Jmat.Real.erf

Percentage Accurate: 78.8% → 86.3%

Time: 46.2s

Alternatives: 16

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 86.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \frac{\frac{1.061405429}{t\_0} + -1.453152027}{t\_0}\\ t_2 := {\left(e^{x}\right)}^{x} \cdot t\_0\\ t_3 := {\left(\frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{t\_0} + 1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{t\_2}\right)}^{2} + 1\\ t_4 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ \frac{{t\_3}^{-1} - \frac{{\left(\frac{\frac{\frac{\left({t\_1}^{2} - 2.020417023103615\right) \cdot {\left(t\_1 - 1.421413741\right)}^{-1}}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{t\_2}\right)}^{4}}{t\_3}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_4} + -1.453152027}{t\_4} + 1.421413741}{t\_4} + -0.284496736}{t\_4} + 0.254829592}{t\_4}, 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1 (/ (+ (/ 1.061405429 t_0) -1.453152027) t_0))
        (t_2 (* (pow (exp x) x) t_0))
        (t_3
         (+
          (pow
           (/
            (+
             (/
              (+
               (/
                (+
                 (/
                  (+
                   (/ -1.061405429 (fma -0.3275911 (fabs x) -1.0))
                   -1.453152027)
                  t_0)
                 1.421413741)
                t_0)
               -0.284496736)
              t_0)
             0.254829592)
            t_2)
           2.0)
          1.0))
        (t_4 (fma (fabs x) 0.3275911 1.0)))
   (/
    (-
     (pow t_3 -1.0)
     (/
      (pow
       (/
        (+
         (/
          (+
           (/
            (*
             (- (pow t_1 2.0) 2.020417023103615)
             (pow (- t_1 1.421413741) -1.0))
            t_0)
           -0.284496736)
          t_0)
         0.254829592)
        t_2)
       4.0)
      t_3))
    (fma
     (pow (exp x) (- x))
     (/
      (+
       (/
        (+
         (/ (+ (/ (+ (/ 1.061405429 t_4) -1.453152027) t_4) 1.421413741) t_4)
         -0.284496736)
        t_4)
       0.254829592)
      t_4)
     1.0))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = ((1.061405429 / t_0) + -1.453152027) / t_0;
	double t_2 = pow(exp(x), x) * t_0;
	double t_3 = pow((((((((((-1.061405429 / fma(-0.3275911, fabs(x), -1.0)) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_2), 2.0) + 1.0;
	double t_4 = fma(fabs(x), 0.3275911, 1.0);
	return (pow(t_3, -1.0) - (pow((((((((pow(t_1, 2.0) - 2.020417023103615) * pow((t_1 - 1.421413741), -1.0)) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_2), 4.0) / t_3)) / fma(pow(exp(x), -x), (((((((((1.061405429 / t_4) + -1.453152027) / t_4) + 1.421413741) / t_4) + -0.284496736) / t_4) + 0.254829592) / t_4), 1.0);
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = Float64(Float64(Float64(1.061405429 / t_0) + -1.453152027) / t_0)
	t_2 = Float64((exp(x) ^ x) * t_0)
	t_3 = Float64((Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1.061405429 / fma(-0.3275911, abs(x), -1.0)) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_2) ^ 2.0) + 1.0)
	t_4 = fma(abs(x), 0.3275911, 1.0)
	return Float64(Float64((t_3 ^ -1.0) - Float64((Float64(Float64(Float64(Float64(Float64(Float64(Float64((t_1 ^ 2.0) - 2.020417023103615) * (Float64(t_1 - 1.421413741) ^ -1.0)) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_2) ^ 4.0) / t_3)) / fma((exp(x) ^ Float64(-x)), Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_4) + -1.453152027) / t_4) + 1.421413741) / t_4) + -0.284496736) / t_4) + 0.254829592) / t_4), 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[(N[(N[(N[(N[(N[(N[(N[(-1.061405429 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(N[(N[Power[t$95$3, -1.0], $MachinePrecision] - N[(N[Power[N[(N[(N[(N[(N[(N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] - 2.020417023103615), $MachinePrecision] * N[Power[N[(t$95$1 - 1.421413741), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$2), $MachinePrecision], 4.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$4), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$4), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$4), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$4), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$4), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 78.7%

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 86.1%

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

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\color{blue}{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\color{blue}{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

   3. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\color{blue}{\frac{-3275911}{10000000} \cdot \left|x\right| + -1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\color{blue}{\left(\mathsf{neg}\left(\frac{3275911}{10000000}\right)\right)} \cdot \left|x\right| + -1} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\color{blue}{\left(\mathsf{neg}\left(\frac{3275911}{10000000} \cdot \left|x\right|\right)\right)} + -1} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\left(\mathsf{neg}\left(\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000}}\right)\right) + -1} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\left(\mathsf{neg}\left(\left|x\right| \cdot \frac{3275911}{10000000}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

   8. distribute-neg-in N/A

      \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\color{blue}{\mathsf{neg}\left(\left(\left|x\right| \cdot \frac{3275911}{10000000} + 1\right)\right)}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

   9. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{\color{blue}{\mathsf{neg}\left(\frac{1061405429}{1000000000}\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

   11. frac-2neg N/A

       \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\color{blue}{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

   12. lift-/.f64 N/A

       \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\color{blue}{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

   13. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\color{blue}{\frac{3275911}{10000000} \cdot \left|x\right| + 1}} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

   14. *-commutative N/A

       \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000}} + 1} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

   15. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\color{blue}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]

8. Applied rewrites 86.2%

   \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\color{blue}{\left({\left(\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} - 2.020417023103615\right) \cdot \frac{1}{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.421413741}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right)} \]

9. Final simplification 86.2%

   \[\leadsto \frac{{\left({\left(\frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} + 1\right)}^{-1} - \frac{{\left(\frac{\frac{\frac{\left({\left(\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} - 2.020417023103615\right) \cdot {\left(\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.421413741\right)}^{-1}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right)} \]
10. Add Preprocessing

Alternative 2: 78.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1}\\ t_1 := e^{\left(-x\right) \cdot x}\\ t_2 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_3 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_4 := \frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{t\_3} + 1.421413741}{t\_3} + -0.284496736}{t\_3} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot t\_3}\\ t_5 := {t\_4}^{2} + 1\\ \mathbf{if}\;\left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot t\_1 \leq 0.999999999:\\ \;\;\;\;1 - \frac{0.254829592 - \mathsf{fma}\left({t\_2}^{-1}, 0.284496736 - \frac{\frac{-1.453152027}{t\_2} - -1.421413741}{t\_2}, \frac{-1.061405429}{{t\_2}^{4}}\right)}{\frac{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_5}^{-1} - \frac{{t\_4}^{4}}{t\_5}}{\mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_2} + -1.453152027}{t\_2} + 1.421413741}{t\_2} + -0.284496736}{t\_2} + 0.254829592}{t\_2}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (* 0.3275911 (fabs x))) -1.0))
        (t_1 (exp (* (- x) x)))
        (t_2 (fma (fabs x) 0.3275911 1.0))
        (t_3 (fma 0.3275911 (fabs x) 1.0))
        (t_4
         (/
          (+
           (/
            (+
             (/
              (+
               (/
                (+
                 (/ -1.061405429 (fma -0.3275911 (fabs x) -1.0))
                 -1.453152027)
                t_3)
               1.421413741)
              t_3)
             -0.284496736)
            t_3)
           0.254829592)
          (* (pow (exp x) x) t_3)))
        (t_5 (+ (pow t_4 2.0) 1.0)))
   (if (<=
        (*
         (*
          t_0
          (+
           0.254829592
           (*
            t_0
            (+
             -0.284496736
             (*
              t_0
              (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
         t_1)
        0.999999999)
     (-
      1.0
      (*
       (/
        (-
         0.254829592
         (fma
          (pow t_2 -1.0)
          (- 0.284496736 (/ (- (/ -1.453152027 t_2) -1.421413741) t_2))
          (/ -1.061405429 (pow t_2 4.0))))
        (/ (fma 0.10731592879921 (* x x) -1.0) (fma 0.3275911 (fabs x) -1.0)))
       t_1))
     (/
      (- (pow t_5 -1.0) (/ (pow t_4 4.0) t_5))
      (fma
       1.0
       (/
        (+
         (/
          (+
           (/ (+ (/ (+ (/ 1.061405429 t_2) -1.453152027) t_2) 1.421413741) t_2)
           -0.284496736)
          t_2)
         0.254829592)
        t_2)
       1.0)))))

