Jmat.Real.erf

Percentage Accurate: 79.4% → 99.7%

Time: 25.2s

Alternatives: 19

Speedup: 121.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, x_m, 1\right) \cdot e^{{x_m}^{2}}\\ t_1 := \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}\\ t_2 := \sqrt{t_1}\\ \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right) + {x_m}^{2} \cdot -0.00011824294398844343\\ \mathbf{else}:\\ \;\;\;\;\left(1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\mathsf{fma}\left(t_2, t_2, -1.453152027\right)}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{t_0}\right)}^{2}\right) \cdot \frac{1}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{t_1 + -1.453152027}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{t_0}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fma 0.3275911 x_m 1.0) (exp (pow x_m 2.0))))
        (t_1 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
        (t_2 (sqrt t_1)))
   (if (<= (fabs x_m) 2e-8)
     (+
      (fma x_m 1.128386358070218 1e-9)
      (* (pow x_m 2.0) -0.00011824294398844343))
     (*
      (-
       1.0
       (pow
        (/
         (+
          0.254829592
          (/
           (+
            -0.284496736
            (/
             (+
              1.421413741
              (/ (fma t_2 t_2 -1.453152027) (fma 0.3275911 x_m 1.0)))
             (fma 0.3275911 x_m 1.0)))
           (fma 0.3275911 x_m 1.0)))
         t_0)
        2.0))
      (/
       1.0
       (+
        1.0
        (/
         (+
          0.254829592
          (/
           (+
            -0.284496736
            (/
             (+ 1.421413741 (/ (+ t_1 -1.453152027) (fma 0.3275911 x_m 1.0)))
             (fma 0.3275911 x_m 1.0)))
           (fma 0.3275911 x_m 1.0)))
         t_0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(0.3275911, x_m, 1.0) * exp(pow(x_m, 2.0));
	double t_1 = 1.061405429 / fma(0.3275911, x_m, 1.0);
	double t_2 = sqrt(t_1);
	double tmp;
	if (fabs(x_m) <= 2e-8) {
		tmp = fma(x_m, 1.128386358070218, 1e-9) + (pow(x_m, 2.0) * -0.00011824294398844343);
	} else {
		tmp = (1.0 - pow(((0.254829592 + ((-0.284496736 + ((1.421413741 + (fma(t_2, t_2, -1.453152027) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / t_0), 2.0)) * (1.0 / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((t_1 + -1.453152027) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / t_0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(fma(0.3275911, x_m, 1.0) * exp((x_m ^ 2.0)))
	t_1 = Float64(1.061405429 / fma(0.3275911, x_m, 1.0))
	t_2 = sqrt(t_1)
	tmp = 0.0
	if (abs(x_m) <= 2e-8)
		tmp = Float64(fma(x_m, 1.128386358070218, 1e-9) + Float64((x_m ^ 2.0) * -0.00011824294398844343));
	else
		tmp = Float64(Float64(1.0 - (Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(fma(t_2, t_2, -1.453152027) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / t_0) ^ 2.0)) * Float64(1.0 / Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(t_1 + -1.453152027) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / t_0))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-8], N[(N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(t$95$2 * t$95$2 + -1.453152027), $MachinePrecision] / 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] / t$95$0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(t$95$1 + -1.453152027), $MachinePrecision] / 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] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

   5. Applied egg-rr 54.2%

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

   6. Taylor expanded in x around 0 96.9%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-commutative 96.9%

         \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

      2. associate-+r+ 96.9%

         \[\leadsto \color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right) + -0.00011824294398844343 \cdot {x}^{2}} \]

      3. +-commutative 96.9%

         \[\leadsto \color{blue}{\left(1.128386358070218 \cdot x + 10^{-9}\right)} + -0.00011824294398844343 \cdot {x}^{2} \]

      4. *-commutative 96.9%

         \[\leadsto \left(\color{blue}{x \cdot 1.128386358070218} + 10^{-9}\right) + -0.00011824294398844343 \cdot {x}^{2} \]

      5. fma-def 96.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} + -0.00011824294398844343 \cdot {x}^{2} \]

      6. *-commutative 96.9%

         \[\leadsto \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \color{blue}{{x}^{2} \cdot -0.00011824294398844343} \]

   8. Simplified 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + {x}^{2} \cdot -0.00011824294398844343} \]

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

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   5. Applied egg-rr 96.9%

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

      1. +-commutative 96.9%

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

      2. fma-udef 96.9%

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

      3. +-commutative 96.9%

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

      4. add-sqr-sqrt 49.3%

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

      5. fma-def 49.3%

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

      6. +-commutative 49.3%

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

      7. fma-udef 49.3%

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

      8. +-commutative 49.3%

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

      9. fma-udef 49.3%

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

   7. Applied egg-rr 49.3%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + {x}^{2} \cdot -0.00011824294398844343\\ \mathbf{else}:\\ \;\;\;\;\left(1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\mathsf{fma}\left(\sqrt{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}, \sqrt{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}, -1.453152027\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\right)}^{2}\right) \cdot \frac{1}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{fma}\left(0.3275911, x_m, 1\right)}\\ \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right) + {x_m}^{2} \cdot -0.00011824294398844343\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \frac{\frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{{t_0}^{2}}}{t_0}}{{\left(e^{x_m}\right)}^{x_m}}}{\mathsf{fma}\left(0.3275911, \left|x_m\right|, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (cbrt (fma 0.3275911 x_m 1.0))))
   (if (<= (fabs x_m) 2e-8)
     (+
      (fma x_m 1.128386358070218 1e-9)
      (* (pow x_m 2.0) -0.00011824294398844343))
     (-
      1.0
      (/
       (/
        (+
         0.254829592
         (/
          (/
           (+
            -0.284496736
            (/
             (+
              1.421413741
              (/
               (+ (/ 1.061405429 (fma 0.3275911 x_m 1.0)) -1.453152027)
               (fma 0.3275911 x_m 1.0)))
             (fma 0.3275911 x_m 1.0)))
           (pow t_0 2.0))
          t_0))
        (pow (exp x_m) x_m))
       (fma 0.3275911 (fabs x_m) 1.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cbrt(fma(0.3275911, x_m, 1.0));
	double tmp;
	if (fabs(x_m) <= 2e-8) {
		tmp = fma(x_m, 1.128386358070218, 1e-9) + (pow(x_m, 2.0) * -0.00011824294398844343);
	} else {
		tmp = 1.0 - (((0.254829592 + (((-0.284496736 + ((1.421413741 + (((1.061405429 / fma(0.3275911, x_m, 1.0)) + -1.453152027) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / pow(t_0, 2.0)) / t_0)) / pow(exp(x_m), x_m)) / fma(0.3275911, fabs(x_m), 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = cbrt(fma(0.3275911, x_m, 1.0))
	tmp = 0.0
	if (abs(x_m) <= 2e-8)
		tmp = Float64(fma(x_m, 1.128386358070218, 1e-9) + Float64((x_m ^ 2.0) * -0.00011824294398844343));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.254829592 + Float64(Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(Float64(1.061405429 / fma(0.3275911, x_m, 1.0)) + -1.453152027) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / (t_0 ^ 2.0)) / t_0)) / (exp(x_m) ^ x_m)) / fma(0.3275911, abs(x_m), 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-8], N[(N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(0.254829592 + N[(N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[Power[N[Exp[x$95$m], $MachinePrecision], x$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

