Jmat.Real.erf

Percentage Accurate: 78.8% → 99.7%

Time: 1.0min

Alternatives: 12

Speedup: 142.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := -0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}\\ t_1 := e^{-{x\_m}^{2}}\\ t_2 := \mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right)\\ \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(x\_m \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x\_m}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x\_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(t\_1 \cdot \frac{0.254829592 + \frac{t\_0}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{t\_2}\right)}^{2}}{\mathsf{fma}\left(t\_1, \frac{0.254829592 + \frac{{\left(\sqrt[3]{t\_0}\right)}^{3}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{t\_2}, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (+
          -0.284496736
          (/
           (+
            1.421413741
            (/
             (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
             (fma 0.3275911 x_m 1.0)))
           (fma 0.3275911 x_m 1.0))))
        (t_1 (exp (- (pow x_m 2.0))))
        (t_2 (fma 0.3275911 (fabs x_m) 1.0)))
   (if (<= (fabs x_m) 2e-7)
     (/
      (+ (pow (* x_m 1.128386358070218) 3.0) 1e-27)
      (+
       (* (pow x_m 2.0) 1.2732557730789702)
       (- 1e-18 (* (* x_m 1.128386358070218) 1e-9))))
     (/
      (-
       1.0
       (pow
        (* t_1 (/ (+ 0.254829592 (/ t_0 (fma 0.3275911 x_m 1.0))) t_2))
        2.0))
      (fma
       t_1
       (/ (+ 0.254829592 (/ (pow (cbrt t_0) 3.0) (fma 0.3275911 x_m 1.0))) t_2)
       1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = -0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0));
	double t_1 = exp(-pow(x_m, 2.0));
	double t_2 = fma(0.3275911, fabs(x_m), 1.0);
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = (pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = (1.0 - pow((t_1 * ((0.254829592 + (t_0 / fma(0.3275911, x_m, 1.0))) / t_2)), 2.0)) / fma(t_1, ((0.254829592 + (pow(cbrt(t_0), 3.0) / fma(0.3275911, x_m, 1.0))) / t_2), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)))
	t_1 = exp(Float64(-(x_m ^ 2.0)))
	t_2 = fma(0.3275911, abs(x_m), 1.0)
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(Float64((Float64(x_m * 1.128386358070218) ^ 3.0) + 1e-27) / Float64(Float64((x_m ^ 2.0) * 1.2732557730789702) + Float64(1e-18 - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = Float64(Float64(1.0 - (Float64(t_1 * Float64(Float64(0.254829592 + Float64(t_0 / fma(0.3275911, x_m, 1.0))) / t_2)) ^ 2.0)) / fma(t_1, Float64(Float64(0.254829592 + Float64((cbrt(t_0) ^ 3.0) / fma(0.3275911, x_m, 1.0))) / t_2), 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-N[Power[x$95$m, 2.0], $MachinePrecision])], $MachinePrecision]}, Block[{t$95$2 = N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 3.0], $MachinePrecision] + 1e-27), $MachinePrecision] / N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision] + N[(1e-18 - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[N[(t$95$1 * N[(N[(0.254829592 + N[(t$95$0 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(N[(0.254829592 + N[(N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.9999999999999999e-7

   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%

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

   6. Applied egg-rr 56.5%

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

   7. Taylor expanded in x around 0 97.3%

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

      1. *-commutative 97.3%

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

   9. Simplified 97.3%

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

       1. expm1-log1p-u 97.3%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-udef 95.3%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]

   11. Applied egg-rr 95.3%

       \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]
   12. Step-by-step derivation

       1. expm1-def 97.3%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-log1p-u 97.3%

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

       3. +-commutative 97.3%

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

       4. flip3-+ 97.3%

          \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + {\left( 10^{-9} \right)}^{3}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

       5. metadata-eval 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + \color{blue}{10^{-27}}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       6. swap-sqr 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{\left(x \cdot x\right) \cdot \left(1.128386358070218 \cdot 1.128386358070218\right)} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       7. unpow2 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{{x}^{2}} \cdot \left(1.128386358070218 \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       8. metadata-eval 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot \color{blue}{1.2732557730789702} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       9. metadata-eval 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(\color{blue}{10^{-18}} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

   13. Applied egg-rr 97.4%

       \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

   if 1.9999999999999999e-7 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 99.2%