C

double code(double x) {
	double t_0 = pow((1.0 + (0.3275911 * fabs(x))), -1.0);
	double t_1 = exp((-x * x));
	double t_2 = fma(fabs(x), 0.3275911, 1.0);
	double t_3 = fma(0.3275911, fabs(x), 1.0);
	double t_4 = ((((((((-1.061405429 / fma(-0.3275911, fabs(x), -1.0)) + -1.453152027) / t_3) + 1.421413741) / t_3) + -0.284496736) / t_3) + 0.254829592) / (pow(exp(x), x) * t_3);
	double t_5 = pow(t_4, 2.0) + 1.0;
	double tmp;
	if (((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * t_1) <= 0.999999999) {
		tmp = 1.0 - (((0.254829592 - fma(pow(t_2, -1.0), (0.284496736 - (((-1.453152027 / t_2) - -1.421413741) / t_2)), (-1.061405429 / pow(t_2, 4.0)))) / (fma(0.10731592879921, (x * x), -1.0) / fma(0.3275911, fabs(x), -1.0))) * t_1);
	} else {
		tmp = (pow(t_5, -1.0) - (pow(t_4, 4.0) / t_5)) / fma(1.0, (((((((((1.061405429 / t_2) + -1.453152027) / t_2) + 1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592) / t_2), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x))) ^ -1.0
	t_1 = exp(Float64(Float64(-x) * x))
	t_2 = fma(abs(x), 0.3275911, 1.0)
	t_3 = fma(0.3275911, abs(x), 1.0)
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1.061405429 / fma(-0.3275911, abs(x), -1.0)) + -1.453152027) / t_3) + 1.421413741) / t_3) + -0.284496736) / t_3) + 0.254829592) / Float64((exp(x) ^ x) * t_3))
	t_5 = Float64((t_4 ^ 2.0) + 1.0)
	tmp = 0.0
	if (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))))))))) * t_1) <= 0.999999999)
		tmp = Float64(1.0 - Float64(Float64(Float64(0.254829592 - fma((t_2 ^ -1.0), Float64(0.284496736 - Float64(Float64(Float64(-1.453152027 / t_2) - -1.421413741) / t_2)), Float64(-1.061405429 / (t_2 ^ 4.0)))) / Float64(fma(0.10731592879921, Float64(x * x), -1.0) / fma(0.3275911, abs(x), -1.0))) * t_1));
	else
		tmp = Float64(Float64((t_5 ^ -1.0) - Float64((t_4 ^ 4.0) / t_5)) / fma(1.0, Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_2) + -1.453152027) / t_2) + 1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592) / t_2), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(N[(N[(N[(-1.061405429 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$3), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$3), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$3), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[t$95$4, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[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] * t$95$1), $MachinePrecision], 0.999999999], N[(1.0 - N[(N[(N[(0.254829592 - N[(N[Power[t$95$2, -1.0], $MachinePrecision] * N[(0.284496736 - N[(N[(N[(-1.453152027 / t$95$2), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(-1.061405429 / N[Power[t$95$2, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.10731592879921 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] / N[(0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[t$95$5, -1.0], $MachinePrecision] - N[(N[Power[t$95$4, 4.0], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision] / N[(1.0 * N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$2), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$2), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$2), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$2), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 0.999999999000000028

   1. Initial program 78.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{\frac{\left(\frac{31853699}{125000000} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4}} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(\frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \frac{1453152027}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}\right)}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   5. Applied rewrites 78.8%

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

   7. Applied rewrites 78.8%

      \[\leadsto 1 - \frac{0.254829592 - \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 0.284496736 - \frac{\frac{-1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, \frac{-1.061405429}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{4}}\right)}{\frac{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   if 0.999999999000000028 < (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))

   1. Initial program 78.7%

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

   4. Applied rewrites 78.7%

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

   6. Applied rewrites 86.1%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{\frac{-1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, -1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{31853699}{125000000}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left(\color{blue}{1}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 83.7%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(1.421413741 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-1.453152027 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x} \leq 0.999999999:\\ \;\;\;\;1 - \frac{0.254829592 - \mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.284496736 - \frac{\frac{-1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, \frac{-1.061405429}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{4}}\right)}{\frac{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}} \cdot e^{\left(-x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(\frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} + 1\right)}^{-1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{x}\right)}^{\left(-x\right)}\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_3 := \frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{t\_2} + 1.421413741}{t\_2} + -0.284496736}{t\_2} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot t\_2}\\ \frac{{\left(\mathsf{fma}\left({\left(\frac{\frac{\frac{\frac{1.061405429}{t\_2} + -1.453152027}{t\_2} + 1.421413741}{t\_2} + -0.284496736}{t\_2} + 0.254829592\right)}^{2}, {\left(\frac{t\_0}{t\_2}\right)}^{2}, 1\right)\right)}^{-1} - \frac{{t\_3}^{4}}{{t\_3}^{2} + 1}}{\mathsf{fma}\left(t\_0, \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_1} + -1.453152027}{t\_1} + 1.421413741}{t\_1} + -0.284496736}{t\_1} + 0.254829592}{t\_1}, 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (exp x) (- x)))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2 (fma 0.3275911 (fabs x) 1.0))
        (t_3
         (/
          (+
           (/
            (+
             (/
              (+
               (/
                (+
                 (/ -1.061405429 (fma -0.3275911 (fabs x) -1.0))
                 -1.453152027)
                t_2)
               1.421413741)
              t_2)
             -0.284496736)
            t_2)
           0.254829592)
          (* (pow (exp x) x) t_2))))
   (/
    (-
     (pow
      (fma
       (pow
        (+
         (/
          (+
           (/ (+ (/ (+ (/ 1.061405429 t_2) -1.453152027) t_2) 1.421413741) t_2)
           -0.284496736)
          t_2)
         0.254829592)
        2.0)
       (pow (/ t_0 t_2) 2.0)
       1.0)
      -1.0)
     (/ (pow t_3 4.0) (+ (pow t_3 2.0) 1.0)))
    (fma
     t_0
     (/
      (+
       (/
        (+
         (/ (+ (/ (+ (/ 1.061405429 t_1) -1.453152027) t_1) 1.421413741) t_1)
         -0.284496736)
        t_1)
       0.254829592)
      t_1)
     1.0))))

