Average Error: 13.3 → 0.2
Time: 18.9s
Precision: binary64
Cost: 109124
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
↓
\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left(t_1 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 - t_1 \cdot \left(-1.421413741 + t_1 \cdot \left(1.453152027 + t_1 \cdot -1.061405429\right)\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)} \leq 0.9995:\\ \;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{\left(0.284496736 + 1.453152027 \cdot \frac{1}{{t_0}^{2}}\right) + \left(1.061405429 \cdot \frac{-1}{{t_0}^{3}} + t_1 \cdot -1.421413741\right)}{t_0}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{{\left(e^{x}\right)}^{x}}\\ \mathbf{else}:\\ \;\;\;\;10^{-9} + \left(\left(x \cdot x\right) \cdot -0.00011824294398844343 + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\right)\\ \end{array} \]
(FPCore (x)
 :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)))))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.3275911 (fabs x)))) (t_1 (/ 1.0 t_0)))
   (if (<=
        (*
         (*
          t_1
          (+
           0.254829592
           (*
            t_1
            (-
             -0.284496736
             (*
              t_1
              (+
               -1.421413741
               (* t_1 (+ 1.453152027 (* t_1 -1.061405429)))))))))
         (exp (* x (- x))))
        0.9995)
     (+
      1.0
      (/
       (/
        (+
         -0.254829592
         (/
          (+
           (+ 0.284496736 (* 1.453152027 (/ 1.0 (pow t_0 2.0))))
           (+ (* 1.061405429 (/ -1.0 (pow t_0 3.0))) (* t_1 -1.421413741)))
          t_0))
        (fma 0.3275911 (fabs x) 1.0))
       (pow (exp x) x)))
     (+
      1e-9
      (+
       (* (* x x) -0.00011824294398844343)
       (sqrt (pow (* x 1.128386358070218) 2.0)))))))
double code(double x) {
	return 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))));
}
↓
double code(double x) {
	double t_0 = 1.0 + (0.3275911 * fabs(x));
	double t_1 = 1.0 / t_0;
	double tmp;
	if (((t_1 * (0.254829592 + (t_1 * (-0.284496736 - (t_1 * (-1.421413741 + (t_1 * (1.453152027 + (t_1 * -1.061405429))))))))) * exp((x * -x))) <= 0.9995) {
		tmp = 1.0 + (((-0.254829592 + (((0.284496736 + (1.453152027 * (1.0 / pow(t_0, 2.0)))) + ((1.061405429 * (-1.0 / pow(t_0, 3.0))) + (t_1 * -1.421413741))) / t_0)) / fma(0.3275911, fabs(x), 1.0)) / pow(exp(x), x));
	} else {
		tmp = 1e-9 + (((x * x) * -0.00011824294398844343) + sqrt(pow((x * 1.128386358070218), 2.0)));
	}
	return tmp;
}

function code(x)
	return Float64(1.0 - Float64(Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * Float64(0.254829592 + Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * Float64(-0.284496736 + Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * Float64(1.421413741 + Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * Float64(-1.453152027 + Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * 1.061405429))))))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end
↓
function code(x)
	t_0 = Float64(1.0 + Float64(0.3275911 * abs(x)))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 - Float64(t_1 * Float64(-1.421413741 + Float64(t_1 * Float64(1.453152027 + Float64(t_1 * -1.061405429))))))))) * exp(Float64(x * Float64(-x)))) <= 0.9995)
		tmp = Float64(1.0 + Float64(Float64(Float64(-0.254829592 + Float64(Float64(Float64(0.284496736 + Float64(1.453152027 * Float64(1.0 / (t_0 ^ 2.0)))) + Float64(Float64(1.061405429 * Float64(-1.0 / (t_0 ^ 3.0))) + Float64(t_1 * -1.421413741))) / t_0)) / fma(0.3275911, abs(x), 1.0)) / (exp(x) ^ x)));
	else
		tmp = Float64(1e-9 + Float64(Float64(Float64(x * x) * -0.00011824294398844343) + sqrt((Float64(x * 1.128386358070218) ^ 2.0))));
	end
	return tmp
end

code[x_] := N[(1.0 - N[(N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.254829592 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.284496736 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.421413741 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.453152027 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]
↓
code[x_] := Block[{t$95$0 = N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 - N[(t$95$1 * N[(-1.421413741 + N[(t$95$1 * N[(1.453152027 + N[(t$95$1 * -1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.9995], N[(1.0 + N[(N[(N[(-0.254829592 + N[(N[(N[(0.284496736 + N[(1.453152027 * N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.061405429 * N[(-1.0 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * -1.421413741), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1e-9 + N[(N[(N[(x * x), $MachinePrecision] * -0.00011824294398844343), $MachinePrecision] + N[Sqrt[N[Power[N[(x * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

