Jmat.Real.erf

Average Error: 13.5 → 0.5

Time: 18.6s

Precision: binary64

Cost: 275588

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \frac{0.254829592 + \frac{-0.284496736 + \frac{\mathsf{fma}\left(1.061405429, {t_0}^{-2}, 1.421413741\right) + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{t_0}}{t_0}}{\frac{t_0}{{\left(e^{-x}\right)}^{x}}}\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{{\left(x \cdot \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right)\right)}^{2}} + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t_1}^{3}}{t_1 + \left(1 + {t_1}^{2}\right)}\\ \end{array} \]

FPCore

(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 (fma 0.3275911 (fabs x) 1.0))
        (t_1
         (/
          (+
           0.254829592
           (/
            (+
             -0.284496736
             (/
              (+
               (fma 1.061405429 (pow t_0 -2.0) 1.421413741)
               (/ -1.453152027 (fma 0.3275911 x 1.0)))
              t_0))
            t_0))
          (/ t_0 (pow (exp (- x)) x)))))
   (if (<= (fabs x) 5e-9)
     (+
      (sqrt (pow (* x (fma x -0.00011824294398844343 1.128386358070218)) 2.0))
      1e-9)
     (/ (- 1.0 (pow t_1 3.0)) (+ t_1 (+ 1.0 (pow t_1 2.0)))))))

C

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 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = (0.254829592 + ((-0.284496736 + ((fma(1.061405429, pow(t_0, -2.0), 1.421413741) + (-1.453152027 / fma(0.3275911, x, 1.0))) / t_0)) / t_0)) / (t_0 / pow(exp(-x), x));
	double tmp;
	if (fabs(x) <= 5e-9) {
		tmp = sqrt(pow((x * fma(x, -0.00011824294398844343, 1.128386358070218)), 2.0)) + 1e-9;
	} else {
		tmp = (1.0 - pow(t_1, 3.0)) / (t_1 + (1.0 + pow(t_1, 2.0)));
	}
	return tmp;
}

Julia

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 = fma(0.3275911, abs(x), 1.0)
	t_1 = Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(fma(1.061405429, (t_0 ^ -2.0), 1.421413741) + Float64(-1.453152027 / fma(0.3275911, x, 1.0))) / t_0)) / t_0)) / Float64(t_0 / (exp(Float64(-x)) ^ x)))
	tmp = 0.0
	if (abs(x) <= 5e-9)
		tmp = Float64(sqrt((Float64(x * fma(x, -0.00011824294398844343, 1.128386358070218)) ^ 2.0)) + 1e-9);
	else
		tmp = Float64(Float64(1.0 - (t_1 ^ 3.0)) / Float64(t_1 + Float64(1.0 + (t_1 ^ 2.0))));
	end
	return tmp
end

Wolfram

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[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(N[(1.061405429 * N[Power[t$95$0, -2.0], $MachinePrecision] + 1.421413741), $MachinePrecision] + N[(-1.453152027 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 / N[Power[N[Exp[(-x)], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-9], N[(N[Sqrt[N[Power[N[(x * N[(x * -0.00011824294398844343 + 1.128386358070218), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + 1e-9), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.0000000000000001e-9

   1. Initial program 27.1

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

   2. Simplified 27.1

      \[\leadsto \color{blue}{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]
      Proof
      (-.f64 1 (/.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))) (*.f64 (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 31853699/125000000 (/.f64 (+.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))) (*.f64 (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 31853699/125000000 (/.f64 (+.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))) (*.f64 (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 31853699/125000000 (/.f64 (+.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))) (*.f64 (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 31853699/125000000 (/.f64 (+.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))) (*.f64 (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 31853699/125000000 (/.f64 (+.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))) (*.f64 (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 31853699/125000000 (/.f64 (+.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))) (*.f64 (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 31853699/125000000 (/.f64 (+.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))) (*.f64 (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 31853699/125000000 (/.f64 (+.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))) (*.f64 (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 31853699/125000000 (/.f64 (+.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))) (*.f64 (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 31853699/125000000 (/.f64 (+.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))) (*.f64 (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 31853699/125000000 (/.f64 (+.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))) (*.f64 (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 31853699/125000000 (/.f64 (+.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))) (*.f64 (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 31853699/125000000 (/.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)))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (*.f64 (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 31853699/125000000 (/.f64 (*.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))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (*.f64 (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 31853699/125000000 (/.f64 (*.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))))))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) (*.f64 (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 31853699/125000000 (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))))))))) (*.f64 (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 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.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 (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 1 (+.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 (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 1 (+.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 (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 1 (+.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 (+.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 1 (+.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 (+.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 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (/.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))))))): 2 points increase in error, 0 points decrease in error
      (-.f64 1 (Rewrite=> associate-*r/_binary64 (/.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 (*.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 (*.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 (*.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))))))): 0 points increase in error, 1 points decrease in error