C

double code(double x) {
	double t_0 = pow(exp(x), -x);
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = fma(0.3275911, fabs(x), 1.0);
	double t_3 = ((((((((-1.061405429 / fma(-0.3275911, fabs(x), -1.0)) + -1.453152027) / t_2) + 1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592) / (pow(exp(x), x) * t_2);
	return (pow(fma(pow(((((((((1.061405429 / t_2) + -1.453152027) / t_2) + 1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592), 2.0), pow((t_0 / t_2), 2.0), 1.0), -1.0) - (pow(t_3, 4.0) / (pow(t_3, 2.0) + 1.0))) / fma(t_0, (((((((((1.061405429 / t_1) + -1.453152027) / t_1) + 1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592) / t_1), 1.0);
}

Julia

function code(x)
	t_0 = exp(x) ^ Float64(-x)
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = fma(0.3275911, abs(x), 1.0)
	t_3 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1.061405429 / fma(-0.3275911, abs(x), -1.0)) + -1.453152027) / t_2) + 1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592) / Float64((exp(x) ^ x) * t_2))
	return Float64(Float64((fma((Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_2) + -1.453152027) / t_2) + 1.421413741) / t_2) + -0.284496736) / t_2) + 0.254829592) ^ 2.0), (Float64(t_0 / t_2) ^ 2.0), 1.0) ^ -1.0) - Float64((t_3 ^ 4.0) / Float64((t_3 ^ 2.0) + 1.0))) / fma(t_0, Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_1) + -1.453152027) / t_1) + 1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592) / t_1), 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(N[(N[(N[(N[(-1.061405429 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$2), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$2), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$2), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[N[(N[Power[N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$2), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$2), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$2), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$2), $MachinePrecision] + 0.254829592), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(t$95$0 / t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision] - N[(N[Power[t$95$3, 4.0], $MachinePrecision] / N[(N[Power[t$95$3, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 78.7%

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 86.1%

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

8. Applied rewrites 86.1%

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

9. Final simplification 86.1%

   \[\leadsto \frac{{\left(\mathsf{fma}\left({\left(\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592\right)}^{2}, {\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2}, 1\right)\right)}^{-1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right)} \]
10. Add Preprocessing

Alternative 4: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_2 := \frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{t\_1} + 1.421413741}{t\_1} + -0.284496736}{t\_1} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot t\_1}\\ t_3 := {t\_2}^{2} + 1\\ \frac{{t\_3}^{-1} - \frac{{t\_2}^{4}}{t\_3}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_0} + -1.453152027}{t\_0} + 1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{t\_0}, 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1 (fma 0.3275911 (fabs x) 1.0))
        (t_2
         (/
          (+
           (/
            (+
             (/
              (+
               (/
                (+
                 (/ -1.061405429 (fma -0.3275911 (fabs x) -1.0))
                 -1.453152027)
                t_1)
               1.421413741)
              t_1)
             -0.284496736)
            t_1)
           0.254829592)
          (* (pow (exp x) x) t_1)))
        (t_3 (+ (pow t_2 2.0) 1.0)))
   (/
    (- (pow t_3 -1.0) (/ (pow t_2 4.0) t_3))
    (fma
     (pow (exp x) (- x))
     (/
      (+
       (/
        (+
         (/ (+ (/ (+ (/ 1.061405429 t_0) -1.453152027) t_0) 1.421413741) t_0)
         -0.284496736)
        t_0)
       0.254829592)
      t_0)
     1.0))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = fma(0.3275911, fabs(x), 1.0);
	double t_2 = ((((((((-1.061405429 / fma(-0.3275911, fabs(x), -1.0)) + -1.453152027) / t_1) + 1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592) / (pow(exp(x), x) * t_1);
	double t_3 = pow(t_2, 2.0) + 1.0;
	return (pow(t_3, -1.0) - (pow(t_2, 4.0) / t_3)) / fma(pow(exp(x), -x), (((((((((1.061405429 / t_0) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_0), 1.0);
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = fma(0.3275911, abs(x), 1.0)
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1.061405429 / fma(-0.3275911, abs(x), -1.0)) + -1.453152027) / t_1) + 1.421413741) / t_1) + -0.284496736) / t_1) + 0.254829592) / Float64((exp(x) ^ x) * t_1))
	t_3 = Float64((t_2 ^ 2.0) + 1.0)
	return Float64(Float64((t_3 ^ -1.0) - Float64((t_2 ^ 4.0) / t_3)) / fma((exp(x) ^ Float64(-x)), Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_0), 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(N[(N[(N[(-1.061405429 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$1), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[t$95$2, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[(N[Power[t$95$3, -1.0], $MachinePrecision] - N[(N[Power[t$95$2, 4.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Exp[x], $MachinePrecision], (-x)], $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 78.7%

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 86.1%

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

7. Final simplification 86.1%

   \[\leadsto \frac{{\left({\left(\frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} + 1\right)}^{-1} - \frac{{\left(\frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{4}}{{\left(\frac{\frac{\frac{\frac{\frac{-1.061405429}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 0.254829592}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}^{2} + 1}}{\mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right)} \]
8. Add Preprocessing

Alternative 5: 78.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1}\\ t_1 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_2 := e^{\left(-x\right) \cdot x}\\ \mathbf{if}\;\left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot t\_2 \leq 0.1:\\ \;\;\;\;1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + \frac{1.421413741 - \frac{1.453152027}{t\_1}}{t\_1}\right)\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, -1\right), x \cdot x, 1\right), \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_1} + -1.453152027}{t\_1} + 1.421413741}{-t\_1} - -0.284496736}{t\_1} - 0.254829592}{t\_1}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (* 0.3275911 (fabs x))) -1.0))
        (t_1 (fma (fabs x) 0.3275911 1.0))
        (t_2 (exp (* (- x) x))))
   (if (<=
        (*
         (*
          t_0
          (+
           0.254829592
           (*
            t_0
            (+
             -0.284496736
             (*
              t_0
              (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
         t_2)
        0.1)
     (-
      1.0
      (*
       (*
        t_0
        (+
         0.254829592
         (* t_0 (+ -0.284496736 (/ (- 1.421413741 (/ 1.453152027 t_1)) t_1)))))
       t_2))
     (fma
      (fma
       (fma (fma -0.16666666666666666 (* x x) 0.5) (* x x) -1.0)
       (* x x)
       1.0)
      (/
       (-
        (/
         (-
          (/
           (+ (/ (+ (/ 1.061405429 t_1) -1.453152027) t_1) 1.421413741)
           (- t_1))
          -0.284496736)
         t_1)
        0.254829592)
       t_1)
      1.0))))

