Jmat.Real.erf

Percentage Accurate: 78.8% → 98.8%

Time: 25.9s

Alternatives: 12

Speedup: 142.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.3275911, x_m, 1\right)\right)}^{3}\\ \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \left(\frac{1.453152027}{t_0} - \frac{\frac{1.421413741}{\mathsf{fma}\left(0.3275911, x_m, 1\right)} + \left(\frac{1.061405429}{t_0} + -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}\right), \frac{{\left(e^{x_m}\right)}^{\left(-x_m\right)}}{\mathsf{fma}\left(0.3275911, \left|x_m\right|, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (pow (fma 0.3275911 x_m 1.0) 3.0)))
   (if (<= (fabs x_m) 2e-17)
     (+ 1e-9 (* x_m 0.3275910996724089))
     (fma
      (+
       -0.254829592
       (-
        (/ 1.453152027 t_0)
        (/
         (+
          (/ 1.421413741 (fma 0.3275911 x_m 1.0))
          (+ (/ 1.061405429 t_0) -0.284496736))
         (fma 0.3275911 x_m 1.0))))
      (/ (pow (exp x_m) (- x_m)) (fma 0.3275911 (fabs x_m) 1.0))
      1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = pow(fma(0.3275911, x_m, 1.0), 3.0);
	double tmp;
	if (fabs(x_m) <= 2e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = fma((-0.254829592 + ((1.453152027 / t_0) - (((1.421413741 / fma(0.3275911, x_m, 1.0)) + ((1.061405429 / t_0) + -0.284496736)) / fma(0.3275911, x_m, 1.0)))), (pow(exp(x_m), -x_m) / fma(0.3275911, fabs(x_m), 1.0)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(0.3275911, x_m, 1.0) ^ 3.0
	tmp = 0.0
	if (abs(x_m) <= 2e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = fma(Float64(-0.254829592 + Float64(Float64(1.453152027 / t_0) - Float64(Float64(Float64(1.421413741 / fma(0.3275911, x_m, 1.0)) + Float64(Float64(1.061405429 / t_0) + -0.284496736)) / fma(0.3275911, x_m, 1.0)))), Float64((exp(x_m) ^ Float64(-x_m)) / fma(0.3275911, abs(x_m), 1.0)), 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], 3.0], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[(N[(-0.254829592 + N[(N[(1.453152027 / t$95$0), $MachinePrecision] - N[(N[(N[(1.421413741 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.061405429 / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000014e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. associate--r+ 57.8%

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

      2. div-sub 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 57.8%

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

      2. expm1-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

      3. log1p-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

      4. add-exp-log 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

      5. +-commutative 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

      6. fma-udef 57.8%

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

      7. add-sqr-sqrt 24.7%

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

      8. fabs-sqr 24.7%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Taylor expanded in x around 0 98.6%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   14. Step-by-step derivation

       1. *-commutative 98.6%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   15. Simplified 98.6%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 2.00000000000000014e-17 < (fabs.f64 x)

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 97.6%

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

      1. associate--r+ 97.6%

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

      2. div-sub 97.6%

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

   7. Simplified 96.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \left(\frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \frac{\frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} + -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right), \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \log \left(1 + \mathsf{expm1}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}\right)\right)}{{\left(e^{x_m}\right)}^{x_m}}}{\mathsf{fma}\left(0.3275911, \left|x_m\right|, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 2e-17)
   (+ 1e-9 (* x_m 0.3275910996724089))
   (-
    1.0
    (/
     (/
      (+
       0.254829592
       (log
        (+
         1.0
         (expm1
          (/
           (+
            -0.284496736
            (/
             (+
              1.421413741
              (/
               (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
               (fma 0.3275911 x_m 1.0)))
             (fma 0.3275911 x_m 1.0)))
           (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) <= 2e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0 - (((0.254829592 + log((1.0 + expm1(((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / 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) <= 2e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.254829592 + log(Float64(1.0 + expm1(Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / 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], 2e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(0.254829592 + N[Log[N[(1.0 + N[(Exp[N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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]] - 1), $MachinePrecision]), $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) < 2.00000000000000014e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. associate--r+ 57.8%

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

      2. div-sub 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 57.8%

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

      2. expm1-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

      3. log1p-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

      4. add-exp-log 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

      5. +-commutative 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

      6. fma-udef 57.8%

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

      7. add-sqr-sqrt 24.7%

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

      8. fabs-sqr 24.7%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Taylor expanded in x around 0 98.6%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   14. Step-by-step derivation