   5. Applied egg-rr 54.2%

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

   6. Taylor expanded in x around 0 96.9%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-commutative 96.9%

         \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

      2. associate-+r+ 96.9%

         \[\leadsto \color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right) + -0.00011824294398844343 \cdot {x}^{2}} \]

      3. +-commutative 96.9%

         \[\leadsto \color{blue}{\left(1.128386358070218 \cdot x + 10^{-9}\right)} + -0.00011824294398844343 \cdot {x}^{2} \]

      4. *-commutative 96.9%

         \[\leadsto \left(\color{blue}{x \cdot 1.128386358070218} + 10^{-9}\right) + -0.00011824294398844343 \cdot {x}^{2} \]

      5. fma-def 96.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} + -0.00011824294398844343 \cdot {x}^{2} \]

      6. *-commutative 96.9%

         \[\leadsto \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \color{blue}{{x}^{2} \cdot -0.00011824294398844343} \]

   8. Simplified 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + {x}^{2} \cdot -0.00011824294398844343} \]

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

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. add-cube-cbrt 99.7%

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

      3. times-frac 99.7%

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

      4. pow2 99.7%

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

      5. add-sqr-sqrt 49.2%

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

      6. fabs-sqr 49.2%

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

      7. add-sqr-sqrt 97.6%

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

   5. Applied egg-rr 97.1%

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

      1. associate-*r/ 97.1%

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

      2. associate-*l/ 97.1%

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

      3. *-lft-identity 97.1%

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

   7. Simplified 97.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + {x}^{2} \cdot -0.00011824294398844343\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \frac{\frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{{\left(\sqrt[3]{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}^{2}}}{\sqrt[3]{\mathsf{fma}\left(0.3275911, x, 1\right)}}}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}\\ \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right) + {x_m}^{2} \cdot -0.00011824294398844343\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\mathsf{fma}\left(t_0, t_0, -1.453152027\right)}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{{\left(e^{x_m}\right)}^{x_m}}}{\mathsf{fma}\left(0.3275911, \left|x_m\right|, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.061405429 (fma 0.3275911 x_m 1.0)))))
   (if (<= (fabs x_m) 2e-8)
     (+
      (fma x_m 1.128386358070218 1e-9)
      (* (pow x_m 2.0) -0.00011824294398844343))
     (-
      1.0
      (/
       (/
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/
            (+
             1.421413741
             (/ (fma t_0 t_0 -1.453152027) (fma 0.3275911 x_m 1.0)))
            (fma 0.3275911 x_m 1.0)))
          (fma 0.3275911 x_m 1.0)))
        (pow (exp x_m) x_m))
       (fma 0.3275911 (fabs x_m) 1.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = sqrt((1.061405429 / fma(0.3275911, x_m, 1.0)));
	double tmp;
	if (fabs(x_m) <= 2e-8) {
		tmp = fma(x_m, 1.128386358070218, 1e-9) + (pow(x_m, 2.0) * -0.00011824294398844343);
	} else {
		tmp = 1.0 - (((0.254829592 + ((-0.284496736 + ((1.421413741 + (fma(t_0, t_0, -1.453152027) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / pow(exp(x_m), x_m)) / fma(0.3275911, fabs(x_m), 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = sqrt(Float64(1.061405429 / fma(0.3275911, x_m, 1.0)))
	tmp = 0.0
	if (abs(x_m) <= 2e-8)
		tmp = Float64(fma(x_m, 1.128386358070218, 1e-9) + Float64((x_m ^ 2.0) * -0.00011824294398844343));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(fma(t_0, t_0, -1.453152027) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / (exp(x_m) ^ x_m)) / fma(0.3275911, abs(x_m), 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-8], N[(N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(t$95$0 * t$95$0 + -1.453152027), $MachinePrecision] / 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] / N[Power[N[Exp[x$95$m], $MachinePrecision], x$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

   5. Applied egg-rr 54.2%

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

   6. Taylor expanded in x around 0 96.9%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-commutative 96.9%

         \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

      2. associate-+r+ 96.9%

         \[\leadsto \color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right) + -0.00011824294398844343 \cdot {x}^{2}} \]

      3. +-commutative 96.9%

         \[\leadsto \color{blue}{\left(1.128386358070218 \cdot x + 10^{-9}\right)} + -0.00011824294398844343 \cdot {x}^{2} \]

      4. *-commutative 96.9%

         \[\leadsto \left(\color{blue}{x \cdot 1.128386358070218} + 10^{-9}\right) + -0.00011824294398844343 \cdot {x}^{2} \]

      5. fma-def 96.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} + -0.00011824294398844343 \cdot {x}^{2} \]

      6. *-commutative 96.9%

         \[\leadsto \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \color{blue}{{x}^{2} \cdot -0.00011824294398844343} \]