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

   8. Applied egg-rr 99.2%

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

   10. Simplified 99.2%

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

       1. add-cube-cbrt 99.3%

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

       2. pow3 99.3%

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

   12. Applied egg-rr 99.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(e^{-{x}^{2}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2}}{\mathsf{fma}\left(e^{-{x}^{2}}, \frac{0.254829592 + \frac{{\left(\sqrt[3]{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}^{3}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x\_m\right| \cdot 0.3275911\\ \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(x\_m \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x\_m}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x\_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{0.254829592 + \frac{\left(1.061405429 \cdot \frac{1}{{t\_0}^{3}} + 1.421413741 \cdot \frac{1}{t\_0}\right) + \left(1.453152027 \cdot \frac{-1}{{t\_0}^{2}} - 0.284496736\right)}{t\_0}}{\mathsf{fma}\left(0.3275911, \left|x\_m\right|, 1\right)} \cdot \frac{-1}{e^{{x\_m}^{2}}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911))))
   (if (<= (fabs x_m) 2e-7)
     (/
      (+ (pow (* x_m 1.128386358070218) 3.0) 1e-27)
      (+
       (* (pow x_m 2.0) 1.2732557730789702)
       (- 1e-18 (* (* x_m 1.128386358070218) 1e-9))))
     (exp
      (log1p
       (*
        (/
         (+
          0.254829592
          (/
           (+
            (+
             (* 1.061405429 (/ 1.0 (pow t_0 3.0)))
             (* 1.421413741 (/ 1.0 t_0)))
            (- (* 1.453152027 (/ -1.0 (pow t_0 2.0))) 0.284496736))
           t_0))
         (fma 0.3275911 (fabs x_m) 1.0))
        (/ -1.0 (exp (pow x_m 2.0)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = (pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = exp(log1p((((0.254829592 + ((((1.061405429 * (1.0 / pow(t_0, 3.0))) + (1.421413741 * (1.0 / t_0))) + ((1.453152027 * (-1.0 / pow(t_0, 2.0))) - 0.284496736)) / t_0)) / fma(0.3275911, fabs(x_m), 1.0)) * (-1.0 / exp(pow(x_m, 2.0))))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(Float64((Float64(x_m * 1.128386358070218) ^ 3.0) + 1e-27) / Float64(Float64((x_m ^ 2.0) * 1.2732557730789702) + Float64(1e-18 - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = exp(log1p(Float64(Float64(Float64(0.254829592 + Float64(Float64(Float64(Float64(1.061405429 * Float64(1.0 / (t_0 ^ 3.0))) + Float64(1.421413741 * Float64(1.0 / t_0))) + Float64(Float64(1.453152027 * Float64(-1.0 / (t_0 ^ 2.0))) - 0.284496736)) / t_0)) / fma(0.3275911, abs(x_m), 1.0)) * Float64(-1.0 / exp((x_m ^ 2.0))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 3.0], $MachinePrecision] + 1e-27), $MachinePrecision] / N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision] + N[(1e-18 - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[1 + N[(N[(N[(0.254829592 + N[(N[(N[(N[(1.061405429 * N[(1.0 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.421413741 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.453152027 * N[(-1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.284496736), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.9999999999999999e-7

   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%

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

   6. Applied egg-rr 56.5%

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

   7. Taylor expanded in x around 0 97.3%

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

      1. *-commutative 97.3%

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

   9. Simplified 97.3%

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

       1. expm1-log1p-u 97.3%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-udef 95.3%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]

   11. Applied egg-rr 95.3%

       \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]
   12. Step-by-step derivation

       1. expm1-def 97.3%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-log1p-u 97.3%

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

       3. +-commutative 97.3%

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

       4. flip3-+ 97.3%

          \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + {\left( 10^{-9} \right)}^{3}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

       5. metadata-eval 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + \color{blue}{10^{-27}}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       6. swap-sqr 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{\left(x \cdot x\right) \cdot \left(1.128386358070218 \cdot 1.128386358070218\right)} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       7. unpow2 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{{x}^{2}} \cdot \left(1.128386358070218 \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       8. metadata-eval 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot \color{blue}{1.2732557730789702} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       9. metadata-eval 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(\color{blue}{10^{-18}} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

   13. Applied egg-rr 97.4%

       \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

   if 1.9999999999999999e-7 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{0.254829592 + \frac{\mathsf{fma}\left(1.061405429, {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-3}, \frac{1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) - \mathsf{fma}\left(1.453152027, {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, 0.284496736\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right) \cdot e^{-{x}^{2}}\right)}} \]
   8. Step-by-step derivation