C

double code(double x) {
	double t_0 = pow((1.0 + (0.3275911 * fabs(x))), -1.0);
	double t_1 = fma(fabs(x), 0.3275911, 1.0);
	double t_2 = exp((-x * x));
	double tmp;
	if (((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * t_2) <= 0.1) {
		tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + ((1.421413741 - (1.453152027 / t_1)) / t_1))))) * t_2);
	} else {
		tmp = fma(fma(fma(fma(-0.16666666666666666, (x * x), 0.5), (x * x), -1.0), (x * x), 1.0), (((((((((1.061405429 / t_1) + -1.453152027) / t_1) + 1.421413741) / -t_1) - -0.284496736) / t_1) - 0.254829592) / t_1), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x))) ^ -1.0
	t_1 = fma(abs(x), 0.3275911, 1.0)
	t_2 = exp(Float64(Float64(-x) * x))
	tmp = 0.0
	if (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))))))))) * t_2) <= 0.1)
		tmp = Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(Float64(1.421413741 - Float64(1.453152027 / t_1)) / t_1))))) * t_2));
	else
		tmp = fma(fma(fma(fma(-0.16666666666666666, Float64(x * x), 0.5), Float64(x * x), -1.0), Float64(x * x), 1.0), Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_1) + -1.453152027) / t_1) + 1.421413741) / Float64(-t_1)) - -0.284496736) / t_1) - 0.254829592) / t_1), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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] * t$95$2), $MachinePrecision], 0.1], N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(N[(1.421413741 - N[(1.453152027 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$1), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.421413741), $MachinePrecision] / (-t$95$1)), $MachinePrecision] - -0.284496736), $MachinePrecision] / t$95$1), $MachinePrecision] - 0.254829592), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 0.10000000000000001

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \color{blue}{\left(\frac{-1453152027}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}\right)}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      2. +-commutative N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \color{blue}{\left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000} + \frac{-1453152027}{1000000000}\right)}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      3. lift-*.f64 N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\color{blue}{\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}} + \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      4. lift-/.f64 N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\color{blue}{\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \cdot \frac{1061405429}{1000000000} + \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      5. associate-*l/ N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\color{blue}{\frac{1 \cdot \frac{1061405429}{1000000000}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} + \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      6. metadata-eval N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{\color{blue}{\frac{1061405429}{1000000000}}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      7. lift-+.f64 N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{\frac{1061405429}{1000000000}}{\color{blue}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} + \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      8. flip-+ N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{\frac{1061405429}{1000000000}}{\color{blue}{\frac{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}} + \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      9. associate-/r/ N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\color{blue}{\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)} \cdot \left(1 - \frac{3275911}{10000000} \cdot \left|x\right|\right)} + \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      10. lower-fma.f64 N/A

          \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \color{blue}{\mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}, 1 - \frac{3275911}{10000000} \cdot \left|x\right|, \frac{-1453152027}{1000000000}\right)}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   4. Applied rewrites 100.0%

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\mathsf{fma}\left(\frac{1.061405429}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, -1.453152027\right)}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   5. Taylor expanded in x around inf

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \color{blue}{\frac{\frac{1421413741}{1000000000} - \frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \color{blue}{\frac{\frac{1421413741}{1000000000} - \frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      2. lower--.f64 N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{\color{blue}{\frac{1421413741}{1000000000} - \frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      3. associate-*r/ N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} - \color{blue}{\frac{\frac{1453152027}{1000000000} \cdot 1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      4. metadata-eval N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} - \frac{\color{blue}{\frac{1453152027}{1000000000}}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      5. lower-/.f64 N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} - \color{blue}{\frac{\frac{1453152027}{1000000000}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      6. +-commutative N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} - \frac{\frac{1453152027}{1000000000}}{\color{blue}{\frac{3275911}{10000000} \cdot \left|x\right| + 1}}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      7. *-commutative N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} - \frac{\frac{1453152027}{1000000000}}{\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000}} + 1}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      8. lower-fma.f64 N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} - \frac{\frac{1453152027}{1000000000}}{\color{blue}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      9. lower-fabs.f64 N/A

         \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} - \frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\color{blue}{\left|x\right|}, \frac{3275911}{10000000}, 1\right)}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      10. +-commutative N/A

          \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} - \frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\color{blue}{\frac{3275911}{10000000} \cdot \left|x\right| + 1}}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      11. *-commutative N/A

          \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} - \frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000}} + 1}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      12. lower-fma.f64 N/A

          \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{\frac{1421413741}{1000000000} - \frac{\frac{1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}{\color{blue}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

      13. lower-fabs.f64 99.4

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

   7. Applied rewrites 99.4%

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

   if 0.10000000000000001 < (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 31853699/125000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -8890523/31250000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal 1421413741/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) (+.f64 #s(literal -1453152027/1000000000 binary64) (*.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 3275911/10000000 binary64) (fabs.f64 x)))) #s(literal 1061405429/1000000000 binary64)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))

   1. Initial program 59.0%

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

   4. Applied rewrites 59.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right) + 1}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1\right) \cdot {x}^{2}} + 1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) - 1, {x}^{2}, 1\right)}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right)}, {x}^{2}, 1\right), \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(1\right)\right), {x}^{2}, 1\right), \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{-1}, {x}^{2}, 1\right), \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{-1}{6} \cdot {x}^{2}, {x}^{2}, -1\right)}, {x}^{2}, 1\right), \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, -1\right), {x}^{2}, 1\right), \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, -1\right), {x}^{2}, 1\right), \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, -1\right), {x}^{2}, 1\right), \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, -1\right), {x}^{2}, 1\right), \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, -1\right), {x}^{2}, 1\right), \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, -1\right), {x}^{2}, 1\right), \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, -1\right), \color{blue}{x \cdot x}, 1\right), \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

      15. lower-*.f64 59.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, -1\right), \color{blue}{x \cdot x}, 1\right), \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{-\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right) \]

   7. Applied rewrites 59.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, -1\right), x \cdot x, 1\right)}, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{-\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(1.421413741 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-1.453152027 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{\left(-x\right) \cdot x} \leq 0.1:\\ \;\;\;\;1 - \left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + {\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(-0.284496736 + \frac{1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)\right) \cdot e^{\left(-x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, -1\right), x \cdot x, 1\right), \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{-\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ 1 - \frac{0.254829592 - \mathsf{fma}\left({t\_0}^{-1}, 0.284496736 - \frac{\frac{-1.453152027}{t\_0} - -1.421413741}{t\_0}, \frac{-1.061405429}{{t\_0}^{4}}\right)}{\frac{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}} \cdot e^{\left(-x\right) \cdot x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (-
    1.0
    (*
     (/
      (-
       0.254829592
       (fma
        (pow t_0 -1.0)
        (- 0.284496736 (/ (- (/ -1.453152027 t_0) -1.421413741) t_0))
        (/ -1.061405429 (pow t_0 4.0))))
      (/ (fma 0.10731592879921 (* x x) -1.0) (fma 0.3275911 (fabs x) -1.0)))
     (exp (* (- x) x))))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 - (((0.254829592 - fma(pow(t_0, -1.0), (0.284496736 - (((-1.453152027 / t_0) - -1.421413741) / t_0)), (-1.061405429 / pow(t_0, 4.0)))) / (fma(0.10731592879921, (x * x), -1.0) / fma(0.3275911, fabs(x), -1.0))) * exp((-x * x)));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(0.254829592 - fma((t_0 ^ -1.0), Float64(0.284496736 - Float64(Float64(Float64(-1.453152027 / t_0) - -1.421413741) / t_0)), Float64(-1.061405429 / (t_0 ^ 4.0)))) / Float64(fma(0.10731592879921, Float64(x * x), -1.0) / fma(0.3275911, abs(x), -1.0))) * exp(Float64(Float64(-x) * x))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[(0.254829592 - N[(N[Power[t$95$0, -1.0], $MachinePrecision] * N[(0.284496736 - N[(N[(N[(-1.453152027 / t$95$0), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-1.061405429 / N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.10731592879921 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] / N[(0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.7%

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

3. Taylor expanded in x around 0

   \[\leadsto 1 - \color{blue}{\frac{\left(\frac{31853699}{125000000} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4}} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(\frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \frac{1453152027}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}\right)}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