   3. Applied egg-rr 27.3

      \[\leadsto \color{blue}{{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   4. Taylor expanded in x around 0 3.4

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + \left(-0.00011824294398844343 \cdot {x}^{2} + 1.128386358070218 \cdot x\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   5. Simplified 3.4

      \[\leadsto {\left({\color{blue}{\left(10^{-9} + \mathsf{fma}\left(1.128386358070218, x, x \cdot \left(x \cdot -0.00011824294398844343\right)\right)\right)}}^{3}\right)}^{0.3333333333333333} \]
      Proof
      (+.f64 1/1000000000 (fma.f64 564193179035109/500000000000000 x (*.f64 x (*.f64 x -2364858879768868679/20000000000000000000000)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/1000000000 (fma.f64 564193179035109/500000000000000 x (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x x) -2364858879768868679/20000000000000000000000)))): 11 points increase in error, 10 points decrease in error
      (+.f64 1/1000000000 (fma.f64 564193179035109/500000000000000 x (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) -2364858879768868679/20000000000000000000000))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/1000000000 (fma.f64 564193179035109/500000000000000 x (Rewrite<= *-commutative_binary64 (*.f64 -2364858879768868679/20000000000000000000000 (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/1000000000 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 564193179035109/500000000000000 x) (*.f64 -2364858879768868679/20000000000000000000000 (pow.f64 x 2))))): 0 points increase in error, 1 points decrease in error
      (+.f64 1/1000000000 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -2364858879768868679/20000000000000000000000 (pow.f64 x 2)) (*.f64 564193179035109/500000000000000 x)))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 0.7

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

   7. Applied egg-rr 0.0

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

   if 5.0000000000000001e-9 < (fabs.f64 x)

   1. Initial program 0.3

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]
      Proof
      (-.f64 1 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (*.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 1061405429/1000000000 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))))))))) (exp.f64 (neg.f64 (*.f64 x x)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (*.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 (Rewrite<= metadata-eval (*.f64 1 1061405429/1000000000)) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))))))))) (exp.f64 (neg.f64 (*.f64 x x)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (*.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 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 x x)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (*.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)))) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) (exp.f64 (neg.f64 (*.f64 x x)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (*.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)))) (Rewrite=> *-lft-identity_binary64 (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 x x)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (*.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 (neg.f64 (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (Rewrite<= associate-*l*_binary64 (*.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 0 0.3

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

   4. Applied egg-rr 0.9

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

   5. Simplified 0.9

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right) - 1.453152027 \cdot \frac{1}{\color{blue}{0.3275911 \cdot x} + 1}\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
      Proof
      (*.f64 3275911/10000000 x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 3275911/10000000 x) 0)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 3275911/10000000 x) (Rewrite<= metadata-eval (-.f64 1 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 3275911/10000000 x) 1) 1)): 62 points increase in error, 75 points decrease in error
      (-.f64 (Rewrite<= fma-udef_binary64 (fma.f64 3275911/10000000 x 1)) 1): 1 points increase in error, 0 points decrease in error