      1. distribute-lft-neg-out 99.9%

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

      2. distribute-rgt-neg-in 99.9%

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

      3. exp-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. metadata-eval 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in x around 0 100.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{0.254829592 + \color{blue}{\frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right)}{1 + 0.3275911 \cdot \left|x\right|}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{-1}{e^{{x}^{2}}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{0.254829592 + \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right) + \left(1.453152027 \cdot \frac{-1}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{2}} - 0.284496736\right)}{1 + \left|x\right| \cdot 0.3275911}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{-1}{e^{{x}^{2}}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x\_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;\left|x\_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(x\_m \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x\_m}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x\_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(t\_1 \cdot \left(\frac{\left(0.284496736 + 1.453152027 \cdot \frac{1}{{t\_0}^{2}}\right) - \left(1.061405429 \cdot \frac{1}{{t\_0}^{3}} + 1.421413741 \cdot t\_1\right)}{1 + x\_m \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x_m) 2e-7)
     (/
      (+ (pow (* x_m 1.128386358070218) 3.0) 1e-27)
      (+
       (* (pow x_m 2.0) 1.2732557730789702)
       (- 1e-18 (* (* x_m 1.128386358070218) 1e-9))))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        t_1
        (-
         (/
          (-
           (+ 0.284496736 (* 1.453152027 (/ 1.0 (pow t_0 2.0))))
           (+ (* 1.061405429 (/ 1.0 (pow t_0 3.0))) (* 1.421413741 t_1)))
          (+ 1.0 (* x_m 0.3275911)))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = (pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * (t_1 * ((((0.284496736 + (1.453152027 * (1.0 / pow(t_0, 2.0)))) - ((1.061405429 * (1.0 / pow(t_0, 3.0))) + (1.421413741 * t_1))) / (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    if (abs(x_m) <= 2d-7) then
        tmp = (((x_m * 1.128386358070218d0) ** 3.0d0) + 1d-27) / (((x_m ** 2.0d0) * 1.2732557730789702d0) + (1d-18 - ((x_m * 1.128386358070218d0) * 1d-9)))
    else
        tmp = 1.0d0 + (exp((x_m * -x_m)) * (t_1 * ((((0.284496736d0 + (1.453152027d0 * (1.0d0 / (t_0 ** 2.0d0)))) - ((1.061405429d0 * (1.0d0 / (t_0 ** 3.0d0))) + (1.421413741d0 * t_1))) / (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 t_0 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (Math.abs(x_m) <= 2e-7) {
		tmp = (Math.pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((Math.pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (Math.exp((x_m * -x_m)) * (t_1 * ((((0.284496736 + (1.453152027 * (1.0 / Math.pow(t_0, 2.0)))) - ((1.061405429 * (1.0 / Math.pow(t_0, 3.0))) + (1.421413741 * t_1))) / (1.0 + (x_m * 0.3275911))) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (math.fabs(x_m) * 0.3275911)
	t_1 = 1.0 / t_0
	tmp = 0
	if math.fabs(x_m) <= 2e-7:
		tmp = (math.pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((math.pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)))
	else:
		tmp = 1.0 + (math.exp((x_m * -x_m)) * (t_1 * ((((0.284496736 + (1.453152027 * (1.0 / math.pow(t_0, 2.0)))) - ((1.061405429 * (1.0 / math.pow(t_0, 3.0))) + (1.421413741 * t_1))) / (1.0 + (x_m * 0.3275911))) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(Float64((Float64(x_m * 1.128386358070218) ^ 3.0) + 1e-27) / Float64(Float64((x_m ^ 2.0) * 1.2732557730789702) + Float64(1e-18 - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(t_1 * Float64(Float64(Float64(Float64(0.284496736 + Float64(1.453152027 * Float64(1.0 / (t_0 ^ 2.0)))) - Float64(Float64(1.061405429 * Float64(1.0 / (t_0 ^ 3.0))) + Float64(1.421413741 * t_1))) / 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)
	t_0 = 1.0 + (abs(x_m) * 0.3275911);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (abs(x_m) <= 2e-7)
		tmp = (((x_m * 1.128386358070218) ^ 3.0) + 1e-27) / (((x_m ^ 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	else
		tmp = 1.0 + (exp((x_m * -x_m)) * (t_1 * ((((0.284496736 + (1.453152027 * (1.0 / (t_0 ^ 2.0)))) - ((1.061405429 * (1.0 / (t_0 ^ 3.0))) + (1.421413741 * t_1))) / (1.0 + (x_m * 0.3275911))) - 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[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 3.0], $MachinePrecision] + 1e-27), $MachinePrecision] / N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision] + N[(1e-18 - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(N[(N[(N[(0.284496736 + N[(1.453152027 * N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1.061405429 * N[(1.0 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.421413741 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.9999999999999999e-7

   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%

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

   6. Applied egg-rr 56.5%

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

   7. Taylor expanded in x around 0 97.3%

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

      1. *-commutative 97.3%

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

   9. Simplified 97.3%

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

       1. expm1-log1p-u 97.3%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-udef 95.3%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]

   11. Applied egg-rr 95.3%

       \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]
   12. Step-by-step derivation

       1. expm1-def 97.3%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-log1p-u 97.3%

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

       3. +-commutative 97.3%

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

       4. flip3-+ 97.3%

          \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + {\left( 10^{-9} \right)}^{3}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

       5. metadata-eval 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + \color{blue}{10^{-27}}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       6. swap-sqr 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{\left(x \cdot x\right) \cdot \left(1.128386358070218 \cdot 1.128386358070218\right)} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       7. unpow2 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{{x}^{2}} \cdot \left(1.128386358070218 \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       8. metadata-eval 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot \color{blue}{1.2732557730789702} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       9. metadata-eval 97.4%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(\color{blue}{10^{-18}} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

   13. Applied egg-rr 97.4%

       \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

   if 1.9999999999999999e-7 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. pow1 99.9%

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

      2. add-sqr-sqrt 48.8%

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

      3. fabs-sqr 48.8%

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

      4. add-sqr-sqrt 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. unpow1 99.3%