5. Applied rewrites 78.8%

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

7. Applied rewrites 78.8%

   \[\leadsto 1 - \frac{0.254829592 - \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 0.284496736 - \frac{\frac{-1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, \frac{-1.061405429}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{4}}\right)}{\frac{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

8. Final simplification 78.8%

   \[\leadsto 1 - \frac{0.254829592 - \mathsf{fma}\left({\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{-1}, 0.284496736 - \frac{\frac{-1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, \frac{-1.061405429}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{4}}\right)}{\frac{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}} \cdot e^{\left(-x\right) \cdot x} \]
9. Add Preprocessing

Alternative 7: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ 1 - \left({\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{-1} \cdot \left(0.254829592 + \left(\frac{\frac{\frac{-1.061405429}{t\_0} + -1.453152027}{t\_1} + 1.421413741}{t\_1 \cdot t\_1} + \frac{0.284496736}{t\_0}\right)\right)\right) \cdot e^{\left(-x\right) \cdot x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.3275911 (fabs x) -1.0))
        (t_1 (fma 0.3275911 (fabs x) 1.0)))
   (-
    1.0
    (*
     (*
      (pow (+ 1.0 (* 0.3275911 (fabs x))) -1.0)
      (+
       0.254829592
       (+
        (/
         (+ (/ (+ (/ -1.061405429 t_0) -1.453152027) t_1) 1.421413741)
         (* t_1 t_1))
        (/ 0.284496736 t_0))))
     (exp (* (- x) x))))))

C

double code(double x) {
	double t_0 = fma(-0.3275911, fabs(x), -1.0);
	double t_1 = fma(0.3275911, fabs(x), 1.0);
	return 1.0 - ((pow((1.0 + (0.3275911 * fabs(x))), -1.0) * (0.254829592 + ((((((-1.061405429 / t_0) + -1.453152027) / t_1) + 1.421413741) / (t_1 * t_1)) + (0.284496736 / t_0)))) * exp((-x * x)));
}

Julia

function code(x)
	t_0 = fma(-0.3275911, abs(x), -1.0)
	t_1 = fma(0.3275911, abs(x), 1.0)
	return Float64(1.0 - Float64(Float64((Float64(1.0 + Float64(0.3275911 * abs(x))) ^ -1.0) * Float64(0.254829592 + Float64(Float64(Float64(Float64(Float64(Float64(-1.061405429 / t_0) + -1.453152027) / t_1) + 1.421413741) / Float64(t_1 * t_1)) + Float64(0.284496736 / t_0)))) * exp(Float64(Float64(-x) * x))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[Power[N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[(0.254829592 + N[(N[(N[(N[(N[(N[(-1.061405429 / t$95$0), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.421413741), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.284496736 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 78.7%

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

   1. lift-+.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \color{blue}{\left(\frac{-1453152027}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}\right)}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   2. +-commutative N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \color{blue}{\left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000} + \frac{-1453152027}{1000000000}\right)}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. lift-*.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\color{blue}{\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}} + \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   4. lift-/.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\color{blue}{\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \cdot \frac{1061405429}{1000000000} + \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   5. associate-*l/ N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\color{blue}{\frac{1 \cdot \frac{1061405429}{1000000000}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} + \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   6. metadata-eval N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{\color{blue}{\frac{1061405429}{1000000000}}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   7. lift-+.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{\frac{1061405429}{1000000000}}{\color{blue}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} + \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   8. flip-+ N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{\frac{1061405429}{1000000000}}{\color{blue}{\frac{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}} + \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   9. associate-/r/ N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\color{blue}{\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)} \cdot \left(1 - \frac{3275911}{10000000} \cdot \left|x\right|\right)} + \frac{-1453152027}{1000000000}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   10. lower-fma.f64 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \color{blue}{\mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}, 1 - \frac{3275911}{10000000} \cdot \left|x\right|, \frac{-1453152027}{1000000000}\right)}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

4. Applied rewrites 78.7%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\mathsf{fma}\left(\frac{1.061405429}{1 - 0.10731592879921 \cdot \left(x \cdot x\right)}, 1 - \left|x\right| \cdot 0.3275911, -1.453152027\right)}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

6. Applied rewrites 78.7%

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

7. Final simplification 78.7%

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

Alternative 8: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ 1 - \frac{0.254829592 - \mathsf{fma}\left({t\_0}^{-4}, -1.061405429, \frac{0.284496736 - \frac{\frac{1.453152027}{\mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)} - -1.421413741}{t\_0}}{t\_0}\right)}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \cdot e^{\left(-x\right) \cdot x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0)))
   (-
    1.0
    (*
     (/
      (-
       0.254829592
       (fma
        (pow t_0 -4.0)
        -1.061405429
        (/
         (-
          0.284496736
          (/
           (- (/ 1.453152027 (fma -0.3275911 (fabs x) -1.0)) -1.421413741)
           t_0))
         t_0)))
      (fma (fabs x) 0.3275911 1.0))
     (exp (* (- x) x))))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	return 1.0 - (((0.254829592 - fma(pow(t_0, -4.0), -1.061405429, ((0.284496736 - (((1.453152027 / fma(-0.3275911, fabs(x), -1.0)) - -1.421413741) / t_0)) / t_0))) / fma(fabs(x), 0.3275911, 1.0)) * exp((-x * x)));
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	return Float64(1.0 - Float64(Float64(Float64(0.254829592 - fma((t_0 ^ -4.0), -1.061405429, Float64(Float64(0.284496736 - Float64(Float64(Float64(1.453152027 / fma(-0.3275911, abs(x), -1.0)) - -1.421413741) / t_0)) / t_0))) / fma(abs(x), 0.3275911, 1.0)) * exp(Float64(Float64(-x) * x))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[(0.254829592 - N[(N[Power[t$95$0, -4.0], $MachinePrecision] * -1.061405429 + N[(N[(0.284496736 - N[(N[(N[(1.453152027 / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - -1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.7%

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

3. Taylor expanded in x around 0

   \[\leadsto 1 - \color{blue}{\frac{\left(\frac{31853699}{125000000} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4}} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(\frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \frac{1453152027}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}\right)}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

5. Applied rewrites 78.8%

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

7. Applied rewrites 78.8%

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

8. Final simplification 78.8%

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

Alternative 9: 78.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ 1 - \frac{\frac{\frac{\frac{\frac{1.061405429}{\frac{\mathsf{fma}\left(x \cdot x, 0.10731592879921, -1\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}} + -1.453152027}{t\_0} + 1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{t\_0} \cdot e^{\left(-x\right) \cdot x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (-
    1.0
    (*
     (/
      (+
       (/
        (+
         (/
          (+
           (/
            (+
             (/
              1.061405429
              (/
               (fma (* x x) 0.10731592879921 -1.0)
               (fma 0.3275911 (fabs x) -1.0)))
             -1.453152027)
            t_0)
           1.421413741)
          t_0)
         -0.284496736)
        t_0)
       0.254829592)
      t_0)
     (exp (* (- x) x))))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 - ((((((((((1.061405429 / (fma((x * x), 0.10731592879921, -1.0) / fma(0.3275911, fabs(x), -1.0))) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_0) * exp((-x * x)));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / Float64(fma(Float64(x * x), 0.10731592879921, -1.0) / fma(0.3275911, abs(x), -1.0))) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_0) * exp(Float64(Float64(-x) * x))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / N[(N[(N[(x * x), $MachinePrecision] * 0.10731592879921 + -1.0), $MachinePrecision] / N[(0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.7%