   8. Simplified 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + {x}^{2} \cdot -0.00011824294398844343} \]

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

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   5. Applied egg-rr 97.1%

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

      1. *-rgt-identity 97.1%

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

      2. *-commutative 97.1%

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

      3. unpow2 97.1%

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

      4. times-frac 97.1%

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

      5. distribute-rgt-in 97.1%

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

      6. associate-*l/ 97.1%

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

      7. *-lft-identity 97.1%

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

   7. Simplified 97.1%

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

      1. +-commutative 96.9%

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

      2. fma-udef 96.9%

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

      3. +-commutative 96.9%

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

      4. add-sqr-sqrt 49.3%

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

      5. fma-def 49.3%

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

      6. +-commutative 49.3%

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

      7. fma-udef 49.3%

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

      8. +-commutative 49.3%

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

      9. fma-udef 49.3%

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

   9. Applied egg-rr 49.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + {x}^{2} \cdot -0.00011824294398844343\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\mathsf{fma}\left(\sqrt{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}, \sqrt{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}, -1.453152027\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right) + \mathsf{fma}\left({x_m}^{3}, -0.37545125292247583, {x_m}^{2} \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{{\left(e^{x_m}\right)}^{x_m}}}{\mathsf{fma}\left(0.3275911, \left|x_m\right|, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.0005)
   (+
    (fma x_m 1.128386358070218 1e-9)
    (fma
     (pow x_m 3.0)
     -0.37545125292247583
     (* (pow x_m 2.0) -0.00011824294398844343)))
   (-
    1.0
    (/
     (/
      (+
       0.254829592
       (/
        (+
         -0.284496736
         (/
          (+
           1.421413741
           (/
            (+ (/ 1.061405429 (fma 0.3275911 x_m 1.0)) -1.453152027)
            (fma 0.3275911 x_m 1.0)))
          (fma 0.3275911 x_m 1.0)))
        (fma 0.3275911 x_m 1.0)))
      (pow (exp x_m) x_m))
     (fma 0.3275911 (fabs x_m) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.0005) {
		tmp = fma(x_m, 1.128386358070218, 1e-9) + fma(pow(x_m, 3.0), -0.37545125292247583, (pow(x_m, 2.0) * -0.00011824294398844343));
	} else {
		tmp = 1.0 - (((0.254829592 + ((-0.284496736 + ((1.421413741 + (((1.061405429 / fma(0.3275911, x_m, 1.0)) + -1.453152027) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / pow(exp(x_m), x_m)) / fma(0.3275911, fabs(x_m), 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.0005)
		tmp = Float64(fma(x_m, 1.128386358070218, 1e-9) + fma((x_m ^ 3.0), -0.37545125292247583, Float64((x_m ^ 2.0) * -0.00011824294398844343)));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(Float64(1.061405429 / fma(0.3275911, x_m, 1.0)) + -1.453152027) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / (exp(x_m) ^ x_m)) / fma(0.3275911, abs(x_m), 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.0005], N[(N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / 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] / N[Power[N[Exp[x$95$m], $MachinePrecision], x$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.0000000000000001e-4

   1. Initial program 57.9%

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

   3. Simplified 57.9%

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

   5. Applied egg-rr 53.8%

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

   6. Taylor expanded in x around 0 96.2%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 96.2%

         \[\leadsto \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) + 10^{-9}} \]

      2. associate-+r+ 96.2%

         \[\leadsto \color{blue}{\left(\left(-0.37545125292247583 \cdot {x}^{3} + -0.00011824294398844343 \cdot {x}^{2}\right) + 1.128386358070218 \cdot x\right)} + 10^{-9} \]

      3. associate-+l+ 96.2%

         \[\leadsto \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + -0.00011824294398844343 \cdot {x}^{2}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right)} \]

      4. *-commutative 96.2%

         \[\leadsto \left(\color{blue}{{x}^{3} \cdot -0.37545125292247583} + -0.00011824294398844343 \cdot {x}^{2}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      5. fma-def 96.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, -0.00011824294398844343 \cdot {x}^{2}\right)} + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      6. *-commutative 96.2%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      7. *-commutative 96.2%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \left(\color{blue}{x \cdot 1.128386358070218} + 10^{-9}\right) \]

      8. fma-def 96.2%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   8. Simplified 96.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   if 5.0000000000000001e-4 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Applied egg-rr 97.8%

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

      1. *-rgt-identity 97.8%

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

      2. *-commutative 97.8%

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

      3. unpow2 97.8%

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

      4. times-frac 97.8%

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

      5. distribute-rgt-in 97.8%

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

      6. associate-*l/ 97.8%

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

      7. *-lft-identity 97.8%

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

   7. Simplified 97.8%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.00058:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right) + \mathsf{fma}\left({x_m}^{3}, -0.37545125292247583, {x_m}^{2} \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.284496736 - \frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)} + -0.254829592}{\mathsf{fma}\left(0.3275911, x_m, 1\right) \cdot e^{{x_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00058)
   (+
    (fma x_m 1.128386358070218 1e-9)
    (fma
     (pow x_m 3.0)
     -0.37545125292247583
     (* (pow x_m 2.0) -0.00011824294398844343)))
   (+
    1.0
    (/
     (+
      (/
       (-
        0.284496736
        (/
         (+
          1.421413741
          (/
           (+ (/ 1.061405429 (fma 0.3275911 x_m 1.0)) -1.453152027)
           (fma 0.3275911 x_m 1.0)))
         (fma 0.3275911 x_m 1.0)))
       (fma 0.3275911 x_m 1.0))
      -0.254829592)
     (* (fma 0.3275911 x_m 1.0) (exp (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00058) {
		tmp = fma(x_m, 1.128386358070218, 1e-9) + fma(pow(x_m, 3.0), -0.37545125292247583, (pow(x_m, 2.0) * -0.00011824294398844343));
	} else {
		tmp = 1.0 + ((((0.284496736 - ((1.421413741 + (((1.061405429 / fma(0.3275911, x_m, 1.0)) + -1.453152027) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) + -0.254829592) / (fma(0.3275911, x_m, 1.0) * exp(pow(x_m, 2.0))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00058)
		tmp = Float64(fma(x_m, 1.128386358070218, 1e-9) + fma((x_m ^ 3.0), -0.37545125292247583, Float64((x_m ^ 2.0) * -0.00011824294398844343)));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(0.284496736 - Float64(Float64(1.421413741 + Float64(Float64(Float64(1.061405429 / fma(0.3275911, x_m, 1.0)) + -1.453152027) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) + -0.254829592) / Float64(fma(0.3275911, x_m, 1.0) * exp((x_m ^ 2.0)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.00058], N[(N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(0.284496736 - N[(N[(1.421413741 + N[(N[(N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / 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] + -0.254829592), $MachinePrecision] / N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e-4

