Jmat.Real.erf

Average Error: 13.4 → 0.2

Time: 16.2s

Precision: binary64

Cost: 98500

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := {\left(e^{x}\right)}^{x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-6}:\\ \;\;\;\;1 - \frac{\frac{\left(\frac{1.061405429}{{t_0}^{4}} + \frac{1.421413741}{{t_0}^{2}}\right) + \left(0.254829592 + \left(\frac{-1.453152027}{{t_0}^{3}} - \frac{0.284496736}{t_0}\right)\right)}{t_1}}{t_0}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{1 - \frac{0.254829592 + \frac{\mathsf{fma}\left(1.061405429, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-3}, \frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \mathsf{fma}\left(1.453152027, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2}, 0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{t_1 \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}}\right)\\ \end{array} \]

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 (fabs x) 0.3275911 1.0)) (t_1 (pow (exp x) x)))
   (if (<= x -2e-6)
     (-
      1.0
      (/
       (/
        (+
         (+ (/ 1.061405429 (pow t_0 4.0)) (/ 1.421413741 (pow t_0 2.0)))
         (+
          0.254829592
          (- (/ -1.453152027 (pow t_0 3.0)) (/ 0.284496736 t_0))))
        t_1)
       t_0))
     (if (<= x 1.5e-6)
       (+ 1e-9 (sqrt (pow (* x 1.128386358070218) 2.0)))
       (log
        (exp
         (-
          1.0
          (/
           (+
            0.254829592
            (/
             (-
              (fma
               1.061405429
               (pow (fma 0.3275911 x 1.0) -3.0)
               (/ 1.421413741 (fma 0.3275911 x 1.0)))
              (fma 1.453152027 (pow (fma 0.3275911 x 1.0) -2.0) 0.284496736))
             (fma 0.3275911 x 1.0)))
           (* t_1 (fma 0.3275911 x 1.0))))))))))

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(fabs(x), 0.3275911, 1.0);
	double t_1 = pow(exp(x), x);
	double tmp;
	if (x <= -2e-6) {
		tmp = 1.0 - (((((1.061405429 / pow(t_0, 4.0)) + (1.421413741 / pow(t_0, 2.0))) + (0.254829592 + ((-1.453152027 / pow(t_0, 3.0)) - (0.284496736 / t_0)))) / t_1) / t_0);
	} else if (x <= 1.5e-6) {
		tmp = 1e-9 + sqrt(pow((x * 1.128386358070218), 2.0));
	} else {
		tmp = log(exp((1.0 - ((0.254829592 + ((fma(1.061405429, pow(fma(0.3275911, x, 1.0), -3.0), (1.421413741 / fma(0.3275911, x, 1.0))) - fma(1.453152027, pow(fma(0.3275911, x, 1.0), -2.0), 0.284496736)) / fma(0.3275911, x, 1.0))) / (t_1 * fma(0.3275911, x, 1.0))))));
	}
	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(abs(x), 0.3275911, 1.0)
	t_1 = exp(x) ^ x
	tmp = 0.0
	if (x <= -2e-6)
		tmp = Float64(1.0 - Float64(Float64(Float64(Float64(Float64(1.061405429 / (t_0 ^ 4.0)) + Float64(1.421413741 / (t_0 ^ 2.0))) + Float64(0.254829592 + Float64(Float64(-1.453152027 / (t_0 ^ 3.0)) - Float64(0.284496736 / t_0)))) / t_1) / t_0));
	elseif (x <= 1.5e-6)
		tmp = Float64(1e-9 + sqrt((Float64(x * 1.128386358070218) ^ 2.0)));
	else
		tmp = log(exp(Float64(1.0 - Float64(Float64(0.254829592 + Float64(Float64(fma(1.061405429, (fma(0.3275911, x, 1.0) ^ -3.0), Float64(1.421413741 / fma(0.3275911, x, 1.0))) - fma(1.453152027, (fma(0.3275911, x, 1.0) ^ -2.0), 0.284496736)) / fma(0.3275911, x, 1.0))) / Float64(t_1 * fma(0.3275911, x, 1.0))))));
	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[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]}, If[LessEqual[x, -2e-6], N[(1.0 - N[(N[(N[(N[(N[(1.061405429 / N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.421413741 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.254829592 + N[(N[(-1.453152027 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.284496736 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.5e-6], N[(1e-9 + N[Sqrt[N[Power[N[(x * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(1.0 - N[(N[(0.254829592 + N[(N[(N[(1.061405429 * N[Power[N[(0.3275911 * x + 1.0), $MachinePrecision], -3.0], $MachinePrecision] + N[(1.421413741 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.453152027 * N[Power[N[(0.3275911 * x + 1.0), $MachinePrecision], -2.0], $MachinePrecision] + 0.284496736), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999991e-6