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

   1. lift-*.f64 N/A

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-1453152027}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}\right)\right)\right)\right)\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   2. *-commutative N/A

      \[\leadsto 1 - \color{blue}{\left(\left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-1453152027}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}\right)\right)\right)\right) \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. lift-/.f64 N/A

      \[\leadsto 1 - \left(\left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-1453152027}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   4. un-div-inv N/A

      \[\leadsto 1 - \color{blue}{\frac{\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-1453152027}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}\right)\right)\right)}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   5. lower-/.f64 78.7

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

4. Applied rewrites 78.7%

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

   1. lift-fma.f64 N/A

      \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000} + 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   2. lift-*.f64 N/A

      \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000}} + 1} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. flip-+ N/A

      \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\color{blue}{\frac{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right) \cdot \left(\left|x\right| \cdot \frac{3275911}{10000000}\right) - 1 \cdot 1}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   4. lower-/.f64 N/A

      \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\color{blue}{\frac{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right) \cdot \left(\left|x\right| \cdot \frac{3275911}{10000000}\right) - 1 \cdot 1}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   5. metadata-eval N/A

      \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right) \cdot \left(\left|x\right| \cdot \frac{3275911}{10000000}\right) - \color{blue}{1}}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   6. sub-neg N/A

      \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\color{blue}{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right) \cdot \left(\left|x\right| \cdot \frac{3275911}{10000000}\right) + \left(\mathsf{neg}\left(1\right)\right)}}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   7. lift-*.f64 N/A

      \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\color{blue}{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right)} \cdot \left(\left|x\right| \cdot \frac{3275911}{10000000}\right) + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   8. lift-*.f64 N/A

      \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right) \cdot \color{blue}{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right)} + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   9. swap-sqr N/A

      \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \frac{3275911}{10000000}\right)} + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   10. metadata-eval N/A

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \color{blue}{\frac{10731592879921}{100000000000000}} + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   11. lift-fabs.f64 N/A

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\color{blue}{\left|x\right|} \cdot \left|x\right|\right) \cdot \frac{10731592879921}{100000000000000} + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   12. lift-fabs.f64 N/A

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right) \cdot \frac{10731592879921}{100000000000000} + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   13. sqr-abs N/A

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{10731592879921}{100000000000000} + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   14. lift-*.f64 N/A

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{10731592879921}{100000000000000} + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   15. metadata-eval N/A

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000} + \color{blue}{-1}}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   16. lower-fma.f64 N/A

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)}}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   17. sub-neg N/A

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)}{\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000} + \left(\mathsf{neg}\left(1\right)\right)}}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   18. lift-*.f64 N/A

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)}{\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000}} + \left(\mathsf{neg}\left(1\right)\right)}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   19. *-commutative N/A

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)}{\color{blue}{\frac{3275911}{10000000} \cdot \left|x\right|} + \left(\mathsf{neg}\left(1\right)\right)}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   20. metadata-eval N/A

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\mathsf{fma}\left(x \cdot x, \frac{10731592879921}{100000000000000}, -1\right)}{\frac{3275911}{10000000} \cdot \left|x\right| + \color{blue}{-1}}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   21. lower-fma.f64 78.7

       \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{1.061405429}{\frac{\mathsf{fma}\left(x \cdot x, 0.10731592879921, -1\right)}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}}} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

6. Applied rewrites 78.7%

   \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{1.061405429}{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, 0.10731592879921, -1\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}}} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

7. Final simplification 78.7%

   \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{1.061405429}{\frac{\mathsf{fma}\left(x \cdot x, 0.10731592879921, -1\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \cdot e^{\left(-x\right) \cdot x} \]
8. Add Preprocessing

Alternative 10: 78.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ 1 - \frac{\frac{\frac{\frac{\frac{\mathsf{fma}\left(-0.3275911, \left|x\right|, 1\right) \cdot 1.061405429}{\mathsf{fma}\left(-0.10731592879921, x \cdot x, 1\right)} + -1.453152027}{t\_0} + 1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{t\_0} \cdot e^{\left(-x\right) \cdot x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (-
    1.0
    (*
     (/
      (+
       (/
        (+
         (/
          (+
           (/
            (+
             (/
              (* (fma -0.3275911 (fabs x) 1.0) 1.061405429)
              (fma -0.10731592879921 (* x x) 1.0))
             -1.453152027)
            t_0)
           1.421413741)
          t_0)
         -0.284496736)
        t_0)
       0.254829592)
      t_0)
     (exp (* (- x) x))))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	return 1.0 - (((((((((((fma(-0.3275911, fabs(x), 1.0) * 1.061405429) / fma(-0.10731592879921, (x * x), 1.0)) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_0) * exp((-x * x)));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(fma(-0.3275911, abs(x), 1.0) * 1.061405429) / fma(-0.10731592879921, Float64(x * x), 1.0)) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_0) * exp(Float64(Float64(-x) * x))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(-0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision] * 1.061405429), $MachinePrecision] / N[(-0.10731592879921 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision] * N[Exp[N[((-x) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.7%

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

   1. lift-*.f64 N/A

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-1453152027}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}\right)\right)\right)\right)\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   2. *-commutative N/A

      \[\leadsto 1 - \color{blue}{\left(\left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-1453152027}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}\right)\right)\right)\right) \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. lift-/.f64 N/A

      \[\leadsto 1 - \left(\left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-1453152027}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   4. un-div-inv N/A

      \[\leadsto 1 - \color{blue}{\frac{\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-1453152027}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}\right)\right)\right)}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   5. lower-/.f64 78.7

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 78.7%

   \[\leadsto 1 - \frac{\frac{\frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(-0.3275911, \left|x\right|, 1\right) \cdot 1.061405429}{\mathsf{fma}\left(-0.10731592879921, x \cdot x, 1\right)}} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

7. Final simplification 78.7%

   \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\mathsf{fma}\left(-0.3275911, \left|x\right|, 1\right) \cdot 1.061405429}{\mathsf{fma}\left(-0.10731592879921, x \cdot x, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} \cdot e^{\left(-x\right) \cdot x} \]
8. Add Preprocessing

Alternative 11: 78.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. lift-*.f64 N/A

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-1453152027}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}\right)\right)\right)\right)\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   2. *-commutative N/A

      \[\leadsto 1 - \color{blue}{\left(\left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-1453152027}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}\right)\right)\right)\right) \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. lift-/.f64 N/A

      \[\leadsto 1 - \left(\left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-1453152027}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   4. un-div-inv N/A

      \[\leadsto 1 - \color{blue}{\frac{\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-1453152027}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \frac{1061405429}{1000000000}\right)\right)\right)}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   5. lower-/.f64 78.7

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

4. Applied rewrites 78.7%

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

   1. lift-*.f64 N/A

      \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\color{blue}{\left|x\right| \cdot \left|x\right|}} \]

   2. lift-fabs.f64 N/A

      \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\color{blue}{\left|x\right|} \cdot \left|x\right|} \]

   3. lift-fabs.f64 N/A

      \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\left|x\right| \cdot \color{blue}{\left|x\right|}} \]

   4. sqr-abs N/A

      \[\leadsto 1 - \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot e^{-\color{blue}{x \cdot x}} \]