   6. Applied egg-rr 0.9

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

   7. Simplified 0.9

      \[\leadsto \color{blue}{\frac{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{\mathsf{fma}\left(1.061405429, {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, 1.421413741\right) - \frac{1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}{{\left(e^{-x}\right)}^{x}}}\right)}^{3}}{\left(1 + {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{\mathsf{fma}\left(1.061405429, {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, 1.421413741\right) - \frac{1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}{{\left(e^{-x}\right)}^{x}}}\right)}^{2}\right) + \frac{0.254829592 + \frac{-0.284496736 + \frac{\mathsf{fma}\left(1.061405429, {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, 1.421413741\right) - \frac{1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}{{\left(e^{-x}\right)}^{x}}}}} \]
      Proof
      (/.f64 (-.f64 1 (pow.f64 (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) (pow.f64 (exp.f64 (neg.f64 x)) x))) 3)) (+.f64 (+.f64 1 (pow.f64 (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) (pow.f64 (exp.f64 (neg.f64 x)) x))) 2)) (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) (pow.f64 (exp.f64 (neg.f64 x)) x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 1 (pow.f64 (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)) (pow.f64 (exp.f64 (neg.f64 x)) x))) 3)) (+.f64 (+.f64 1 (pow.f64 (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) (pow.f64 (exp.f64 (neg.f64 x)) x))) 2)) (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) (pow.f64 (exp.f64 (neg.f64 x)) x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 1 (pow.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) 3)) (+.f64 (+.f64 1 (pow.f64 (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) (pow.f64 (exp.f64 (neg.f64 x)) x))) 2)) (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) (pow.f64 (exp.f64 (neg.f64 x)) x))))): 1 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 1 (pow.f64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (Rewrite<= fma-udef_binary64 (fma.f64 3275911/10000000 (fabs.f64 x) 1))) 3)) (+.f64 (+.f64 1 (pow.f64 (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) (pow.f64 (exp.f64 (neg.f64 x)) x))) 2)) (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) (pow.f64 (exp.f64 (neg.f64 x)) x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 1 (pow.f64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) 3)) (+.f64 (+.f64 1 (pow.f64 (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)) (pow.f64 (exp.f64 (neg.f64 x)) x))) 2)) (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) (pow.f64 (exp.f64 (neg.f64 x)) x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 1 (pow.f64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) 3)) (+.f64 (+.f64 1 (pow.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) 2)) (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) (pow.f64 (exp.f64 (neg.f64 x)) x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 1 (pow.f64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) 3)) (+.f64 (+.f64 1 (pow.f64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (Rewrite<= fma-udef_binary64 (fma.f64 3275911/10000000 (fabs.f64 x) 1))) 2)) (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) (pow.f64 (exp.f64 (neg.f64 x)) x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 1 (pow.f64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) 3)) (+.f64 (+.f64 1 (pow.f64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) 2)) (/.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (/.f64 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)) (pow.f64 (exp.f64 (neg.f64 x)) x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 1 (pow.f64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) 3)) (+.f64 (+.f64 1 (pow.f64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) 2)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 1 (pow.f64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) 3)) (+.f64 (+.f64 1 (pow.f64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) 2)) (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (Rewrite<= fma-udef_binary64 (fma.f64 3275911/10000000 (fabs.f64 x) 1))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 1 (pow.f64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) 3)) (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 (pow.f64 (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) 2) (/.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (-.f64 (fma.f64 1061405429/1000000000 (pow.f64 (fma.f64 3275911/10000000 (fabs.f64 x) 1) -2) 1421413741/1000000000) (/.f64 1453152027/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (pow.f64 (exp.f64 (neg.f64 x)) x)) (fma.f64 3275911/10000000 (fabs.f64 x) 1)))))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{{\left(x \cdot \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right)\right)}^{2}} + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{\mathsf{fma}\left(1.061405429, {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, 1.421413741\right) + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}{{\left(e^{-x}\right)}^{x}}}\right)}^{3}}{\frac{0.254829592 + \frac{-0.284496736 + \frac{\mathsf{fma}\left(1.061405429, {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, 1.421413741\right) + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}{{\left(e^{-x}\right)}^{x}}} + \left(1 + {\left(\frac{0.254829592 + \frac{-0.284496736 + \frac{\mathsf{fma}\left(1.061405429, {\left(\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\right)}^{-2}, 1.421413741\right) + \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}{{\left(e^{-x}\right)}^{x}}}\right)}^{2}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	250052