   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. Taylor expanded in x around 0 0.1

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

   3. Simplified 0.1

      \[\leadsto \color{blue}{1 - \frac{\frac{\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}}\right) + \left(0.254829592 - \left(\frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + \frac{1.453152027}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{3}}\right)\right)}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}} \]
      Proof
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (/.f64 1061405429/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 4)) (/.f64 1421413741/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 2))) (-.f64 31853699/125000000 (+.f64 (/.f64 8890523/31250000 (fma.f64 (fabs.f64 x) 3275911/10000000 1)) (/.f64 1453152027/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1061405429/1000000000 1)) (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 4)) (/.f64 1421413741/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 2))) (-.f64 31853699/125000000 (+.f64 (/.f64 8890523/31250000 (fma.f64 (fabs.f64 x) 3275911/10000000 1)) (/.f64 1453152027/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (/.f64 (*.f64 1061405429/1000000000 1) (pow.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (fabs.f64 x) 3275911/10000000) 1)) 4)) (/.f64 1421413741/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 2))) (-.f64 31853699/125000000 (+.f64 (/.f64 8890523/31250000 (fma.f64 (fabs.f64 x) 3275911/10000000 1)) (/.f64 1453152027/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (/.f64 (*.f64 1061405429/1000000000 1) (pow.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 3275911/10000000 (fabs.f64 x))) 1) 4)) (/.f64 1421413741/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 2))) (-.f64 31853699/125000000 (+.f64 (/.f64 8890523/31250000 (fma.f64 (fabs.f64 x) 3275911/10000000 1)) (/.f64 1453152027/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4)))) (/.f64 1421413741/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 2))) (-.f64 31853699/125000000 (+.f64 (/.f64 8890523/31250000 (fma.f64 (fabs.f64 x) 3275911/10000000 1)) (/.f64 1453152027/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 1 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (/.f64 (Rewrite<= metadata-eval (*.f64 1421413741/1000000000 1)) (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 2))) (-.f64 31853699/125000000 (+.f64 (/.f64 8890523/31250000 (fma.f64 (fabs.f64 x) 3275911/10000000 1)) (/.f64 1453152027/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (/.f64 (*.f64 1421413741/1000000000 1) (pow.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (fabs.f64 x) 3275911/10000000) 1)) 2))) (-.f64 31853699/125000000 (+.f64 (/.f64 8890523/31250000 (fma.f64 (fabs.f64 x) 3275911/10000000 1)) (/.f64 1453152027/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (/.f64 (*.f64 1421413741/1000000000 1) (pow.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 3275911/10000000 (fabs.f64 x))) 1) 2))) (-.f64 31853699/125000000 (+.f64 (/.f64 8890523/31250000 (fma.f64 (fabs.f64 x) 3275911/10000000 1)) (/.f64 1453152027/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (/.f64 (*.f64 1421413741/1000000000 1) (pow.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 2))) (-.f64 31853699/125000000 (+.f64 (/.f64 8890523/31250000 (fma.f64 (fabs.f64 x) 3275911/10000000 1)) (/.f64 1453152027/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (Rewrite<= associate-*r/_binary64 (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2))))) (-.f64 31853699/125000000 (+.f64 (/.f64 8890523/31250000 (fma.f64 (fabs.f64 x) 3275911/10000000 1)) (/.f64 1453152027/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 1 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2)))) (-.f64 31853699/125000000 (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 8890523/31250000 1)) (fma.f64 (fabs.f64 x) 3275911/10000000 1)) (/.f64 1453152027/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2)))) (-.f64 31853699/125000000 (+.f64 (/.f64 (*.f64 8890523/31250000 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (fabs.f64 x) 3275911/10000000) 1))) (/.f64 1453152027/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2)))) (-.f64 31853699/125000000 (+.f64 (/.f64 (*.f64 8890523/31250000 1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 3275911/10000000 (fabs.f64 x))) 1)) (/.f64 1453152027/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2)))) (-.f64 31853699/125000000 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (/.f64 1453152027/1000000000 (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2)))) (-.f64 31853699/125000000 (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (/.f64 (Rewrite<= metadata-eval (*.f64 1453152027/1000000000 1)) (pow.f64 (fma.f64 (fabs.f64 x) 3275911/10000000 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2)))) (-.f64 31853699/125000000 (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (/.f64 (*.f64 1453152027/1000000000 1) (pow.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (fabs.f64 x) 3275911/10000000) 1)) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2)))) (-.f64 31853699/125000000 (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (/.f64 (*.f64 1453152027/1000000000 1) (pow.