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|}
↓
\begin{array}{l}
t_0 := 1 + 0.3275911 \cdot \left|x\right|\\
t_1 := \frac{1}{t_0}\\
\mathbf{if}\;\left(t_1 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 - t_1 \cdot \left(-1.421413741 + t_1 \cdot \left(1.453152027 + t_1 \cdot -1.061405429\right)\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)} \leq 0.9995:\\
\;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{\left(0.284496736 + 1.453152027 \cdot \frac{1}{{t_0}^{2}}\right) + \left(1.061405429 \cdot \frac{-1}{{t_0}^{3}} + t_1 \cdot -1.421413741\right)}{t_0}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{{\left(e^{x}\right)}^{x}}\\

\mathbf{else}:\\
\;\;\;\;10^{-9} + \left(\left(x \cdot x\right) \cdot -0.00011824294398844343 + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\right)\\


\end{array}

Error

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) < 0.99950000000000006

    1. Initial program 0.1

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

    2. Simplified0.1

      \[\leadsto \color{blue}{1 + \frac{\frac{-0.254829592 + \frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{{\left(e^{x}\right)}^{x}}} \]
      Proof
      (+.f64 1 (/.f64 (/.f64 (+.f64 -31853699/125000000 (/.f64 (-.f64 8890523/31250000 (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 1061405429/1000000000 (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (Rewrite<= metadata-eval (neg.f64 31853699/125000000)) (/.f64 (-.f64 8890523/31250000 (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 1061405429/1000000000 (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (Rewrite<= metadata-eval (neg.f64 -8890523/31250000)) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 1061405429/1000000000 (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 (Rewrite<= metadata-eval (*.f64 1 1061405429/1000000000)) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 (*.f64 1 1061405429/1000000000) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 (*.f64 1 1061405429/1000000000) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (*.f64 1 (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (*.f64 1 (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (*.f64 1 (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (*.f64 1 (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 -8890523/31250000) (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (neg.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (neg.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (neg.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (neg.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 1 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (exp.f64 (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))): 0 points increase in error, 1 points decrease in error
      (+.f64 1 (/.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) 1)) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) (/.f64 1 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (*.f64 (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 1 points increase in error, 2 points decrease in error
      (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 0 points increase in error, 0 points decrease in error

    3. Taylor expanded in x around inf 0.1

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

    if 0.99950000000000006 < (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))

    1. Initial program 26.9

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

    2. Simplified26.9

      \[\leadsto \color{blue}{1 + \frac{\frac{-0.254829592 + \frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{{\left(e^{x}\right)}^{x}}} \]
      Proof
      (+.f64 1 (/.f64 (/.f64 (+.f64 -31853699/125000000 (/.f64 (-.f64 8890523/31250000 (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 1061405429/1000000000 (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (Rewrite<= metadata-eval (neg.f64 31853699/125000000)) (/.f64 (-.f64 8890523/31250000 (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 1061405429/1000000000 (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (Rewrite<= metadata-eval (neg.f64 -8890523/31250000)) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 1061405429/1000000000 (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 (Rewrite<= metadata-eval (*.f64 1 1061405429/1000000000)) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 (*.f64 1 1061405429/1000000000) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 (*.f64 1 1061405429/1000000000) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (*.f64 1 (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (/.f64 (*.f64 1 (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (+.f64 1421413741/1000000000 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (*.f64 1 (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (/.f64 (*.f64 1 (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (neg.f64 -8890523/31250000) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 -8890523/31250000) (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (neg.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (/.f64 (neg.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (neg.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (+.f64 (neg.f64 31853699/125000000) (neg.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 1 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (pow.f64 (exp.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (exp.f64 (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))): 0 points increase in error, 1 points decrease in error
      (+.f64 1 (/.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) 1)) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) (/.f64 1 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (*.f64 (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 1 points increase in error, 2 points decrease in error
      (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 0 points increase in error, 0 points decrease in error