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in x around 0 65.3%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 65.3%

         \[\leadsto \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) + 10^{-9}} \]

      2. associate-+r+ 65.3%

         \[\leadsto \color{blue}{\left(\left(-0.37545125292247583 \cdot {x}^{3} + -0.00011824294398844343 \cdot {x}^{2}\right) + 1.128386358070218 \cdot x\right)} + 10^{-9} \]

      3. associate-+l+ 65.3%

         \[\leadsto \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + -0.00011824294398844343 \cdot {x}^{2}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right)} \]

      4. *-commutative 65.3%

         \[\leadsto \left(\color{blue}{{x}^{3} \cdot -0.37545125292247583} + -0.00011824294398844343 \cdot {x}^{2}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      5. fma-def 65.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, -0.00011824294398844343 \cdot {x}^{2}\right)} + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      6. *-commutative 65.3%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      7. *-commutative 65.3%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \left(\color{blue}{x \cdot 1.128386358070218} + 10^{-9}\right) \]

      8. fma-def 65.3%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   8. Simplified 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   if 5.8e-4 < 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%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00058:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.284496736 - \frac{1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)} + -0.254829592}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot e^{{x}^{2}}}\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x_m\right| \cdot 0.3275911}\\ \mathbf{if}\;x_m \leq 0.00058:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right) + \mathsf{fma}\left({x_m}^{3}, -0.37545125292247583, {x_m}^{2} \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 - t_0 \cdot \left(\left(0.254829592 + t_0 \cdot \left(-0.284496736 + t_0 \cdot \left(1.421413741 + t_0 \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x_m \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{x_m \cdot \left(-x_m\right)}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x_m) 0.3275911)))))
   (if (<= x_m 0.00058)
     (+
      (fma x_m 1.128386358070218 1e-9)
      (fma
       (pow x_m 3.0)
       -0.37545125292247583
       (* (pow x_m 2.0) -0.00011824294398844343)))
     (-
      1.0
      (*
       t_0
       (*
        (+
         0.254829592
         (*
          t_0
          (+
           -0.284496736
           (*
            t_0
            (+
             1.421413741
             (*
              t_0
              (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* x_m 0.3275911))))))))))
        (exp (* x_m (- x_m)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (fabs(x_m) * 0.3275911));
	double tmp;
	if (x_m <= 0.00058) {
		tmp = fma(x_m, 1.128386358070218, 1e-9) + fma(pow(x_m, 3.0), -0.37545125292247583, (pow(x_m, 2.0) * -0.00011824294398844343));
	} else {
		tmp = 1.0 - (t_0 * ((0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911)))))))))) * exp((x_m * -x_m))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))
	tmp = 0.0
	if (x_m <= 0.00058)
		tmp = Float64(fma(x_m, 1.128386358070218, 1e-9) + fma((x_m ^ 3.0), -0.37545125292247583, Float64((x_m ^ 2.0) * -0.00011824294398844343)));
	else
		tmp = Float64(1.0 - Float64(t_0 * Float64(Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * Float64(-1.453152027 + Float64(1.061405429 / Float64(1.0 + Float64(x_m * 0.3275911)))))))))) * exp(Float64(x_m * Float64(-x_m))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00058], N[(N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(t$95$0 * N[(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[(1.061405429 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e-4

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in x around 0 65.3%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 65.3%

         \[\leadsto \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) + 10^{-9}} \]

      2. associate-+r+ 65.3%

         \[\leadsto \color{blue}{\left(\left(-0.37545125292247583 \cdot {x}^{3} + -0.00011824294398844343 \cdot {x}^{2}\right) + 1.128386358070218 \cdot x\right)} + 10^{-9} \]

      3. associate-+l+ 65.3%

         \[\leadsto \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + -0.00011824294398844343 \cdot {x}^{2}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right)} \]

      4. *-commutative 65.3%

         \[\leadsto \left(\color{blue}{{x}^{3} \cdot -0.37545125292247583} + -0.00011824294398844343 \cdot {x}^{2}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      5. fma-def 65.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, -0.00011824294398844343 \cdot {x}^{2}\right)} + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      6. *-commutative 65.3%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      7. *-commutative 65.3%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \left(\color{blue}{x \cdot 1.128386358070218} + 10^{-9}\right) \]

      8. fma-def 65.3%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   8. Simplified 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   if 5.8e-4 < 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%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}\right)} \]

      3. log1p-udef 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-udef 100.0%

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

      6. add-exp-log 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. fabs-sqr 100.0%

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

      9. add-sqr-sqrt 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. associate--l+ 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)\right)} \]

      4. +-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00058:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\right)\\ \end{array} \]

Alternative 7: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x_m \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right) + \mathsf{fma}\left({x_m}^{3}, -0.37545125292247583, {x_m}^{2} \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 - 1.453152027 \cdot \frac{1}{1 + x_m \cdot 0.3275911}\right)\right)\right)\right) \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

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 (<= x_m 1.0)
     (+
      (fma x_m 1.128386358070218 1e-9)
      (fma
       (pow x_m 3.0)
       -0.37545125292247583
       (* (pow x_m 2.0) -0.00011824294398844343)))
     (+
      1.0
      (*
       (*
        (exp (* x_m (- x_m)))
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (-
             1.421413741
             (* 1.453152027 (/ 1.0 (+ 1.0 (* x_m 0.3275911))))))))))
       (/ -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 / t_0;
	double tmp;
	if (x_m <= 1.0) {
		tmp = fma(x_m, 1.128386358070218, 1e-9) + fma(pow(x_m, 3.0), -0.37545125292247583, (pow(x_m, 2.0) * -0.00011824294398844343));
	} else {
		tmp = 1.0 + ((exp((x_m * -x_m)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 - (1.453152027 * (1.0 / (1.0 + (x_m * 0.3275911)))))))))) * (-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 / t_0)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(fma(x_m, 1.128386358070218, 1e-9) + fma((x_m ^ 3.0), -0.37545125292247583, Float64((x_m ^ 2.0) * -0.00011824294398844343)));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 - Float64(1.453152027 * Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911)))))))))) * Float64(-1.0 / t_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]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 1.0], N[(N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 - N[(1.453152027 * N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in x around 0 65.3%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 65.3%

         \[\leadsto \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) + 10^{-9}} \]