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := \frac{-1.453152027}{\mathsf{fma}\left(0.3275911, x, 1\right)}\\ t_2 := \frac{0.254829592 + \frac{-0.284496736 + \frac{\mathsf{fma}\left(1.061405429, {t_0}^{-2}, 1.421413741\right) + t_1}{t_0}}{t_0}}{\frac{t_0}{{\left(e^{-x}\right)}^{x}}}\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{{\left(x \cdot \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right)\right)}^{2}} + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{{\left(\frac{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}{0.254829592 + \frac{-0.284496736 + \frac{\mathsf{fma}\left(1.061405429, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2}, 1.421413741\right) + t_1}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}\right)}^{3}}}{t_2 + \left(1 + {t_2}^{2}\right)}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	107012

\[\begin{array}{l} t_0 := \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{{\left(x \cdot \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right)\right)}^{2}} + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t_0}^{2}}{1 + t_0}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	88900

\[\begin{array}{l} t_0 := \frac{1}{1 + \left|x\right| \cdot 0.3275911}\\ \mathbf{if}\;\left(t_0 \cdot \left(0.254829592 + t_0 \cdot \left(-0.284496736 + t_0 \cdot \left(1.421413741 + t_0 \cdot \left(-1.453152027 + 1.061405429 \cdot t_0\right)\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)} \leq 0.99998:\\ \;\;\;\;1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(x \cdot \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right)\right)}^{2}} + 10^{-9}\\ \end{array} \]

Alternative 4
Error	0.5
Cost	66436

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{{\left(x \cdot \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right)\right)}^{2}} + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

Alternative 5
Error	0.5
Cost	66180

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{{\left(x \cdot \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right)\right)}^{2}} + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{-0.254829592 + \frac{0.284496736 + \frac{-1.421413741 - \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}\right)}\\ \end{array} \]

Alternative 6
Error	0.3
Cost	35912

\[\begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := 1 + \left|x\right| \cdot 0.3275911\\ t_2 := \frac{1}{t_1}\\ \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{{\left(x \cdot \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right)\right)}^{2}} + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(0.254829592 + t_2 \cdot \left(-0.284496736 + t_2 \cdot \left(1.453152027 \cdot \frac{-1}{t_0} - \left(-1.421413741 - 1.061405429 \cdot \frac{1}{{t_0}^{2}}\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\right) \cdot \frac{-1}{t_1}\\ \end{array} \]

Alternative 7
Error	0.3
Cost	29320

\[\begin{array}{l} t_0 := \frac{1}{1 + x \cdot 0.3275911}\\ t_1 := 1 + \left|x\right| \cdot 0.3275911\\ t_2 := \frac{1}{t_1}\\ \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{{\left(x \cdot \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right)\right)}^{2}} + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 + t_2 \cdot \left(e^{x \cdot \left(-x\right)} \cdot \left(-0.254829592 + t_0 \cdot \left(0.284496736 - t_2 \cdot \left(1.421413741 + t_0 \cdot \left(-1.453152027 + \frac{1.061405429}{t_1}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 8
Error	0.4
Cost	19976

\[\begin{array}{l} \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\sqrt{{\left(x \cdot \mathsf{fma}\left(x, -0.00011824294398844343, 1.128386358070218\right)\right)}^{2}} + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \]

Alternative 9
Error	0.8
Cost	13960

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.00011824294398844343, x \cdot x, 10^{-9}\right) + x \cdot 1.128386358070218\right) + -0.37545125292247583 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \]

Alternative 10
Error	0.9
Cost	13768

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + x \cdot \left(1.128386358070218 + \log \left(1 + \mathsf{expm1}\left(x \cdot -0.00011824294398844343\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \]

Alternative 11
Error	0.9
Cost	6792

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\left(10^{-9} + x \cdot 1.128386358070218\right) + x \cdot \left(x \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;{1}^{0.3333333333333333}\\ \end{array} \]

Alternative 12
Error	29.6
Cost	64

\[10^{-9} \]

Reproduce

herbie shell --seed 2022338 
(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)))))))