    3. Applied egg-rr27.5

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

    4. Taylor expanded in x around 0 1.3

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

    5. Applied egg-rr1.3

      \[\leadsto 10^{-9} + \left(\color{blue}{\left(0 + -0.00011824294398844343 \cdot \left(x \cdot x\right)\right)} + 1.128386358070218 \cdot x\right) \]

    6. Applied egg-rr0.3

      \[\leadsto 10^{-9} + \left(\left(0 + -0.00011824294398844343 \cdot \left(x \cdot x\right)\right) + \color{blue}{\sqrt{{\left(1.128386358070218 \cdot x\right)}^{2}}}\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot -1.061405429\right)\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)} \leq 0.9995:\\ \;\;\;\;1 + \frac{\frac{-0.254829592 + \frac{\left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right) + \left(1.061405429 \cdot \frac{-1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot -1.421413741\right)}{1 + 0.3275911 \cdot \left|x\right|}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{{\left(e^{x}\right)}^{x}}\\ \mathbf{else}:\\ \;\;\;\;10^{-9} + \left(\left(x \cdot x\right) \cdot -0.00011824294398844343 + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error0.2
Cost90244
\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ t_2 := e^{x \cdot \left(-x\right)}\\ \mathbf{if}\;\left(t_1 \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 - t_1 \cdot \left(-1.421413741 + t_1 \cdot \left(1.453152027 + t_1 \cdot -1.061405429\right)\right)\right)\right)\right) \cdot t_2 \leq 0.9995:\\ \;\;\;\;1 + t_2 \cdot \left(\left(0.254829592 + t_1 \cdot \left(-0.284496736 - t_1 \cdot \left(\frac{-1.061405429}{{t_0}^{2}} - \left(1.421413741 + t_1 \cdot -1.453152027\right)\right)\right)\right) \cdot \frac{-1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9} + \left(\left(x \cdot x\right) \cdot -0.00011824294398844343 + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\right)\\ \end{array} \]
Alternative 2
Error0.2
Cost83908
\[\begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ t_1 := t_0 \cdot \left(-1.421413741 + t_0 \cdot \left(1.453152027 + t_0 \cdot -1.061405429\right)\right)\\ t_2 := e^{x \cdot \left(-x\right)}\\ \mathbf{if}\;\left(t_0 \cdot \left(0.254829592 + t_0 \cdot \left(-0.284496736 - t_1\right)\right)\right) \cdot t_2 \leq 0.9995:\\ \;\;\;\;1 + t_2 \cdot \left(t_0 \cdot \left(-0.254829592 + t_0 \cdot \left(0.284496736 + t_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;10^{-9} + \left(\left(x \cdot x\right) \cdot -0.00011824294398844343 + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\right)\\ \end{array} \]
Alternative 3
Error0.4
Cost20100
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;10^{-9} + \left(\left(x \cdot x\right) \cdot -0.00011824294398844343 + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 4
Error0.9
Cost13448
\[\begin{array}{l} \mathbf{if}\;x \leq -47137.75285091917:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.820343214013506 \cdot 10^{-11}:\\ \;\;\;\;10^{-9} + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 5
Error1.3
Cost840
\[\begin{array}{l} \mathbf{if}\;x \leq -7.725753466292351 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.820343214013506 \cdot 10^{-11}:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 6
Error1.7
Cost584
\[\begin{array}{l} \mathbf{if}\;x \leq -47137.75285091917:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.820343214013506 \cdot 10^{-11}:\\ \;\;\;\;10^{-9} + x \cdot -0.6936284084460227\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 7
Error1.3
Cost584
\[\begin{array}{l} \mathbf{if}\;x \leq -7.725753466292351 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.820343214013506 \cdot 10^{-11}:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 8
Error1.6
Cost328
\[\begin{array}{l} \mathbf{if}\;x \leq -0.000825127979350158:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.820343214013506 \cdot 10^{-11}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 9
Error30.0
Cost64
\[10^{-9} \]

Error

Reproduce

herbie shell --seed 2022318 
(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)))))))