      2. associate-+r+ 65.3%

         \[\leadsto \color{blue}{\left(\left(-0.37545125292247583 \cdot {x}^{3} + -0.00011824294398844343 \cdot {x}^{2}\right) + 1.128386358070218 \cdot x\right)} + 10^{-9} \]

      3. associate-+l+ 65.3%

         \[\leadsto \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + -0.00011824294398844343 \cdot {x}^{2}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right)} \]

      4. *-commutative 65.3%

         \[\leadsto \left(\color{blue}{{x}^{3} \cdot -0.37545125292247583} + -0.00011824294398844343 \cdot {x}^{2}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      5. fma-def 65.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, -0.00011824294398844343 \cdot {x}^{2}\right)} + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      6. *-commutative 65.3%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      7. *-commutative 65.3%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \left(\color{blue}{x \cdot 1.128386358070218} + 10^{-9}\right) \]

      8. fma-def 65.3%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   8. Simplified 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   if 1 < 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%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}\right)} \]

      3. log1p-udef 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-udef 100.0%

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

      6. add-exp-log 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. fabs-sqr 100.0%

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

      9. add-sqr-sqrt 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. associate--l+ 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)\right)} \]

      4. +-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}\right)} \]

      3. log1p-udef 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-udef 100.0%

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

      6. add-exp-log 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. fabs-sqr 100.0%

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

      9. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. fma-udef 100.0%

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

       2. associate--l+ 100.0%

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

       3. metadata-eval 100.0%

          \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)\right)} \]

       4. +-rgt-identity 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 - 1.453152027 \cdot \frac{1}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 8: 99.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right) + \mathsf{fma}\left({x_m}^{3}, -0.37545125292247583, {x_m}^{2} \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 - 0.254829592 \cdot \frac{1}{e^{{x_m}^{2}} \cdot \left(1 + x_m \cdot 0.3275911\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.98)
   (+
    (fma x_m 1.128386358070218 1e-9)
    (fma
     (pow x_m 3.0)
     -0.37545125292247583
     (* (pow x_m 2.0) -0.00011824294398844343)))
   (-
    1.0
    (*
     0.254829592
     (/ 1.0 (* (exp (pow x_m 2.0)) (+ 1.0 (* x_m 0.3275911))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.98) {
		tmp = fma(x_m, 1.128386358070218, 1e-9) + fma(pow(x_m, 3.0), -0.37545125292247583, (pow(x_m, 2.0) * -0.00011824294398844343));
	} else {
		tmp = 1.0 - (0.254829592 * (1.0 / (exp(pow(x_m, 2.0)) * (1.0 + (x_m * 0.3275911)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.98)
		tmp = Float64(fma(x_m, 1.128386358070218, 1e-9) + fma((x_m ^ 3.0), -0.37545125292247583, Float64((x_m ^ 2.0) * -0.00011824294398844343)));
	else
		tmp = Float64(1.0 - Float64(0.254829592 * Float64(1.0 / Float64(exp((x_m ^ 2.0)) * Float64(1.0 + Float64(x_m * 0.3275911))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.98], N[(N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.37545125292247583 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.254829592 * N[(1.0 / N[(N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.97999999999999998

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in x around 0 65.3%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 65.3%

         \[\leadsto \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right) + 10^{-9}} \]

      2. associate-+r+ 65.3%

         \[\leadsto \color{blue}{\left(\left(-0.37545125292247583 \cdot {x}^{3} + -0.00011824294398844343 \cdot {x}^{2}\right) + 1.128386358070218 \cdot x\right)} + 10^{-9} \]

      3. associate-+l+ 65.3%

         \[\leadsto \color{blue}{\left(-0.37545125292247583 \cdot {x}^{3} + -0.00011824294398844343 \cdot {x}^{2}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right)} \]

      4. *-commutative 65.3%

         \[\leadsto \left(\color{blue}{{x}^{3} \cdot -0.37545125292247583} + -0.00011824294398844343 \cdot {x}^{2}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      5. fma-def 65.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, -0.00011824294398844343 \cdot {x}^{2}\right)} + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      6. *-commutative 65.3%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, \color{blue}{{x}^{2} \cdot -0.00011824294398844343}\right) + \left(1.128386358070218 \cdot x + 10^{-9}\right) \]

      7. *-commutative 65.3%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \left(\color{blue}{x \cdot 1.128386358070218} + 10^{-9}\right) \]

      8. fma-def 65.3%

         \[\leadsto \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   8. Simplified 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right) + \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   if 0.97999999999999998 < 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%

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

   5. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. times-frac 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. associate-*l/ 100.0%

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

      7. *-lft-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}\right)} \]

      3. log1p-udef 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-udef 100.0%

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

      6. add-exp-log 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. fabs-sqr 100.0%

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

      9. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. fma-udef 100.0%

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

       2. associate--l+ 100.0%

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

       3. metadata-eval 100.0%

          \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)\right)} \]

       4. +-rgt-identity 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \mathsf{fma}\left({x}^{3}, -0.37545125292247583, {x}^{2} \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + x \cdot 0.3275911\right)}\\ \end{array} \]