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

   9. Simplified 99.3%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{\left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{2}}\right) - \left(1.061405429 \cdot \frac{1}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right)}{1 + x \cdot 0.3275911} - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-6}:\\ \;\;\;\;\frac{{\left(x\_m \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x\_m}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x\_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(\frac{1}{1 + x\_m \cdot 0.3275911} \cdot \left(\frac{-1}{\frac{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x\_m, 1\right)}}} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1e-6)
   (/
    (+ (pow (* x_m 1.128386358070218) 3.0) 1e-27)
    (+
     (* (pow x_m 2.0) 1.2732557730789702)
     (- 1e-18 (* (* x_m 1.128386358070218) 1e-9))))
   (+
    1.0
    (*
     (exp (* x_m (- x_m)))
     (*
      (/ 1.0 (+ 1.0 (* x_m 0.3275911)))
      (-
       (/
        -1.0
        (/
         (fma 0.3275911 x_m 1.0)
         (+
          -0.284496736
          (/
           (+
            1.421413741
            (/
             (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
             (fma 0.3275911 x_m 1.0)))
           (fma 0.3275911 x_m 1.0)))))
       0.254829592))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1e-6) {
		tmp = (pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((1.0 / (1.0 + (x_m * 0.3275911))) * ((-1.0 / (fma(0.3275911, x_m, 1.0) / (-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))))) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1e-6)
		tmp = Float64(Float64((Float64(x_m * 1.128386358070218) ^ 3.0) + 1e-27) / Float64(Float64((x_m ^ 2.0) * 1.2732557730789702) + Float64(1e-18 - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911))) * Float64(Float64(-1.0 / Float64(fma(0.3275911, x_m, 1.0) / Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))))) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1e-6], N[(N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 3.0], $MachinePrecision] + 1e-27), $MachinePrecision] / N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision] + N[(1e-18 - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 / N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] / N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999955e-7

   1. Initial program 72.7%

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

   3. Simplified 72.7%

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

   6. Applied egg-rr 37.5%

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

   7. Taylor expanded in x around 0 63.1%

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

      1. *-commutative 63.1%

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

   9. Simplified 63.1%

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

       1. expm1-log1p-u 62.7%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-udef 61.4%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]

   11. Applied egg-rr 61.4%

       \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]
   12. Step-by-step derivation

       1. expm1-def 62.7%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-log1p-u 63.1%

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

       3. +-commutative 63.1%

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

       4. flip3-+ 62.9%

          \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + {\left( 10^{-9} \right)}^{3}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

       5. metadata-eval 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + \color{blue}{10^{-27}}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       6. swap-sqr 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{\left(x \cdot x\right) \cdot \left(1.128386358070218 \cdot 1.128386358070218\right)} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       7. unpow2 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{{x}^{2}} \cdot \left(1.128386358070218 \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       8. metadata-eval 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot \color{blue}{1.2732557730789702} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       9. metadata-eval 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(\color{blue}{10^{-18}} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

   13. Applied egg-rr 62.9%

       \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

   if 9.99999999999999955e-7 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. pow1 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. fabs-sqr 99.9%

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

      4. add-sqr-sqrt 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. unpow1 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-6}:\\ \;\;\;\;\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + x \cdot 0.3275911} \cdot \left(\frac{-1}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}} - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x\_m\right| \cdot 0.3275911\\ t_1 := 1 + x\_m \cdot 0.3275911\\ t_2 := \frac{1}{t\_1}\\ \mathbf{if}\;x\_m \leq 8.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(x\_m \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x\_m}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x\_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(\left(0.254829592 + \frac{1}{t\_0} \cdot \left(-0.284496736 + t\_2 \cdot \left(1.421413741 + t\_2 \cdot \left(-1.453152027 + \frac{1.061405429}{t\_1}\right)\right)\right)\right) \cdot \frac{-1}{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911)))
        (t_1 (+ 1.0 (* x_m 0.3275911)))
        (t_2 (/ 1.0 t_1)))
   (if (<= x_m 8.3e-7)
     (/
      (+ (pow (* x_m 1.128386358070218) 3.0) 1e-27)
      (+
       (* (pow x_m 2.0) 1.2732557730789702)
       (- 1e-18 (* (* x_m 1.128386358070218) 1e-9))))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (+
         0.254829592
         (*
          (/ 1.0 t_0)
          (+
           -0.284496736
           (*
            t_2
            (+ 1.421413741 (* t_2 (+ -1.453152027 (/ 1.061405429 t_1))))))))
        (/ -1.0 t_0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_1 = 1.0 + (x_m * 0.3275911);
	double t_2 = 1.0 / t_1;
	double tmp;
	if (x_m <= 8.3e-7) {
		tmp = (pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((0.254829592 + ((1.0 / t_0) * (-0.284496736 + (t_2 * (1.421413741 + (t_2 * (-1.453152027 + (1.061405429 / t_1)))))))) * (-1.0 / t_0)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    t_1 = 1.0d0 + (x_m * 0.3275911d0)
    t_2 = 1.0d0 / t_1
    if (x_m <= 8.3d-7) then
        tmp = (((x_m * 1.128386358070218d0) ** 3.0d0) + 1d-27) / (((x_m ** 2.0d0) * 1.2732557730789702d0) + (1d-18 - ((x_m * 1.128386358070218d0) * 1d-9)))
    else
        tmp = 1.0d0 + (exp((x_m * -x_m)) * ((0.254829592d0 + ((1.0d0 / t_0) * ((-0.284496736d0) + (t_2 * (1.421413741d0 + (t_2 * ((-1.453152027d0) + (1.061405429d0 / t_1)))))))) * ((-1.0d0) / t_0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double t_1 = 1.0 + (x_m * 0.3275911);
	double t_2 = 1.0 / t_1;
	double tmp;
	if (x_m <= 8.3e-7) {
		tmp = (Math.pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((Math.pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (Math.exp((x_m * -x_m)) * ((0.254829592 + ((1.0 / t_0) * (-0.284496736 + (t_2 * (1.421413741 + (t_2 * (-1.453152027 + (1.061405429 / t_1)))))))) * (-1.0 / t_0)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (math.fabs(x_m) * 0.3275911)
	t_1 = 1.0 + (x_m * 0.3275911)
	t_2 = 1.0 / t_1
	tmp = 0
	if x_m <= 8.3e-7:
		tmp = (math.pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((math.pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)))
	else:
		tmp = 1.0 + (math.exp((x_m * -x_m)) * ((0.254829592 + ((1.0 / t_0) * (-0.284496736 + (t_2 * (1.421413741 + (t_2 * (-1.453152027 + (1.061405429 / t_1)))))))) * (-1.0 / t_0)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 + Float64(x_m * 0.3275911))
	t_2 = Float64(1.0 / t_1)
	tmp = 0.0
	if (x_m <= 8.3e-7)
		tmp = Float64(Float64((Float64(x_m * 1.128386358070218) ^ 3.0) + 1e-27) / Float64(Float64((x_m ^ 2.0) * 1.2732557730789702) + Float64(1e-18 - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(0.254829592 + Float64(Float64(1.0 / t_0) * Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(t_2 * Float64(-1.453152027 + Float64(1.061405429 / t_1)))))))) * Float64(-1.0 / t_0))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (abs(x_m) * 0.3275911);
	t_1 = 1.0 + (x_m * 0.3275911);
	t_2 = 1.0 / t_1;
	tmp = 0.0;
	if (x_m <= 8.3e-7)
		tmp = (((x_m * 1.128386358070218) ^ 3.0) + 1e-27) / (((x_m ^ 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	else
		tmp = 1.0 + (exp((x_m * -x_m)) * ((0.254829592 + ((1.0 / t_0) * (-0.284496736 + (t_2 * (1.421413741 + (t_2 * (-1.453152027 + (1.061405429 / t_1)))))))) * (-1.0 / t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, If[LessEqual[x$95$m, 8.3e-7], N[(N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 3.0], $MachinePrecision] + 1e-27), $MachinePrecision] / N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision] + N[(1e-18 - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(0.254829592 + N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(t$95$2 * N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.29999999999999994e-7