       1. *-commutative 98.6%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   15. Simplified 98.6%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 2.00000000000000014e-17 < (fabs.f64 x)

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

      1. log1p-expm1-u 97.6%

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

      2. log1p-udef 97.7%

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

   6. Applied egg-rr 96.7%

      \[\leadsto 1 - \frac{\frac{0.254829592 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \log \left(1 + \mathsf{expm1}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.3275911, x_m, 1\right)\right)}^{3}\\ \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{\left(\frac{1.453152027}{t_0} - \frac{\frac{1.421413741}{\mathsf{fma}\left(0.3275911, x_m, 1\right)} + \left(\frac{1.061405429}{t_0} + -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}\right) - 0.254829592}{{\left(e^{x_m}\right)}^{x_m}}}{\mathsf{fma}\left(0.3275911, \left|x_m\right|, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (pow (fma 0.3275911 x_m 1.0) 3.0)))
   (if (<= (fabs x_m) 2e-17)
     (+ 1e-9 (* x_m 0.3275910996724089))
     (+
      1.0
      (/
       (/
        (-
         (-
          (/ 1.453152027 t_0)
          (/
           (+
            (/ 1.421413741 (fma 0.3275911 x_m 1.0))
            (+ (/ 1.061405429 t_0) -0.284496736))
           (fma 0.3275911 x_m 1.0)))
         0.254829592)
        (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 = pow(fma(0.3275911, x_m, 1.0), 3.0);
	double tmp;
	if (fabs(x_m) <= 2e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0 + (((((1.453152027 / t_0) - (((1.421413741 / fma(0.3275911, x_m, 1.0)) + ((1.061405429 / t_0) + -0.284496736)) / fma(0.3275911, x_m, 1.0))) - 0.254829592) / 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 = fma(0.3275911, x_m, 1.0) ^ 3.0
	tmp = 0.0
	if (abs(x_m) <= 2e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(Float64(1.453152027 / t_0) - Float64(Float64(Float64(1.421413741 / fma(0.3275911, x_m, 1.0)) + Float64(Float64(1.061405429 / t_0) + -0.284496736)) / fma(0.3275911, x_m, 1.0))) - 0.254829592) / (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], 3.0], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(N[(1.453152027 / t$95$0), $MachinePrecision] - N[(N[(N[(1.421413741 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.061405429 / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.254829592), $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) < 2.00000000000000014e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. associate--r+ 57.8%

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

      2. div-sub 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 57.8%

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

      2. expm1-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

      3. log1p-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

      4. add-exp-log 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

      5. +-commutative 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

      6. fma-udef 57.8%

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

      7. add-sqr-sqrt 24.7%

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

      8. fabs-sqr 24.7%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Taylor expanded in x around 0 98.6%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   14. Step-by-step derivation

       1. *-commutative 98.6%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   15. Simplified 98.6%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 2.00000000000000014e-17 < (fabs.f64 x)

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 97.6%

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

      1. associate--r+ 97.6%

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

      2. div-sub 97.6%

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

   7. Simplified 96.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{\left(\frac{1.453152027}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} - \frac{\frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)} + \left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{3}} + -0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - 0.254829592}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 + e^{-x_m \cdot x_m} \cdot \left(\left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot {\left(\sqrt{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}\right)}^{2}\right)\right) \cdot \frac{-1}{t_0}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x_m) 2e-17)
     (+ 1e-9 (* x_m 0.3275910996724089))
     (+
      1.0
      (*
       (exp (- (* x_m x_m)))
       (*
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (pow
             (sqrt
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
                (fma 0.3275911 x_m 1.0))))
             2.0)))))
        (/ -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 (fabs(x_m) <= 2e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0 + (exp(-(x_m * x_m)) * ((0.254829592 + (t_1 * (-0.284496736 + (t_1 * pow(sqrt((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)))), 2.0))))) * (-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 (abs(x_m) <= 2e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(-Float64(x_m * x_m))) * Float64(Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * (sqrt(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)))) ^ 2.0))))) * 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[N[Abs[x$95$m], $MachinePrecision], 2e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[(-N[(x$95$m * x$95$m), $MachinePrecision])], $MachinePrecision] * N[(N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[Power[N[Sqrt[N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000014e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. associate--r+ 57.8%