Alternative 9: 99.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.98:\\ \;\;\;\;10^{-9} + \left({x_m}^{3} \cdot -0.37545125292247583 + \left({x_m}^{2} \cdot -0.00011824294398844343 + x_m \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - 0.254829592 \cdot \frac{1}{e^{{x_m}^{2}} \cdot \left(1 + x_m \cdot 0.3275911\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.98)
   (+
    1e-9
    (+
     (* (pow x_m 3.0) -0.37545125292247583)
     (+ (* (pow x_m 2.0) -0.00011824294398844343) (* x_m 1.128386358070218))))
   (-
    1.0
    (*
     0.254829592
     (/ 1.0 (* (exp (pow x_m 2.0)) (+ 1.0 (* x_m 0.3275911))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.98) {
		tmp = 1e-9 + ((pow(x_m, 3.0) * -0.37545125292247583) + ((pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 - (0.254829592 * (1.0 / (exp(pow(x_m, 2.0)) * (1.0 + (x_m * 0.3275911)))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.98d0) then
        tmp = 1d-9 + (((x_m ** 3.0d0) * (-0.37545125292247583d0)) + (((x_m ** 2.0d0) * (-0.00011824294398844343d0)) + (x_m * 1.128386358070218d0)))
    else
        tmp = 1.0d0 - (0.254829592d0 * (1.0d0 / (exp((x_m ** 2.0d0)) * (1.0d0 + (x_m * 0.3275911d0)))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.98) {
		tmp = 1e-9 + ((Math.pow(x_m, 3.0) * -0.37545125292247583) + ((Math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)));
	} else {
		tmp = 1.0 - (0.254829592 * (1.0 / (Math.exp(Math.pow(x_m, 2.0)) * (1.0 + (x_m * 0.3275911)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.98:
		tmp = 1e-9 + ((math.pow(x_m, 3.0) * -0.37545125292247583) + ((math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)))
	else:
		tmp = 1.0 - (0.254829592 * (1.0 / (math.exp(math.pow(x_m, 2.0)) * (1.0 + (x_m * 0.3275911)))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.98)
		tmp = Float64(1e-9 + Float64(Float64((x_m ^ 3.0) * -0.37545125292247583) + Float64(Float64((x_m ^ 2.0) * -0.00011824294398844343) + Float64(x_m * 1.128386358070218))));
	else
		tmp = Float64(1.0 - Float64(0.254829592 * Float64(1.0 / Float64(exp((x_m ^ 2.0)) * Float64(1.0 + Float64(x_m * 0.3275911))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.98)
		tmp = 1e-9 + (((x_m ^ 3.0) * -0.37545125292247583) + (((x_m ^ 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218)));
	else
		tmp = 1.0 - (0.254829592 * (1.0 / (exp((x_m ^ 2.0)) * (1.0 + (x_m * 0.3275911)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.98], N[(1e-9 + N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.37545125292247583), $MachinePrecision] + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.254829592 * N[(1.0 / N[(N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.97999999999999998

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in x around 0 65.3%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.37545125292247583 \cdot {x}^{3} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)} \]

   if 0.97999999999999998 < 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%

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

   5. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. times-frac 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. associate-*l/ 100.0%

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

      7. *-lft-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}\right)} \]

      3. log1p-udef 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-udef 100.0%

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

      6. add-exp-log 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. fabs-sqr 100.0%

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

      9. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. fma-udef 100.0%

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

       2. associate--l+ 100.0%

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

       3. metadata-eval 100.0%

          \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)\right)} \]

       4. +-rgt-identity 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.98:\\ \;\;\;\;10^{-9} + \left({x}^{3} \cdot -0.37545125292247583 + \left({x}^{2} \cdot -0.00011824294398844343 + x \cdot 1.128386358070218\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + x \cdot 0.3275911\right)}\\ \end{array} \]

Alternative 10: 99.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right) + {x_m}^{2} \cdot -0.00011824294398844343\\ \mathbf{else}:\\ \;\;\;\;1 - 0.254829592 \cdot \frac{1}{e^{{x_m}^{2}} \cdot \left(1 + x_m \cdot 0.3275911\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.8)
   (+
    (fma x_m 1.128386358070218 1e-9)
    (* (pow x_m 2.0) -0.00011824294398844343))
   (-
    1.0
    (*
     0.254829592
     (/ 1.0 (* (exp (pow x_m 2.0)) (+ 1.0 (* x_m 0.3275911))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.8) {
		tmp = fma(x_m, 1.128386358070218, 1e-9) + (pow(x_m, 2.0) * -0.00011824294398844343);
	} else {
		tmp = 1.0 - (0.254829592 * (1.0 / (exp(pow(x_m, 2.0)) * (1.0 + (x_m * 0.3275911)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.8)
		tmp = Float64(fma(x_m, 1.128386358070218, 1e-9) + Float64((x_m ^ 2.0) * -0.00011824294398844343));
	else
		tmp = Float64(1.0 - Float64(0.254829592 * Float64(1.0 / Float64(exp((x_m ^ 2.0)) * Float64(1.0 + Float64(x_m * 0.3275911))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.8], N[(N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.254829592 * N[(1.0 / N[(N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in x around 0 64.6%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-commutative 64.6%

         \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

      2. associate-+r+ 64.6%

         \[\leadsto \color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right) + -0.00011824294398844343 \cdot {x}^{2}} \]

      3. +-commutative 64.6%

         \[\leadsto \color{blue}{\left(1.128386358070218 \cdot x + 10^{-9}\right)} + -0.00011824294398844343 \cdot {x}^{2} \]

      4. *-commutative 64.6%

         \[\leadsto \left(\color{blue}{x \cdot 1.128386358070218} + 10^{-9}\right) + -0.00011824294398844343 \cdot {x}^{2} \]

      5. fma-def 64.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} + -0.00011824294398844343 \cdot {x}^{2} \]

      6. *-commutative 64.6%

         \[\leadsto \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \color{blue}{{x}^{2} \cdot -0.00011824294398844343} \]

   8. Simplified 64.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + {x}^{2} \cdot -0.00011824294398844343} \]

   if 0.80000000000000004 < 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%

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

   5. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. times-frac 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. associate-*l/ 100.0%

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

      7. *-lft-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)\right)}\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}\right)} \]

      3. log1p-udef 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-udef 100.0%

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

      6. add-exp-log 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. fabs-sqr 100.0%

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

      9. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. fma-udef 100.0%

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

       2. associate--l+ 100.0%

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

       3. metadata-eval 100.0%

          \[\leadsto 1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)\right)} \]

