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.254829592), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000016e-5

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 56.8%

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

      1. +-commutative 56.8%

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

      2. associate-+l+ 56.8%

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

   8. Simplified 56.8%

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

   9. Taylor expanded in x around 0 97.9%

      \[\leadsto \frac{\color{blue}{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}}{{\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2} + \left(1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}\right)} \]

   10. Taylor expanded in x around 0 96.5%

       \[\leadsto \frac{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}{\color{blue}{2.999999997 + -3.3851590719538813 \cdot x}} \]
   11. Step-by-step derivation

       1. *-commutative 96.5%

          \[\leadsto \frac{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}{2.999999997 + \color{blue}{x \cdot -3.3851590719538813}} \]

   12. Simplified 96.5%

       \[\leadsto \frac{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}{\color{blue}{2.999999997 + x \cdot -3.3851590719538813}} \]

   13. Taylor expanded in x around 0 97.9%

       \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824251945160904 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   14. Step-by-step derivation

       1. unpow2 97.9%

          \[\leadsto 10^{-9} + \left(-0.00011824251945160904 \cdot \color{blue}{\left(x \cdot x\right)} + 1.128386358070218 \cdot x\right) \]

       2. associate-*r* 97.9%

          \[\leadsto 10^{-9} + \left(\color{blue}{\left(-0.00011824251945160904 \cdot x\right) \cdot x} + 1.128386358070218 \cdot x\right) \]

       3. distribute-rgt-out 97.9%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot \left(-0.00011824251945160904 \cdot x + 1.128386358070218\right)} \]

   15. Simplified 97.9%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(-0.00011824251945160904 \cdot x + 1.128386358070218\right)} \]

   if 2.00000000000000016e-5 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. expm1-log1p-u 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

       2. log1p-define 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

       3. +-commutative 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

       4. fma-undefine 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

       5. expm1-undefine 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

       6. add-exp-log 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

       7. add-sqr-sqrt 0.3%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

       8. fabs-sqr 0.3%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

       9. add-sqr-sqrt 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   11. Applied egg-rr 100.0%

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

       1. fma-undefine 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 2.5%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Simplified 100.0%

       \[\leadsto 1 + -0.254829592 \cdot \frac{e^{-1 \cdot {x}^{2}}}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 97.6%

       \[\leadsto 1 + -0.254829592 \cdot \frac{\color{blue}{1}}{1 + 0.3275911 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;10^{-9} + x \cdot \left(x \cdot -0.00011824251945160904 + 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{1 + x \cdot 0.3275911} \cdot -0.254829592\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.0% accurate, 61.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.65:\\ \;\;\;\;10^{-9} + x\_m \cdot \left(x\_m \cdot -0.00011824251945160904 + 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;0.745170408\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.65)
   (+ 1e-9 (* x_m (+ (* x_m -0.00011824251945160904) 1.128386358070218)))
   0.745170408))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.65) {
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218));
	} else {
		tmp = 0.745170408;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.65d0) then
        tmp = 1d-9 + (x_m * ((x_m * (-0.00011824251945160904d0)) + 1.128386358070218d0))
    else
        tmp = 0.745170408d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.65) {
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218));
	} else {
		tmp = 0.745170408;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.65:
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218))
	else:
		tmp = 0.745170408
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.65)
		tmp = Float64(1e-9 + Float64(x_m * Float64(Float64(x_m * -0.00011824251945160904) + 1.128386358070218)));
	else
		tmp = 0.745170408;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.65)
		tmp = 1e-9 + (x_m * ((x_m * -0.00011824251945160904) + 1.128386358070218));
	else
		tmp = 0.745170408;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.65], N[(1e-9 + N[(x$95$m * N[(N[(x$95$m * -0.00011824251945160904), $MachinePrecision] + 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.745170408]

Derivation

1. Split input into 2 regimes

2. if x < 0.650000000000000022

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

   6. Applied egg-rr 72.6%

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

      1. +-commutative 72.6%

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

      2. associate-+l+ 72.6%

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

   8. Simplified 72.6%

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

   9. Taylor expanded in x around 0 62.5%

      \[\leadsto \frac{\color{blue}{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}}{{\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2} + \left(1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{e^{{x}^{2}} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}\right)} \]

   10. Taylor expanded in x around 0 61.6%

       \[\leadsto \frac{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}{\color{blue}{2.999999997 + -3.3851590719538813 \cdot x}} \]
   11. Step-by-step derivation

       1. *-commutative 61.6%

          \[\leadsto \frac{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}{2.999999997 + \color{blue}{x \cdot -3.3851590719538813}} \]

   12. Simplified 61.6%

       \[\leadsto \frac{2.999999997 \cdot 10^{-9} + \left(-3.820122044248399 \cdot {x}^{2} + 3.385159067440336 \cdot x\right)}{\color{blue}{2.999999997 + x \cdot -3.3851590719538813}} \]

   13. Taylor expanded in x around 0 62.5%

       \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824251945160904 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   14. Step-by-step derivation

       1. unpow2 62.5%

          \[\leadsto 10^{-9} + \left(-0.00011824251945160904 \cdot \color{blue}{\left(x \cdot x\right)} + 1.128386358070218 \cdot x\right) \]