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

      2. div-sub 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 57.8%

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

      2. expm1-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

      3. log1p-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

      4. add-exp-log 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

      5. +-commutative 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

      6. fma-udef 57.8%

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

      7. add-sqr-sqrt 24.7%

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

      8. fabs-sqr 24.7%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Taylor expanded in x around 0 98.6%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   14. Step-by-step derivation

       1. *-commutative 98.6%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   15. Simplified 98.6%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 2.00000000000000014e-17 < (fabs.f64 x)

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

      1. associate-*l/ 97.6%

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

      2. *-un-lft-identity 97.6%

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

      3. +-commutative 97.6%

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

      4. fma-udef 97.6%

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

      5. +-commutative 97.6%

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

      6. fma-udef 97.6%

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

      7. add-sqr-sqrt 97.7%

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

      8. pow2 97.7%

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

   6. Applied egg-rr 97.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 + e^{-x \cdot x} \cdot \left(\left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot {\left(\sqrt{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}^{2}\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \frac{-1}{\frac{\mathsf{fma}\left(0.3275911, x_m, 1\right)}{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}}, \frac{{\left(e^{x_m}\right)}^{\left(-x_m\right)}}{1 + x_m \cdot 0.3275911}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 2e-17)
   (+ 1e-9 (* x_m 0.3275910996724089))
   (fma
    (+
     -0.254829592
     (/
      -1.0
      (/
       (fma 0.3275911 x_m 1.0)
       (+
        -0.284496736
        (/
         (+
          1.421413741
          (/
           (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
           (fma 0.3275911 x_m 1.0)))
         (fma 0.3275911 x_m 1.0))))))
    (/ (pow (exp x_m) (- x_m)) (+ 1.0 (* x_m 0.3275911)))
    1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 2e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = fma((-0.254829592 + (-1.0 / (fma(0.3275911, x_m, 1.0) / (-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)))))), (pow(exp(x_m), -x_m) / (1.0 + (x_m * 0.3275911))), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 2e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = fma(Float64(-0.254829592 + Float64(-1.0 / Float64(fma(0.3275911, x_m, 1.0) / Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)))))), Float64((exp(x_m) ^ Float64(-x_m)) / Float64(1.0 + Float64(x_m * 0.3275911))), 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], 2e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[(N[(-0.254829592 + N[(-1.0 / N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] / N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision] / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000014e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. associate--r+ 57.8%

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

      2. div-sub 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 57.8%

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

      2. expm1-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

      3. log1p-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

      4. add-exp-log 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

      5. +-commutative 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

      6. fma-udef 57.8%

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

      7. add-sqr-sqrt 24.7%

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

      8. fabs-sqr 24.7%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Taylor expanded in x around 0 98.6%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   14. Step-by-step derivation

       1. *-commutative 98.6%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   15. Simplified 98.6%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 2.00000000000000014e-17 < (fabs.f64 x)

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

      1. add-cube-cbrt 97.6%

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

      2. pow3 97.6%

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

   6. Applied egg-rr 96.8%

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

      1. rem-cube-cbrt 96.7%

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

      2. clear-num 96.7%

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

   8. Applied egg-rr 96.7%

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

      1. fma-udef 96.7%

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

   10. Applied egg-rr 96.7%

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

       1. expm1-log1p-u 93.6%

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

       2. expm1-udef 93.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

       3. log1p-udef 93.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

       4. add-exp-log 93.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

       5. +-commutative 93.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

       6. fma-udef 93.6%

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

       7. add-sqr-sqrt 39.9%

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

       8. fabs-sqr 39.9%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

       9. add-sqr-sqrt 93.5%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   12. Applied egg-rr 96.7%

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

       1. fma-udef 93.5%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 93.5%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 93.5%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 93.5%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Simplified 96.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \frac{-1}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{1 + x \cdot 0.3275911}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{{\left(e^{x_m}\right)}^{x_m}}}{1 + x_m \cdot 0.3275911}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 2e-17)
   (+ 1e-9 (* x_m 0.3275910996724089))
   (-
    1.0
    (/
     (/
      (+
       0.254829592
       (/
        (+
         -0.284496736
         (/
          (+
           1.421413741
           (/
            (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
            (fma 0.3275911 x_m 1.0)))
          (fma 0.3275911 x_m 1.0)))
        (fma 0.3275911 x_m 1.0)))
      (pow (exp x_m) x_m))
     (+ 1.0 (* x_m 0.3275911))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 2e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0 - (((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / pow(exp(x_m), x_m)) / (1.0 + (x_m * 0.3275911)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 2e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / 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)) / 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[N[Abs[x$95$m], $MachinePrecision], 2e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000014e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. associate--r+ 57.8%