       4. +-rgt-identity 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + {x}^{2} \cdot -0.00011824294398844343\\ \mathbf{else}:\\ \;\;\;\;1 - 0.254829592 \cdot \frac{1}{e^{{x}^{2}} \cdot \left(1 + x \cdot 0.3275911\right)}\\ \end{array} \]

Alternative 11: 99.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right) + {x_m}^{2} \cdot -0.00011824294398844343\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x_m + {x_m}^{3}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.0)
   (+
    (fma x_m 1.128386358070218 1e-9)
    (* (pow x_m 2.0) -0.00011824294398844343))
   (- 1.0 (/ 0.7778892405807117 (+ x_m (pow x_m 3.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = fma(x_m, 1.128386358070218, 1e-9) + (pow(x_m, 2.0) * -0.00011824294398844343);
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x_m + pow(x_m, 3.0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(fma(x_m, 1.128386358070218, 1e-9) + Float64((x_m ^ 2.0) * -0.00011824294398844343));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x_m + (x_m ^ 3.0))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.0], N[(N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x$95$m + N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in x around 0 64.6%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-commutative 64.6%

         \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

      2. associate-+r+ 64.6%

         \[\leadsto \color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right) + -0.00011824294398844343 \cdot {x}^{2}} \]

      3. +-commutative 64.6%

         \[\leadsto \color{blue}{\left(1.128386358070218 \cdot x + 10^{-9}\right)} + -0.00011824294398844343 \cdot {x}^{2} \]

      4. *-commutative 64.6%

         \[\leadsto \left(\color{blue}{x \cdot 1.128386358070218} + 10^{-9}\right) + -0.00011824294398844343 \cdot {x}^{2} \]

      5. fma-def 64.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} + -0.00011824294398844343 \cdot {x}^{2} \]

      6. *-commutative 64.6%

         \[\leadsto \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \color{blue}{{x}^{2} \cdot -0.00011824294398844343} \]

   8. Simplified 64.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + {x}^{2} \cdot -0.00011824294398844343} \]

   if 1 < 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%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + {x}^{2} \cdot -0.00011824294398844343\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x + {x}^{3}}\\ \end{array} \]

Alternative 12: 99.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.88:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right) + {x_m}^{2} \cdot -0.00011824294398844343\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x_m \cdot e^{{x_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88)
   (+
    (fma x_m 1.128386358070218 1e-9)
    (* (pow x_m 2.0) -0.00011824294398844343))
   (- 1.0 (/ 0.7778892405807117 (* x_m (exp (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = fma(x_m, 1.128386358070218, 1e-9) + (pow(x_m, 2.0) * -0.00011824294398844343);
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x_m * exp(pow(x_m, 2.0))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.88)
		tmp = Float64(fma(x_m, 1.128386358070218, 1e-9) + Float64((x_m ^ 2.0) * -0.00011824294398844343));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x_m * exp((x_m ^ 2.0)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.88], N[(N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision] + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x$95$m * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in x around 0 64.6%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-commutative 64.6%

         \[\leadsto 10^{-9} + \color{blue}{\left(1.128386358070218 \cdot x + -0.00011824294398844343 \cdot {x}^{2}\right)} \]

      2. associate-+r+ 64.6%

         \[\leadsto \color{blue}{\left(10^{-9} + 1.128386358070218 \cdot x\right) + -0.00011824294398844343 \cdot {x}^{2}} \]

      3. +-commutative 64.6%

         \[\leadsto \color{blue}{\left(1.128386358070218 \cdot x + 10^{-9}\right)} + -0.00011824294398844343 \cdot {x}^{2} \]

      4. *-commutative 64.6%

         \[\leadsto \left(\color{blue}{x \cdot 1.128386358070218} + 10^{-9}\right) + -0.00011824294398844343 \cdot {x}^{2} \]

      5. fma-def 64.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} + -0.00011824294398844343 \cdot {x}^{2} \]

      6. *-commutative 64.6%

         \[\leadsto \mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + \color{blue}{{x}^{2} \cdot -0.00011824294398844343} \]

   8. Simplified 64.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + {x}^{2} \cdot -0.00011824294398844343} \]

   if 0.880000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right) + {x}^{2} \cdot -0.00011824294398844343\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x \cdot e^{{x}^{2}}}\\ \end{array} \]

Alternative 13: 99.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1:\\ \;\;\;\;10^{-9} + \left({x_m}^{2} \cdot -0.00011824294398844343 + x_m \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x_m + {x_m}^{3}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.0)
   (+
    1e-9
    (+ (* (pow x_m 2.0) -0.00011824294398844343) (* x_m 1.128386358070218)))
   (- 1.0 (/ 0.7778892405807117 (+ x_m (pow x_m 3.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = 1e-9 + ((pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x_m + pow(x_m, 3.0)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.0d0) then
        tmp = 1d-9 + (((x_m ** 2.0d0) * (-0.00011824294398844343d0)) + (x_m * 1.128386358070218d0))
    else
        tmp = 1.0d0 - (0.7778892405807117d0 / (x_m + (x_m ** 3.0d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = 1e-9 + ((Math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x_m + Math.pow(x_m, 3.0)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = 1e-9 + ((math.pow(x_m, 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218))
	else:
		tmp = 1.0 - (0.7778892405807117 / (x_m + math.pow(x_m, 3.0)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(1e-9 + Float64(Float64((x_m ^ 2.0) * -0.00011824294398844343) + Float64(x_m * 1.128386358070218)));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x_m + (x_m ^ 3.0))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = 1e-9 + (((x_m ^ 2.0) * -0.00011824294398844343) + (x_m * 1.128386358070218));
	else
		tmp = 1.0 - (0.7778892405807117 / (x_m + (x_m ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.0], N[(1e-9 + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision] + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x$95$m + N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in x around 0 64.6%

      \[\leadsto \color{blue}{10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)} \]

   if 1 < 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%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;10^{-9} + \left({x}^{2} \cdot -0.00011824294398844343 + x \cdot 1.128386358070218\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x + {x}^{3}}\\ \end{array} \]