       2. associate-*r* 62.5%

          \[\leadsto 10^{-9} + \left(\color{blue}{\left(-0.00011824251945160904 \cdot x\right) \cdot x} + 1.128386358070218 \cdot x\right) \]

       3. distribute-rgt-out 62.5%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot \left(-0.00011824251945160904 \cdot x + 1.128386358070218\right)} \]

   15. Simplified 62.5%

       \[\leadsto \color{blue}{10^{-9} + x \cdot \left(-0.00011824251945160904 \cdot x + 1.128386358070218\right)} \]

   if 0.650000000000000022 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. expm1-log1p-u 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

       2. log1p-define 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

       3. +-commutative 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

       4. fma-undefine 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

       5. expm1-undefine 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

       6. add-exp-log 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

       7. add-sqr-sqrt 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

       8. fabs-sqr 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

       9. add-sqr-sqrt 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   11. Applied egg-rr 100.0%

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

       1. fma-undefine 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Simplified 100.0%

       \[\leadsto 1 + -0.254829592 \cdot \frac{e^{-1 \cdot {x}^{2}}}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 20.3%

       \[\leadsto 1 + \color{blue}{-0.254829592} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.65:\\ \;\;\;\;10^{-9} + x \cdot \left(x \cdot -0.00011824251945160904 + 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;0.745170408\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.9% accurate, 85.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.65:\\ \;\;\;\;10^{-9} + x\_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;0.745170408\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.65) (+ 1e-9 (* x_m 1.128386358070218)) 0.745170408))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.65) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 0.745170408;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.65d0) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 0.745170408d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.65) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 0.745170408;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.65:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 0.745170408
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.65)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = 0.745170408;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.65)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 0.745170408;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.65], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 0.745170408]

Derivation

1. Split input into 2 regimes

2. if x < 0.650000000000000022

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

   6. Applied egg-rr 37.1%

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

   7. Taylor expanded in x around 0 62.6%

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

      1. *-commutative 62.6%

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

   9. Simplified 62.6%

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

   if 0.650000000000000022 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. expm1-log1p-u 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

       2. log1p-define 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

       3. +-commutative 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

       4. fma-undefine 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

       5. expm1-undefine 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

       6. add-exp-log 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

       7. add-sqr-sqrt 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

       8. fabs-sqr 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

       9. add-sqr-sqrt 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   11. Applied egg-rr 100.0%

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

       1. fma-undefine 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Simplified 100.0%

       \[\leadsto 1 + -0.254829592 \cdot \frac{e^{-1 \cdot {x}^{2}}}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 20.3%

       \[\leadsto 1 + \color{blue}{-0.254829592} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.65:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;0.745170408\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.1% accurate, 142.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;0.745170408\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.4e-5) 1e-9 0.745170408))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.4e-5) {
		tmp = 1e-9;
	} else {
		tmp = 0.745170408;
	}
	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 <= 2.4d-5) then
        tmp = 1d-9
    else
        tmp = 0.745170408d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.4e-5) {
		tmp = 1e-9;
	} else {
		tmp = 0.745170408;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.4e-5:
		tmp = 1e-9
	else:
		tmp = 0.745170408
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.4e-5)
		tmp = 1e-9;
	else
		tmp = 0.745170408;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.4e-5)
		tmp = 1e-9;
	else
		tmp = 0.745170408;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.4e-5], 1e-9, 0.745170408]

Derivation

1. Split input into 2 regimes

2. if x < 2.4000000000000001e-5

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

   6. Applied egg-rr 37.1%

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

   7. Taylor expanded in x around 0 65.1%

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

   if 2.4000000000000001e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. expm1-log1p-u 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}} \]

       2. log1p-define 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)} \]

       3. +-commutative 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)}\right)} \]

       4. fma-undefine 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \mathsf{expm1}\left(\log \color{blue}{\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}\right)} \]

       5. expm1-undefine 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\log \left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)} - 1\right)}} \]

       6. add-exp-log 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1\right)} \]

       7. add-sqr-sqrt 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 1\right) - 1\right)} \]

       8. fabs-sqr 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

       9. add-sqr-sqrt 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   11. Applied egg-rr 100.0%

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

       1. fma-undefine 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 0.6%

          \[\leadsto \left(1 + \left(-0.3754899882585643 \cdot \frac{{x}^{3}}{1 + 0.3275911 \cdot \left|x\right|} + \left(0.36953108532122814 \cdot \frac{{x}^{2}}{1 + 0.3275911 \cdot \left|x\right|} + 0.8007952583978091 \cdot \frac{x}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right) - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Simplified 100.0%

       \[\leadsto 1 + -0.254829592 \cdot \frac{e^{-1 \cdot {x}^{2}}}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Taylor expanded in x around 0 20.3%

       \[\leadsto 1 + \color{blue}{-0.254829592} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;0.745170408\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.6% accurate, 856.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

3. Simplified 80.2%

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

6. Applied egg-rr 27.4%

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

7. Taylor expanded in x around 0 51.0%

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

8. Final simplification 51.0%

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

Reproduce

herbie shell --seed 2024044 
(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)))))))