   5. lift-*.f64 78.7

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

6. Applied rewrites 78.7%

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

7. Final simplification 78.7%

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

Alternative 12: 77.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{1.061405429}{\frac{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}} + -1.453152027}{t\_0} + 1.421413741}{-t\_0} - -0.284496736}{t\_0} - 0.254829592}{t\_0}, 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (fma
    1.0
    (/
     (-
      (/
       (-
        (/
         (+
          (/
           (+
            (/
             1.061405429
             (/
              (fma 0.10731592879921 (* x x) -1.0)
              (fma 0.3275911 (fabs x) -1.0)))
            -1.453152027)
           t_0)
          1.421413741)
         (- t_0))
        -0.284496736)
       t_0)
      0.254829592)
     t_0)
    1.0)))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	return fma(1.0, (((((((((1.061405429 / (fma(0.10731592879921, (x * x), -1.0) / fma(0.3275911, fabs(x), -1.0))) + -1.453152027) / t_0) + 1.421413741) / -t_0) - -0.284496736) / t_0) - 0.254829592) / t_0), 1.0);
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return fma(1.0, Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / Float64(fma(0.10731592879921, Float64(x * x), -1.0) / fma(0.3275911, abs(x), -1.0))) + -1.453152027) / t_0) + 1.421413741) / Float64(-t_0)) - -0.284496736) / t_0) - 0.254829592) / t_0), 1.0)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 * N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / N[(N[(0.10731592879921 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] / N[(0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.421413741), $MachinePrecision] / (-t$95$0)), $MachinePrecision] - -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - 0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Initial program 78.7%

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]
6. Step-by-step derivation

7. Applied rewrites 75.7%

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

   1. lift-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000} + 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\color{blue}{\frac{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right) \cdot \left(\left|x\right| \cdot \frac{3275911}{10000000}\right) - 1 \cdot 1}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   3. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\color{blue}{\frac{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right) \cdot \left(\left|x\right| \cdot \frac{3275911}{10000000}\right) - 1 \cdot 1}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right) \cdot \left(\left|x\right| \cdot \frac{3275911}{10000000}\right) - \color{blue}{1}}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\color{blue}{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right) \cdot \left(\left|x\right| \cdot \frac{3275911}{10000000}\right) + \left(\mathsf{neg}\left(1\right)\right)}}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right) \cdot \color{blue}{\left(\frac{3275911}{10000000} \cdot \left|x\right|\right)} + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\color{blue}{\left(\frac{3275911}{10000000} \cdot \left|x\right|\right)} \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   8. swap-sqr N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\color{blue}{\left(\frac{3275911}{10000000} \cdot \frac{3275911}{10000000}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)} + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\color{blue}{\frac{10731592879921}{100000000000000}} \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\color{blue}{\left(\frac{-3275911}{10000000} \cdot \frac{-3275911}{10000000}\right)} \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   11. lift-fabs.f64 N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\frac{-3275911}{10000000} \cdot \frac{-3275911}{10000000}\right) \cdot \left(\color{blue}{\left|x\right|} \cdot \left|x\right|\right) + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   12. lift-fabs.f64 N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\frac{-3275911}{10000000} \cdot \frac{-3275911}{10000000}\right) \cdot \left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right) + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   13. sqr-abs N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\frac{-3275911}{10000000} \cdot \frac{-3275911}{10000000}\right) \cdot \color{blue}{\left(x \cdot x\right)} + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\frac{-3275911}{10000000} \cdot \frac{-3275911}{10000000}\right) \cdot \color{blue}{\left(x \cdot x\right)} + \left(\mathsf{neg}\left(1\right)\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\left(\frac{-3275911}{10000000} \cdot \frac{-3275911}{10000000}\right) \cdot \left(x \cdot x\right) + \color{blue}{-1}}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\color{blue}{\mathsf{fma}\left(\frac{-3275911}{10000000} \cdot \frac{-3275911}{10000000}, x \cdot x, -1\right)}}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\mathsf{fma}\left(\color{blue}{\frac{10731592879921}{100000000000000}}, x \cdot x, -1\right)}{\left|x\right| \cdot \frac{3275911}{10000000} - 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   18. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\mathsf{fma}\left(\frac{10731592879921}{100000000000000}, x \cdot x, -1\right)}{\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000} + \left(\mathsf{neg}\left(1\right)\right)}}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\mathsf{fma}\left(\frac{10731592879921}{100000000000000}, x \cdot x, -1\right)}{\color{blue}{\frac{3275911}{10000000} \cdot \left|x\right|} + \left(\mathsf{neg}\left(1\right)\right)}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\frac{\mathsf{fma}\left(\frac{10731592879921}{100000000000000}, x \cdot x, -1\right)}{\frac{3275911}{10000000} \cdot \left|x\right| + \color{blue}{-1}}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]

   21. lower-fma.f64 75.8

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{1.061405429}{\frac{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}}} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{-\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right) \]

9. Applied rewrites 75.8%

   \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{1.061405429}{\color{blue}{\frac{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}}} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{-\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right) \]

10. Final simplification 75.8%

    \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{1.061405429}{\frac{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, -1\right)}} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{-\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right) \]
11. Add Preprocessing

Alternative 13: 77.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{1.061405429}{t\_0} + -1.453152027}{t\_0} + 1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{\frac{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}{\mathsf{fma}\left(-0.3275911, \left|x\right|, 1\right)}}, 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (fma
    1.0
    (/
     (+
      (/
       (+
        (/ (+ (/ (+ (/ 1.061405429 t_0) -1.453152027) t_0) 1.421413741) t_0)
        -0.284496736)
       t_0)
      0.254829592)
     (/ (fma 0.10731592879921 (* x x) -1.0) (fma -0.3275911 (fabs x) 1.0)))
    1.0)))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	return fma(1.0, (((((((((1.061405429 / t_0) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / (fma(0.10731592879921, (x * x), -1.0) / fma(-0.3275911, fabs(x), 1.0))), 1.0);
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return fma(1.0, Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.061405429 / t_0) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / Float64(fma(0.10731592879921, Float64(x * x), -1.0) / fma(-0.3275911, abs(x), 1.0))), 1.0)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 * N[(N[(N[(N[(N[(N[(N[(N[(N[(1.061405429 / t$95$0), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(N[(0.10731592879921 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] / N[(-0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Initial program 78.7%

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]
6. Step-by-step derivation

7. Applied rewrites 75.7%

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

   1. lift-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\color{blue}{\mathsf{neg}\left(\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)\right)}}, 1\right) \]

   2. lift-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\mathsf{neg}\left(\color{blue}{\left(\left|x\right| \cdot \frac{3275911}{10000000} + 1\right)}\right)}, 1\right) \]

   3. distribute-neg-in N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\color{blue}{\left(\mathsf{neg}\left(\left|x\right| \cdot \frac{3275911}{10000000}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}, 1\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\left(\mathsf{neg}\left(\color{blue}{\frac{3275911}{10000000} \cdot \left|x\right|}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\color{blue}{\left(\mathsf{neg}\left(\frac{3275911}{10000000}\right)\right) \cdot \left|x\right|} + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\color{blue}{\frac{-3275911}{10000000}} \cdot \left|x\right| + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\color{blue}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1}}, 1\right) \]