   1. Initial program 72.7%

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

   3. Simplified 72.7%

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

   6. Applied egg-rr 37.5%

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

   7. Taylor expanded in x around 0 63.1%

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

      1. *-commutative 63.1%

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

   9. Simplified 63.1%

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

       1. expm1-log1p-u 62.7%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-udef 61.4%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]

   11. Applied egg-rr 61.4%

       \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]
   12. Step-by-step derivation

       1. expm1-def 62.7%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-log1p-u 63.1%

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

       3. +-commutative 63.1%

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

       4. flip3-+ 62.9%

          \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + {\left( 10^{-9} \right)}^{3}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

       5. metadata-eval 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + \color{blue}{10^{-27}}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       6. swap-sqr 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{\left(x \cdot x\right) \cdot \left(1.128386358070218 \cdot 1.128386358070218\right)} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       7. unpow2 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{{x}^{2}} \cdot \left(1.128386358070218 \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       8. metadata-eval 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot \color{blue}{1.2732557730789702} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       9. metadata-eval 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(\color{blue}{10^{-18}} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

   13. Applied egg-rr 62.9%

       \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

   if 8.29999999999999994e-7 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. pow1 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. fabs-sqr 99.9%

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

      4. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. unpow1 99.9%

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

   8. Simplified 99.9%

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

      1. pow1 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. fabs-sqr 99.9%

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

      4. add-sqr-sqrt 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. unpow1 99.9%