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

      2. div-sub 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 57.8%

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

      2. expm1-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

      3. log1p-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

      4. add-exp-log 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

      5. +-commutative 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

      6. fma-udef 57.8%

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

      7. add-sqr-sqrt 24.7%

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

      8. fabs-sqr 24.7%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Taylor expanded in x around 0 98.6%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   14. Step-by-step derivation

       1. *-commutative 98.6%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   15. Simplified 98.6%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 2.00000000000000014e-17 < (fabs.f64 x)

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

   6. Applied egg-rr 96.7%

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

      1. unpow2 96.7%

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

      2. *-lft-identity 96.7%

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

      3. times-frac 96.7%

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

      4. distribute-lft-in 96.7%

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

      5. associate-*l/ 96.7%

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

      6. *-lft-identity 96.7%

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

   8. Simplified 96.7%

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

      1. fma-udef 96.7%

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

   10. Applied egg-rr 96.7%

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

       1. expm1-log1p-u 93.6%

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

       2. expm1-udef 93.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

       3. log1p-udef 93.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

       4. add-exp-log 93.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

       5. +-commutative 93.6%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

       6. fma-udef 93.6%

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

       7. add-sqr-sqrt 39.9%

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

       8. fabs-sqr 39.9%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

       9. add-sqr-sqrt 93.5%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   12. Applied egg-rr 96.7%

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

       1. fma-udef 93.5%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 93.5%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 93.5%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 93.5%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   14. Simplified 96.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{{\left(e^{x}\right)}^{x}}}{1 + x \cdot 0.3275911}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 + e^{-x_m \cdot x_m} \cdot \left(t_1 \cdot \left(t_1 \cdot \left(t_1 \cdot \left(\left(-1.453152027 + \frac{1.061405429}{1 + x_m \cdot 0.3275911}\right) \cdot \frac{-1}{t_0} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x_m) 2e-17)
     (+ 1e-9 (* x_m 0.3275910996724089))
     (+
      1.0
      (*
       (exp (- (* x_m x_m)))
       (*
        t_1
        (-
         (*
          t_1
          (-
           (*
            t_1
            (-
             (*
              (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* x_m 0.3275911))))
              (/ -1.0 t_0))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x_m) <= 2e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0 + (exp(-(x_m * x_m)) * (t_1 * ((t_1 * ((t_1 * (((-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911)))) * (-1.0 / t_0)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    if (abs(x_m) <= 2d-17) then
        tmp = 1d-9 + (x_m * 0.3275910996724089d0)
    else
        tmp = 1.0d0 + (exp(-(x_m * x_m)) * (t_1 * ((t_1 * ((t_1 * ((((-1.453152027d0) + (1.061405429d0 / (1.0d0 + (x_m * 0.3275911d0)))) * ((-1.0d0) / t_0)) - 1.421413741d0)) - (-0.284496736d0))) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (Math.abs(x_m) <= 2e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0 + (Math.exp(-(x_m * x_m)) * (t_1 * ((t_1 * ((t_1 * (((-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911)))) * (-1.0 / t_0)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (math.fabs(x_m) * 0.3275911)
	t_1 = 1.0 / t_0
	tmp = 0
	if math.fabs(x_m) <= 2e-17:
		tmp = 1e-9 + (x_m * 0.3275910996724089)
	else:
		tmp = 1.0 + (math.exp(-(x_m * x_m)) * (t_1 * ((t_1 * ((t_1 * (((-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911)))) * (-1.0 / t_0)) - 1.421413741)) - -0.284496736)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x_m) <= 2e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(-Float64(x_m * x_m))) * Float64(t_1 * Float64(Float64(t_1 * Float64(Float64(t_1 * Float64(Float64(Float64(-1.453152027 + Float64(1.061405429 / Float64(1.0 + Float64(x_m * 0.3275911)))) * Float64(-1.0 / t_0)) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (abs(x_m) * 0.3275911);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (abs(x_m) <= 2e-17)
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	else
		tmp = 1.0 + (exp(-(x_m * x_m)) * (t_1 * ((t_1 * ((t_1 * (((-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911)))) * (-1.0 / t_0)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[(-N[(x$95$m * x$95$m), $MachinePrecision])], $MachinePrecision] * N[(t$95$1 * N[(N[(t$95$1 * N[(N[(t$95$1 * N[(N[(N[(-1.453152027 + N[(1.061405429 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000014e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. associate--r+ 57.8%