Alternative 14: 99.0% accurate, 7.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x_m + {x_m}^{3}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.0)
   (fma x_m 1.128386358070218 1e-9)
   (- 1.0 (/ 0.7778892405807117 (+ x_m (pow x_m 3.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = fma(x_m, 1.128386358070218, 1e-9);
	} else {
		tmp = 1.0 - (0.7778892405807117 / (x_m + pow(x_m, 3.0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = fma(x_m, 1.128386358070218, 1e-9);
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / Float64(x_m + (x_m ^ 3.0))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.0], N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / N[(x$95$m + N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in x around 0 64.6%

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

      1. +-commutative 64.6%

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

      2. *-commutative 64.6%

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

      3. fma-def 64.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   8. Simplified 64.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   if 1 < 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%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x + {x}^{3}}\\ \end{array} \]

Alternative 15: 98.3% accurate, 7.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.7:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.7)
   (fma x_m 1.128386358070218 1e-9)
   (+ 1.0 (/ -0.254829592 (fma x_m 0.3275911 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.7) {
		tmp = fma(x_m, 1.128386358070218, 1e-9);
	} else {
		tmp = 1.0 + (-0.254829592 / fma(x_m, 0.3275911, 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.7)
		tmp = fma(x_m, 1.128386358070218, 1e-9);
	else
		tmp = Float64(1.0 + Float64(-0.254829592 / fma(x_m, 0.3275911, 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.7], N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision], N[(1.0 + N[(-0.254829592 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in x around 0 64.6%

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

      1. +-commutative 64.6%

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

      2. *-commutative 64.6%

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

      3. fma-def 64.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   8. Simplified 64.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   if 0.69999999999999996 < 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%

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

   5. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. times-frac 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. associate-*l/ 100.0%

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

      7. *-lft-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   9. Taylor expanded in x around 0 98.6%

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

       1. sub-neg 98.6%

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

       2. associate-*r/ 98.6%

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

       3. metadata-eval 98.6%

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

       4. distribute-neg-frac 98.6%

          \[\leadsto 1 + \color{blue}{\frac{-0.254829592}{1 + 0.3275911 \cdot \left|x\right|}} \]

       5. metadata-eval 98.6%

          \[\leadsto 1 + \frac{\color{blue}{-0.254829592}}{1 + 0.3275911 \cdot \left|x\right|} \]

       6. +-commutative 98.6%

          \[\leadsto 1 + \frac{-0.254829592}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \]

       7. *-commutative 98.6%

          \[\leadsto 1 + \frac{-0.254829592}{\color{blue}{\left|x\right| \cdot 0.3275911} + 1} \]

       8. fma-def 98.6%

          \[\leadsto 1 + \frac{-0.254829592}{\color{blue}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \]

       9. unpow1 98.6%

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

       10. sqr-pow 98.6%

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

       11. fabs-sqr 98.6%

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

       12. sqr-pow 98.6%

           \[\leadsto 1 + \frac{-0.254829592}{\mathsf{fma}\left(\color{blue}{{x}^{1}}, 0.3275911, 1\right)} \]

       13. unpow1 98.6%

           \[\leadsto 1 + \frac{-0.254829592}{\mathsf{fma}\left(\color{blue}{x}, 0.3275911, 1\right)} \]

   11. Simplified 98.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592}{\mathsf{fma}\left(x, 0.3275911, 1\right)}\\ \end{array} \]

Alternative 16: 98.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.65:\\ \;\;\;\;\mathsf{fma}\left(x_m, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.65)
   (fma x_m 1.128386358070218 1e-9)
   (- 1.0 (/ 0.7778892405807117 x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.65) {
		tmp = fma(x_m, 1.128386358070218, 1e-9);
	} else {
		tmp = 1.0 - (0.7778892405807117 / x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.65)
		tmp = fma(x_m, 1.128386358070218, 1e-9);
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.65], N[(x$95$m * 1.128386358070218 + 1e-9), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in x around 0 64.6%

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

      1. +-commutative 64.6%

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

      2. *-commutative 64.6%

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

      3. fma-def 64.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   8. Simplified 64.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)} \]

   if 1.6499999999999999 < 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%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 98.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;\mathsf{fma}\left(x, 1.128386358070218, 10^{-9}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x}\\ \end{array} \]

Alternative 17: 54.4% accurate, 121.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 900000000:\\ \;\;\;\;10^{-9} + x_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 900000000.0) (+ 1e-9 (* x_m 1.128386358070218)) 1e-9))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 900000000.0) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1e-9;
	}
	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 <= 900000000.0d0) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 1d-9
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 900000000.0) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1e-9;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 900000000.0:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 1e-9
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 900000000.0)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = 1e-9;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 900000000.0)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 1e-9;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 900000000.0], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1e-9]

Derivation

1. Split input into 2 regimes

2. if x < 9e8

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

   5. Applied egg-rr 67.7%

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

   6. Taylor expanded in x around 0 64.3%

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

      1. *-commutative 64.3%

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

   8. Simplified 64.3%

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

   if 9e8 < 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%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 11.1%

      \[\leadsto \color{blue}{10^{-9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 900000000:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;10^{-9}\\ \end{array} \]

Alternative 18: 98.3% accurate, 121.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.65:\\ \;\;\;\;10^{-9} + x_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.65)
   (+ 1e-9 (* x_m 1.128386358070218))
   (- 1.0 (/ 0.7778892405807117 x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.65) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 - (0.7778892405807117 / x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.65d0) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 1.0d0 - (0.7778892405807117d0 / x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.65) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0 - (0.7778892405807117 / x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.65:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 1.0 - (0.7778892405807117 / x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.65)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = Float64(1.0 - Float64(0.7778892405807117 / x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.65)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 1.0 - (0.7778892405807117 / x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.65], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.7778892405807117 / x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in x around 0 64.6%

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

      1. *-commutative 64.6%

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

   8. Simplified 64.6%

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

   if 1.6499999999999999 < 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%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 98.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.7778892405807117}{x}\\ \end{array} \]

Alternative 19: 52.6% accurate, 856.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

3. Simplified 78.7%

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

5. Applied egg-rr 75.6%

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

6. Taylor expanded in x around 0 53.6%

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

7. Final simplification 53.6%

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

Reproduce

herbie shell --seed 2023322 
(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)))))))