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

   12. Simplified 99.9%

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

       1. pow1 99.9%

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

       2. add-sqr-sqrt 99.9%

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

       3. fabs-sqr 99.9%

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

       4. add-sqr-sqrt 99.9%

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

   14. Applied egg-rr 99.9%

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

       1. unpow1 99.9%

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

   16. Simplified 99.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(x\_m \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x\_m}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x\_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(\frac{1}{1 + \left|x\_m\right| \cdot 0.3275911} \cdot \left(\frac{-1}{1.3419749235962346 + \left({x\_m}^{2} \cdot 0.41439251223535706 + x\_m \cdot 1.4421495346696274\right)} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 8.5e-7)
   (/
    (+ (pow (* x_m 1.128386358070218) 3.0) 1e-27)
    (+
     (* (pow x_m 2.0) 1.2732557730789702)
     (- 1e-18 (* (* x_m 1.128386358070218) 1e-9))))
   (+
    1.0
    (*
     (exp (* x_m (- x_m)))
     (*
      (/ 1.0 (+ 1.0 (* (fabs x_m) 0.3275911)))
      (-
       (/
        -1.0
        (+
         1.3419749235962346
         (+ (* (pow x_m 2.0) 0.41439251223535706) (* x_m 1.4421495346696274))))
       0.254829592))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 8.5e-7) {
		tmp = (pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((1.0 / (1.0 + (fabs(x_m) * 0.3275911))) * ((-1.0 / (1.3419749235962346 + ((pow(x_m, 2.0) * 0.41439251223535706) + (x_m * 1.4421495346696274)))) - 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 (x_m <= 8.5d-7) then
        tmp = (((x_m * 1.128386358070218d0) ** 3.0d0) + 1d-27) / (((x_m ** 2.0d0) * 1.2732557730789702d0) + (1d-18 - ((x_m * 1.128386358070218d0) * 1d-9)))
    else
        tmp = 1.0d0 + (exp((x_m * -x_m)) * ((1.0d0 / (1.0d0 + (abs(x_m) * 0.3275911d0))) * (((-1.0d0) / (1.3419749235962346d0 + (((x_m ** 2.0d0) * 0.41439251223535706d0) + (x_m * 1.4421495346696274d0)))) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 8.5e-7) {
		tmp = (Math.pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((Math.pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (Math.exp((x_m * -x_m)) * ((1.0 / (1.0 + (Math.abs(x_m) * 0.3275911))) * ((-1.0 / (1.3419749235962346 + ((Math.pow(x_m, 2.0) * 0.41439251223535706) + (x_m * 1.4421495346696274)))) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 8.5e-7:
		tmp = (math.pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((math.pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)))
	else:
		tmp = 1.0 + (math.exp((x_m * -x_m)) * ((1.0 / (1.0 + (math.fabs(x_m) * 0.3275911))) * ((-1.0 / (1.3419749235962346 + ((math.pow(x_m, 2.0) * 0.41439251223535706) + (x_m * 1.4421495346696274)))) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 8.5e-7)
		tmp = Float64(Float64((Float64(x_m * 1.128386358070218) ^ 3.0) + 1e-27) / Float64(Float64((x_m ^ 2.0) * 1.2732557730789702) + Float64(1e-18 - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911))) * Float64(Float64(-1.0 / Float64(1.3419749235962346 + Float64(Float64((x_m ^ 2.0) * 0.41439251223535706) + Float64(x_m * 1.4421495346696274)))) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 8.5e-7)
		tmp = (((x_m * 1.128386358070218) ^ 3.0) + 1e-27) / (((x_m ^ 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	else
		tmp = 1.0 + (exp((x_m * -x_m)) * ((1.0 / (1.0 + (abs(x_m) * 0.3275911))) * ((-1.0 / (1.3419749235962346 + (((x_m ^ 2.0) * 0.41439251223535706) + (x_m * 1.4421495346696274)))) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 8.5e-7], N[(N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 3.0], $MachinePrecision] + 1e-27), $MachinePrecision] / N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision] + N[(1e-18 - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 / N[(1.3419749235962346 + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.41439251223535706), $MachinePrecision] + N[(x$95$m * 1.4421495346696274), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.50000000000000014e-7

   1. Initial program 72.7%

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

   3. Simplified 72.7%

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

   6. Applied egg-rr 37.5%

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

   7. Taylor expanded in x around 0 63.1%

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

      1. *-commutative 63.1%

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

   9. Simplified 63.1%

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

       1. expm1-log1p-u 62.7%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-udef 61.4%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]

   11. Applied egg-rr 61.4%

       \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]
   12. Step-by-step derivation

       1. expm1-def 62.7%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-log1p-u 63.1%

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

       3. +-commutative 63.1%

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

       4. flip3-+ 62.9%

          \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + {\left( 10^{-9} \right)}^{3}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

       5. metadata-eval 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + \color{blue}{10^{-27}}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       6. swap-sqr 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{\left(x \cdot x\right) \cdot \left(1.128386358070218 \cdot 1.128386358070218\right)} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       7. unpow2 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{{x}^{2}} \cdot \left(1.128386358070218 \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       8. metadata-eval 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot \color{blue}{1.2732557730789702} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       9. metadata-eval 62.9%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(\color{blue}{10^{-18}} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