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

      2. div-sub 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 57.8%

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

      2. expm1-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

      3. log1p-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

      4. add-exp-log 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

      5. +-commutative 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

      6. fma-udef 57.8%

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

      7. add-sqr-sqrt 24.7%

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

      8. fabs-sqr 24.7%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Taylor expanded in x around 0 98.6%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   14. Step-by-step derivation

       1. *-commutative 98.6%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   15. Simplified 98.6%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 2.00000000000000014e-17 < (fabs.f64 x)

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

      1. expm1-log1p-u 93.6%

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

      2. expm1-udef 93.6%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

      3. log1p-udef 93.6%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

      4. add-exp-log 93.6%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

      5. +-commutative 93.6%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

      6. fma-udef 93.6%

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

      7. add-sqr-sqrt 39.9%

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

      8. fabs-sqr 39.9%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 93.5%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   6. Applied egg-rr 96.9%

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

      1. fma-udef 93.5%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

      2. associate--l+ 93.5%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

      3. metadata-eval 93.5%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

      4. +-rgt-identity 93.5%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   8. Simplified 96.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 + e^{-x \cdot x} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \frac{-1}{1.3419749235962346 + \left(0.41439251223535706 \cdot {x_m}^{2} + x_m \cdot 1.4421495346696274\right)}, \frac{{\left(e^{x_m}\right)}^{\left(-x_m\right)}}{1 + \left|x_m\right| \cdot 0.3275911}, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 2e-17)
   (+ 1e-9 (* x_m 0.3275910996724089))
   (fma
    (+
     -0.254829592
     (/
      -1.0
      (+
       1.3419749235962346
       (+ (* 0.41439251223535706 (pow x_m 2.0)) (* x_m 1.4421495346696274)))))
    (/ (pow (exp x_m) (- x_m)) (+ 1.0 (* (fabs x_m) 0.3275911)))
    1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 2e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = fma((-0.254829592 + (-1.0 / (1.3419749235962346 + ((0.41439251223535706 * pow(x_m, 2.0)) + (x_m * 1.4421495346696274))))), (pow(exp(x_m), -x_m) / (1.0 + (fabs(x_m) * 0.3275911))), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 2e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = fma(Float64(-0.254829592 + Float64(-1.0 / Float64(1.3419749235962346 + Float64(Float64(0.41439251223535706 * (x_m ^ 2.0)) + Float64(x_m * 1.4421495346696274))))), Float64((exp(x_m) ^ Float64(-x_m)) / Float64(1.0 + Float64(abs(x_m) * 0.3275911))), 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], 2e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[(N[(-0.254829592 + N[(-1.0 / N[(1.3419749235962346 + N[(N[(0.41439251223535706 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 1.4421495346696274), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x$95$m], $MachinePrecision], (-x$95$m)], $MachinePrecision] / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000014e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. associate--r+ 57.8%

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

      2. div-sub 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 57.8%

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

      2. expm1-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

      3. log1p-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

      4. add-exp-log 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

      5. +-commutative 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

      6. fma-udef 57.8%

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

      7. add-sqr-sqrt 24.7%

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

      8. fabs-sqr 24.7%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Taylor expanded in x around 0 98.6%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   14. Step-by-step derivation

       1. *-commutative 98.6%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   15. Simplified 98.6%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 2.00000000000000014e-17 < (fabs.f64 x)