   8. flip-- N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\color{blue}{\frac{\left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) - 1 \cdot 1}{\frac{-3275911}{10000000} \cdot \left|x\right| + 1}}}, 1\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\color{blue}{\left(\left|x\right| \cdot \frac{-3275911}{10000000}\right)} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right|\right) - 1 \cdot 1}{\frac{-3275911}{10000000} \cdot \left|x\right| + 1}}, 1\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\left(\left|x\right| \cdot \frac{-3275911}{10000000}\right) \cdot \color{blue}{\left(\left|x\right| \cdot \frac{-3275911}{10000000}\right)} - 1 \cdot 1}{\frac{-3275911}{10000000} \cdot \left|x\right| + 1}}, 1\right) \]

   11. swap-sqr N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \frac{-3275911}{10000000}\right)} - 1 \cdot 1}{\frac{-3275911}{10000000} \cdot \left|x\right| + 1}}, 1\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \color{blue}{\frac{10731592879921}{100000000000000}} - 1 \cdot 1}{\frac{-3275911}{10000000} \cdot \left|x\right| + 1}}, 1\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \color{blue}{\left(\frac{3275911}{10000000} \cdot \frac{3275911}{10000000}\right)} - 1 \cdot 1}{\frac{-3275911}{10000000} \cdot \left|x\right| + 1}}, 1\right) \]

   14. swap-sqr N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\color{blue}{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right) \cdot \left(\left|x\right| \cdot \frac{3275911}{10000000}\right)} - 1 \cdot 1}{\frac{-3275911}{10000000} \cdot \left|x\right| + 1}}, 1\right) \]

   15. lift-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\frac{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right) \cdot \left(\left|x\right| \cdot \frac{3275911}{10000000}\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, 1\right)}}}, 1\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{\color{blue}{\frac{\left(\left|x\right| \cdot \frac{3275911}{10000000}\right) \cdot \left(\left|x\right| \cdot \frac{3275911}{10000000}\right) - 1 \cdot 1}{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, 1\right)}}}, 1\right) \]

9. Applied rewrites 75.7%

   \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{\color{blue}{\frac{\mathsf{fma}\left(0.10731592879921, x \cdot x, -1\right)}{\mathsf{fma}\left(-0.3275911, \left|x\right|, 1\right)}}}, 1\right) \]
10. Add Preprocessing

Alternative 14: 77.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\mathsf{fma}\left(-0.3275911, \left|x\right|, 1\right) \cdot 1.061405429}{\mathsf{fma}\left(-0.10731592879921, x \cdot x, 1\right)} + -1.453152027}{t\_0} + 1.421413741}{-t\_0} - -0.284496736}{t\_0} - 0.254829592}{t\_0}, 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0)))
   (fma
    1.0
    (/
     (-
      (/
       (-
        (/
         (+
          (/
           (+
            (/
             (* (fma -0.3275911 (fabs x) 1.0) 1.061405429)
             (fma -0.10731592879921 (* x x) 1.0))
            -1.453152027)
           t_0)
          1.421413741)
         (- t_0))
        -0.284496736)
       t_0)
      0.254829592)
     t_0)
    1.0)))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	return fma(1.0, ((((((((((fma(-0.3275911, fabs(x), 1.0) * 1.061405429) / fma(-0.10731592879921, (x * x), 1.0)) + -1.453152027) / t_0) + 1.421413741) / -t_0) - -0.284496736) / t_0) - 0.254829592) / t_0), 1.0);
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	return fma(1.0, Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(fma(-0.3275911, abs(x), 1.0) * 1.061405429) / fma(-0.10731592879921, Float64(x * x), 1.0)) + -1.453152027) / t_0) + 1.421413741) / Float64(-t_0)) - -0.284496736) / t_0) - 0.254829592) / t_0), 1.0)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, N[(1.0 * N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(-0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision] * 1.061405429), $MachinePrecision] / N[(-0.10731592879921 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.421413741), $MachinePrecision] / (-t$95$0)), $MachinePrecision] - -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - 0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Initial program 78.7%

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 78.7%

   \[\leadsto \mathsf{fma}\left({\left(e^{x}\right)}^{\left(-x\right)}, \frac{\frac{\frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(-0.3275911, \left|x\right|, 1\right) \cdot 1.061405429}{\mathsf{fma}\left(-0.10731592879921, x \cdot x, 1\right)}} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{-\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right) \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{\frac{\frac{\frac{\frac{\mathsf{fma}\left(\frac{-3275911}{10000000}, \left|x\right|, 1\right) \cdot \frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{-10731592879921}{100000000000000}, x \cdot x, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]
8. Step-by-step derivation

9. Applied rewrites 75.7%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{\frac{\frac{\frac{\frac{\mathsf{fma}\left(-0.3275911, \left|x\right|, 1\right) \cdot 1.061405429}{\mathsf{fma}\left(-0.10731592879921, x \cdot x, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 0.254829592}{-\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right) \]

10. Final simplification 75.7%

    \[\leadsto \mathsf{fma}\left(1, \frac{\frac{\frac{\frac{\frac{\mathsf{fma}\left(-0.3275911, \left|x\right|, 1\right) \cdot 1.061405429}{\mathsf{fma}\left(-0.10731592879921, x \cdot x, 1\right)} + -1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + 1.421413741}{-\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - -0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} - 0.254829592}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, 1\right) \]
11. Add Preprocessing

Alternative 15: 77.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \mathsf{fma}\left(-0.3275911, \left|x\right|, -1\right)\\ \mathsf{fma}\left(\frac{\frac{\frac{\frac{\frac{-1.061405429}{t\_1} + -1.453152027}{t\_0} + 1.421413741}{t\_0} + -0.284496736}{t\_0} + 0.254829592}{t\_1}, 1, 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0))
        (t_1 (fma -0.3275911 (fabs x) -1.0)))
   (fma
    (/
     (+
      (/
       (+
        (/ (+ (/ (+ (/ -1.061405429 t_1) -1.453152027) t_0) 1.421413741) t_0)
        -0.284496736)
       t_0)
      0.254829592)
     t_1)
    1.0
    1.0)))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = fma(-0.3275911, fabs(x), -1.0);
	return fma((((((((((-1.061405429 / t_1) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_1), 1.0, 1.0);
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = fma(-0.3275911, abs(x), -1.0)
	return fma(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1.061405429 / t_1) + -1.453152027) / t_0) + 1.421413741) / t_0) + -0.284496736) / t_0) + 0.254829592) / t_1), 1.0, 1.0)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(-0.3275911 * N[Abs[x], $MachinePrecision] + -1.0), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(-1.061405429 / t$95$1), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] + -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.254829592), $MachinePrecision] / t$95$1), $MachinePrecision] * 1.0 + 1.0), $MachinePrecision]]]

Derivation

1. Initial program 78.7%

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]
6. Step-by-step derivation

7. Applied rewrites 75.7%

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

   1. lift-fma.f64 N/A

      \[\leadsto \color{blue}{1 \cdot \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + 1} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot 1} + 1 \]

   3. lower-fma.f64 75.7

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

9. Applied rewrites 75.7%

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

Alternative 16: 77.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}, 1\right) \]
6. Step-by-step derivation

7. Applied rewrites 75.7%

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

   1. lift-fma.f64 N/A

      \[\leadsto \color{blue}{1 \cdot \frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + 1} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{1421413741}{1000000000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{-8890523}{31250000}}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} + \frac{31853699}{125000000}}{-\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)} \cdot 1} + 1 \]

   3. lower-fma.f64 75.7

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

9. Applied rewrites 75.7%

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

11. Applied rewrites 75.7%

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

Reproduce

herbie shell --seed 2024318 
(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)))))))