   13. Applied egg-rr 62.9%

       \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

   if 8.50000000000000014e-7 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 98.2%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{\color{blue}{1.3419749235962346 + \left(0.41439251223535706 \cdot {x}^{2} + 1.4421495346696274 \cdot x\right)}}\right)\right) \cdot e^{-x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{-1}{1.3419749235962346 + \left({x}^{2} \cdot 0.41439251223535706 + x \cdot 1.4421495346696274\right)} - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.29:\\ \;\;\;\;\frac{{\left(x\_m \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x\_m}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x\_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x\_m \cdot \left(-x\_m\right)} \cdot \left(\frac{1}{1 + \left|x\_m\right| \cdot 0.3275911} \cdot \left(\frac{-1}{1.3419749235962346 + x\_m \cdot 1.4421495346696274} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.29)
   (/
    (+ (pow (* x_m 1.128386358070218) 3.0) 1e-27)
    (+
     (* (pow x_m 2.0) 1.2732557730789702)
     (- 1e-18 (* (* x_m 1.128386358070218) 1e-9))))
   (+
    1.0
    (*
     (exp (* x_m (- x_m)))
     (*
      (/ 1.0 (+ 1.0 (* (fabs x_m) 0.3275911)))
      (-
       (/ -1.0 (+ 1.3419749235962346 (* x_m 1.4421495346696274)))
       0.254829592))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.29) {
		tmp = (pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((1.0 / (1.0 + (fabs(x_m) * 0.3275911))) * ((-1.0 / (1.3419749235962346 + (x_m * 1.4421495346696274))) - 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 (x_m <= 0.29d0) then
        tmp = (((x_m * 1.128386358070218d0) ** 3.0d0) + 1d-27) / (((x_m ** 2.0d0) * 1.2732557730789702d0) + (1d-18 - ((x_m * 1.128386358070218d0) * 1d-9)))
    else
        tmp = 1.0d0 + (exp((x_m * -x_m)) * ((1.0d0 / (1.0d0 + (abs(x_m) * 0.3275911d0))) * (((-1.0d0) / (1.3419749235962346d0 + (x_m * 1.4421495346696274d0))) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.29) {
		tmp = (Math.pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((Math.pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (Math.exp((x_m * -x_m)) * ((1.0 / (1.0 + (Math.abs(x_m) * 0.3275911))) * ((-1.0 / (1.3419749235962346 + (x_m * 1.4421495346696274))) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.29:
		tmp = (math.pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((math.pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)))
	else:
		tmp = 1.0 + (math.exp((x_m * -x_m)) * ((1.0 / (1.0 + (math.fabs(x_m) * 0.3275911))) * ((-1.0 / (1.3419749235962346 + (x_m * 1.4421495346696274))) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.29)
		tmp = Float64(Float64((Float64(x_m * 1.128386358070218) ^ 3.0) + 1e-27) / Float64(Float64((x_m ^ 2.0) * 1.2732557730789702) + Float64(1e-18 - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911))) * Float64(Float64(-1.0 / Float64(1.3419749235962346 + Float64(x_m * 1.4421495346696274))) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.29)
		tmp = (((x_m * 1.128386358070218) ^ 3.0) + 1e-27) / (((x_m ^ 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	else
		tmp = 1.0 + (exp((x_m * -x_m)) * ((1.0 / (1.0 + (abs(x_m) * 0.3275911))) * ((-1.0 / (1.3419749235962346 + (x_m * 1.4421495346696274))) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.29], N[(N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 3.0], $MachinePrecision] + 1e-27), $MachinePrecision] / N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision] + N[(1e-18 - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 / N[(1.3419749235962346 + N[(x$95$m * 1.4421495346696274), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.28999999999999998

   1. Initial program 72.8%

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

   3. Simplified 72.9%

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

   6. Applied egg-rr 37.4%

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

   7. Taylor expanded in x around 0 62.9%

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

      1. *-commutative 62.9%

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

   9. Simplified 62.9%

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

       1. expm1-log1p-u 62.6%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-udef 61.3%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]

   11. Applied egg-rr 61.3%

       \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]
   12. Step-by-step derivation

       1. expm1-def 62.6%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-log1p-u 62.9%

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

       3. +-commutative 62.9%

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

       4. flip3-+ 62.8%

          \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + {\left( 10^{-9} \right)}^{3}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

       5. metadata-eval 62.8%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + \color{blue}{10^{-27}}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       6. swap-sqr 62.8%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{\left(x \cdot x\right) \cdot \left(1.128386358070218 \cdot 1.128386358070218\right)} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       7. unpow2 62.8%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{{x}^{2}} \cdot \left(1.128386358070218 \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       8. metadata-eval 62.8%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot \color{blue}{1.2732557730789702} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       9. metadata-eval 62.8%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(\color{blue}{10^{-18}} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

   13. Applied egg-rr 62.8%

       \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

   if 0.28999999999999998 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 98.9%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{\color{blue}{1.3419749235962346 + 1.4421495346696274 \cdot x}}\right)\right) \cdot e^{-x \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1.3419749235962346 + \color{blue}{x \cdot 1.4421495346696274}}\right)\right) \cdot e^{-x \cdot x} \]

   9. Simplified 98.9%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{\color{blue}{1.3419749235962346 + x \cdot 1.4421495346696274}}\right)\right) \cdot e^{-x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.29:\\ \;\;\;\;\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{-1}{1.3419749235962346 + x \cdot 1.4421495346696274} - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.9:\\ \;\;\;\;\frac{{\left(x\_m \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x\_m}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x\_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.9)
   (/
    (+ (pow (* x_m 1.128386358070218) 3.0) 1e-27)
    (+
     (* (pow x_m 2.0) 1.2732557730789702)
     (- 1e-18 (* (* x_m 1.128386358070218) 1e-9))))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = (pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.9d0) then
        tmp = (((x_m * 1.128386358070218d0) ** 3.0d0) + 1d-27) / (((x_m ** 2.0d0) * 1.2732557730789702d0) + (1d-18 - ((x_m * 1.128386358070218d0) * 1d-9)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = (Math.pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((Math.pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.9:
		tmp = (math.pow((x_m * 1.128386358070218), 3.0) + 1e-27) / ((math.pow(x_m, 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.9)
		tmp = Float64(Float64((Float64(x_m * 1.128386358070218) ^ 3.0) + 1e-27) / Float64(Float64((x_m ^ 2.0) * 1.2732557730789702) + Float64(1e-18 - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.9)
		tmp = (((x_m * 1.128386358070218) ^ 3.0) + 1e-27) / (((x_m ^ 2.0) * 1.2732557730789702) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.9], N[(N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 3.0], $MachinePrecision] + 1e-27), $MachinePrecision] / N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision] + N[(1e-18 - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 73.0%