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

      1. add-cube-cbrt 97.6%

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

      2. pow3 97.6%

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

   6. Applied egg-rr 96.8%

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

      1. rem-cube-cbrt 96.7%

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

      2. clear-num 96.7%

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

   8. Applied egg-rr 96.7%

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

      1. fma-udef 96.7%

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

   10. Applied egg-rr 96.7%

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

   11. Taylor expanded in x around 0 96.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.254829592 + \frac{-1}{1.3419749235962346 + \left(0.41439251223535706 \cdot {x}^{2} + x \cdot 1.4421495346696274\right)}, \frac{{\left(e^{x}\right)}^{\left(-x\right)}}{1 + \left|x\right| \cdot 0.3275911}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \left(0.745170407 + x_m \cdot -0.8007952583978091\right)}{{\left(e^{x_m}\right)}^{x_m}}}{\mathsf{fma}\left(0.3275911, \left|x_m\right|, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 2e-17)
   (+ 1e-9 (* x_m 0.3275910996724089))
   (-
    1.0
    (/
     (/
      (+ 0.254829592 (+ 0.745170407 (* x_m -0.8007952583978091)))
      (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) <= 2e-17) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0 - (((0.254829592 + (0.745170407 + (x_m * -0.8007952583978091))) / 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) <= 2e-17)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.254829592 + Float64(0.745170407 + Float64(x_m * -0.8007952583978091))) / (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], 2e-17], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(0.254829592 + N[(0.745170407 + N[(x$95$m * -0.8007952583978091), $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) < 2.00000000000000014e-17

   1. Initial program 57.8%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. associate--r+ 57.8%

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

      2. div-sub 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in x around 0 57.8%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 57.8%

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

      2. expm1-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

      3. log1p-udef 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

      4. add-exp-log 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

      5. +-commutative 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

      6. fma-udef 57.8%

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

      7. add-sqr-sqrt 24.7%

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

      8. fabs-sqr 24.7%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 57.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 57.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 57.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Taylor expanded in x around 0 98.6%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   14. Step-by-step derivation

       1. *-commutative 98.6%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   15. Simplified 98.6%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 2.00000000000000014e-17 < (fabs.f64 x)

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

   6. Applied egg-rr 96.7%

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

      1. unpow2 96.7%

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

      2. *-lft-identity 96.7%

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

      3. times-frac 96.7%

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

      4. distribute-lft-in 96.7%

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

      5. associate-*l/ 96.7%

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

      6. *-lft-identity 96.7%

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

   8. Simplified 96.7%

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

   9. Taylor expanded in x around 0 96.2%

      \[\leadsto 1 - \frac{\frac{0.254829592 + \color{blue}{\left(0.745170407 + -0.8007952583978091 \cdot x\right)}}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]
   10. Step-by-step derivation

       1. *-commutative 96.2%

          \[\leadsto 1 - \frac{\frac{0.254829592 + \left(0.745170407 + \color{blue}{x \cdot -0.8007952583978091}\right)}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]

   11. Simplified 96.2%

       \[\leadsto 1 - \frac{\frac{0.254829592 + \color{blue}{\left(0.745170407 + x \cdot -0.8007952583978091\right)}}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \left(0.745170407 + x \cdot -0.8007952583978091\right)}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.02:\\ \;\;\;\;10^{-9} + x_m \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.02) (+ 1e-9 (* x_m 0.3275910996724089)) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.02) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (abs(x_m) <= 0.02d0) then
        tmp = 1d-9 + (x_m * 0.3275910996724089d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.02) {
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.02:
		tmp = 1e-9 + (x_m * 0.3275910996724089)
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.02)
		tmp = Float64(1e-9 + Float64(x_m * 0.3275910996724089));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.02)
		tmp = 1e-9 + (x_m * 0.3275910996724089);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.02], N[(1e-9 + N[(x$95$m * 0.3275910996724089), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0200000000000000004

   1. Initial program 58.1%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in x around 0 58.1%

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

      1. associate--r+ 58.1%

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

      2. div-sub 58.2%

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

   7. Simplified 57.4%

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

   8. Taylor expanded in x around 0 56.0%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 56.0%

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

      2. expm1-udef 56.0%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

      3. log1p-udef 56.0%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

      4. add-exp-log 56.0%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

      5. +-commutative 56.0%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

      6. fma-udef 56.0%

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

      7. add-sqr-sqrt 24.1%

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

      8. fabs-sqr 24.1%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 55.8%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 55.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 55.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 55.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 55.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 55.8%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 55.8%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Taylor expanded in x around 0 93.8%

       \[\leadsto \color{blue}{10^{-9} + 0.3275910996724089 \cdot x} \]
   14. Step-by-step derivation

       1. *-commutative 93.8%

          \[\leadsto 10^{-9} + \color{blue}{x \cdot 0.3275910996724089} \]