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

   3. Simplified 73.0%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in x around 0 62.7%

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

      1. *-commutative 62.7%

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

   9. Simplified 62.7%

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

       1. expm1-log1p-u 62.3%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-udef 61.0%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]

   11. Applied egg-rr 61.0%

       \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]
   12. Step-by-step derivation

       1. expm1-def 62.3%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-log1p-u 62.7%

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

       3. +-commutative 62.7%

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

       4. flip3-+ 62.6%

          \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + {\left( 10^{-9} \right)}^{3}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

       5. metadata-eval 62.6%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + \color{blue}{10^{-27}}}{\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       6. swap-sqr 62.6%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{\left(x \cdot x\right) \cdot \left(1.128386358070218 \cdot 1.128386358070218\right)} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       7. unpow2 62.6%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{\color{blue}{{x}^{2}} \cdot \left(1.128386358070218 \cdot 1.128386358070218\right) + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       8. metadata-eval 62.6%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot \color{blue}{1.2732557730789702} + \left(10^{-9} \cdot 10^{-9} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

       9. metadata-eval 62.6%

          \[\leadsto 1 \cdot \frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(\color{blue}{10^{-18}} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)} \]

   13. Applied egg-rr 62.6%

       \[\leadsto 1 \cdot \color{blue}{\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}} \]

   if 0.900000000000000022 < 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 0.6%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;\frac{{\left(x \cdot 1.128386358070218\right)}^{3} + 10^{-27}}{{x}^{2} \cdot 1.2732557730789702 + \left(10^{-18} - \left(x \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.9:\\ \;\;\;\;10^{-9} + \sqrt[3]{{\left(x\_m \cdot 1.128386358070218\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.9) (+ 1e-9 (cbrt (pow (* x_m 1.128386358070218) 3.0))) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = 1e-9 + cbrt(pow((x_m * 1.128386358070218), 3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = 1e-9 + Math.cbrt(Math.pow((x_m * 1.128386358070218), 3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.9)
		tmp = Float64(1e-9 + cbrt((Float64(x_m * 1.128386358070218) ^ 3.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.9], N[(1e-9 + N[Power[N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 73.0%

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

   3. Simplified 73.0%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in x around 0 62.7%

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

      1. *-commutative 62.7%

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

   9. Simplified 62.7%

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

       1. expm1-log1p-u 62.3%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-udef 61.0%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]

   11. Applied egg-rr 61.0%

       \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 1.128386358070218\right)} - 1\right)}\right) \]
   12. Step-by-step derivation

       1. expm1-def 62.3%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 1.128386358070218\right)\right)}\right) \]

       2. expm1-log1p-u 62.7%

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

       3. add-cbrt-cube 62.7%

          \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\sqrt[3]{\left(\left(x \cdot 1.128386358070218\right) \cdot \left(x \cdot 1.128386358070218\right)\right) \cdot \left(x \cdot 1.128386358070218\right)}}\right) \]

       4. pow3 62.7%

          \[\leadsto 1 \cdot \left(10^{-9} + \sqrt[3]{\color{blue}{{\left(x \cdot 1.128386358070218\right)}^{3}}}\right) \]

   13. Applied egg-rr 62.7%

       \[\leadsto 1 \cdot \left(10^{-9} + \color{blue}{\sqrt[3]{{\left(x \cdot 1.128386358070218\right)}^{3}}}\right) \]

   if 0.900000000000000022 < 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 0.6%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;10^{-9} + \sqrt[3]{{\left(x \cdot 1.128386358070218\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.3% accurate, 85.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.9:\\ \;\;\;\;x\_m \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.9) (+ (* x_m 1.128386358070218) 1e-9) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.9d0) then
        tmp = (x_m * 1.128386358070218d0) + 1d-9
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = (x_m * 1.128386358070218) + 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.9:
		tmp = (x_m * 1.128386358070218) + 1e-9
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.9)
		tmp = Float64(Float64(x_m * 1.128386358070218) + 1e-9);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.9)
		tmp = (x_m * 1.128386358070218) + 1e-9;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.9], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 73.0%

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

   3. Simplified 73.0%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in x around 0 62.7%

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

      1. *-commutative 62.7%

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

   9. Simplified 62.7%

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

   if 0.900000000000000022 < 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 0.6%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;x \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.7% accurate, 142.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 72.7%

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

   3. Simplified 72.7%

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

   6. Applied egg-rr 37.5%

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

   7. Taylor expanded in x around 0 65.0%

      \[\leadsto 1 \cdot \color{blue}{10^{-9}} \]

   if 2.79999999999999996e-5 < x

   1. Initial program 99.9%

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

   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 1.0%

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

   7. Taylor expanded in x around inf 97.5%

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

4. Final simplification 73.2%

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

Alternative 12: 54.0% accurate, 856.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

3. Simplified 79.6%

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

6. Applied egg-rr 28.2%

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

7. Taylor expanded in x around 0 51.3%

   \[\leadsto 1 \cdot \color{blue}{10^{-9}} \]

8. Final simplification 51.3%

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

Reproduce

herbie shell --seed 2024031 
(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)))))))