   15. Simplified 93.8%

       \[\leadsto \color{blue}{10^{-9} + x \cdot 0.3275910996724089} \]

   if 0.0200000000000000004 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate--r+ 100.0%

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

      2. div-sub 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 97.9%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 97.9%

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

      2. expm1-udef 97.9%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

      3. log1p-udef 97.9%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

      4. add-exp-log 97.9%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

      5. +-commutative 97.9%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

      6. fma-udef 97.9%

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

      7. add-sqr-sqrt 41.6%

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

      8. fabs-sqr 41.6%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 97.9%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 97.9%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 97.9%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 97.9%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 97.9%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 97.9%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 97.9%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Taylor expanded in x around inf 100.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;10^{-9} + x \cdot 0.3275910996724089\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.7% accurate, 142.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in x around 0 73.4%

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

      1. associate--r+ 73.4%

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

      2. div-sub 73.5%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in x around 0 71.4%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 71.4%

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

      2. expm1-udef 71.4%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

      3. log1p-udef 71.4%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

      4. add-exp-log 71.4%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

      5. +-commutative 71.4%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

      6. fma-udef 71.4%

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

      7. add-sqr-sqrt 15.3%

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

      8. fabs-sqr 15.3%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 71.3%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 71.3%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 71.3%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 71.3%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 71.3%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 71.3%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 71.3%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Taylor expanded in x around 0 63.7%

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

   if 2.79999999999999996e-5 < 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%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate--r+ 99.6%

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

      2. div-sub 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 97.1%

      \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 97.1%

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

      2. expm1-udef 97.1%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

      3. log1p-udef 97.1%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

      4. add-exp-log 97.1%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

      5. +-commutative 97.1%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

      6. fma-udef 97.1%

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

      7. add-sqr-sqrt 97.1%

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

      8. fabs-sqr 97.1%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

      9. add-sqr-sqrt 97.1%

         \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

   10. Applied egg-rr 97.1%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 97.1%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

       2. associate--l+ 97.1%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

       3. metadata-eval 97.1%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

       4. +-rgt-identity 97.1%

          \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   12. Simplified 97.1%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

   13. Taylor expanded in x around inf 98.4%

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

4. Final simplification 71.2%

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

Alternative 12: 53.9% accurate, 856.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Simplified 79.1%

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

5. Taylor expanded in x around 0 79.1%

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

   1. associate--r+ 79.1%

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

   2. div-sub 79.1%

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

7. Simplified 78.7%

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

8. Taylor expanded in x around 0 76.9%

   \[\leadsto \color{blue}{1 - 0.999999999 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \]
9. Step-by-step derivation

   1. expm1-log1p-u 76.9%

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

   2. expm1-udef 76.9%

      \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3275911 \cdot \left|x\right|\right)} - 1\right)}} \]

   3. log1p-udef 76.9%

      \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(e^{\color{blue}{\log \left(1 + 0.3275911 \cdot \left|x\right|\right)}} - 1\right)} \]

   4. add-exp-log 76.9%

      \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)} - 1\right)} \]

   5. +-commutative 76.9%

      \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} - 1\right)} \]

   6. fma-udef 76.9%

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

   7. add-sqr-sqrt 32.9%

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

   8. fabs-sqr 32.9%

      \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, 1\right) - 1\right)} \]

   9. add-sqr-sqrt 76.9%

      \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right) - 1\right)} \]

10. Applied egg-rr 76.9%

    \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{fma}\left(0.3275911, x, 1\right) - 1\right)}} \]
11. Step-by-step derivation

    1. fma-udef 76.9%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(\color{blue}{\left(0.3275911 \cdot x + 1\right)} - 1\right)} \]

    2. associate--l+ 76.9%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{\left(0.3275911 \cdot x + \left(1 - 1\right)\right)}} \]

    3. metadata-eval 76.9%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \left(0.3275911 \cdot x + \color{blue}{0}\right)} \]

    4. +-rgt-identity 76.9%

       \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

12. Simplified 76.9%

    \[\leadsto 1 - 0.999999999 \cdot \frac{1}{1 + \color{blue}{0.3275911 \cdot x}} \]

13. Taylor expanded in x around 0 52.4%

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

14. Final simplification 52.4%

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

Reproduce

herbie shell --seed 2024017 
(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)))))))