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 3275911/10000000 (fabs.f64 x))) 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2)))) (-.f64 31853699/125000000 (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (Rewrite<= associate-*r/_binary64 (*.f64 1453152027/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3))))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 1 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2)))) 31853699/125000000) (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (*.f64 1453152027/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3)))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 123 points increase in error, 2 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (-.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)) 2)))) 31853699/125000000) (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (*.f64 1453152027/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 2)))))) (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (*.f64 1453152027/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 2))))) (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (*.f64 1453152027/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3))))))) (pow.f64 (exp.f64 x) x)) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (*.f64 1 (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 2))))) (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (*.f64 1453152027/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3)))))) (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 x x)))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (*.f64 1 (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 2))))) (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (*.f64 1453152027/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3)))))) (exp.f64 (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (/.f64 (*.f64 1 (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 2))))) (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (*.f64 1453152027/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3)))))) (exp.f64 (Rewrite<= unpow2_binary64 (pow.f64 (fabs.f64 x) 2)))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (exp.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 2))))) (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (*.f64 1453152027/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3))))))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 1 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2)))) (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 2))))) (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (*.f64 1453152027/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3)))))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 1 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 2)))) 31853699/125000000)) (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (*.f64 1453152027/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3)))))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (+.f64 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 2)))) 31853699/125000000) (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (*.f64 1453152027/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3)))))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2)))))) (+.f64 (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (*.f64 1453152027/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3)))))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2))))) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 1453152027/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3))) (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))))))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2))))) (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 1453152027/1000000000 1) (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 3))) (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 1 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2))))) (+.f64 (/.f64 (*.f64 1453152027/1000000000 1) (pow.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 3)) (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2))))) (+.f64 (/.f64 (*.f64 1453152027/1000000000 1) (Rewrite=> unpow3_binary64 (*.f64 (*.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 1 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2))))) (+.f64 (/.f64 (*.f64 1453152027/1000000000 1) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2)) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2))))) (+.f64 (/.f64 (*.f64 1453152027/1000000000 1) (*.f64 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2))))) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1453152027/1000000000 (/.f64 1 (*.f64 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2) (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))))) (*.f64 8890523/31250000 (/.f64 1 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 1 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2))))) (+.f64 (*.f64 1453152027/1000000000 (/.f64 1 (*.f64 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2) (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (*.f64 8890523/31250000 (/.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))))))) (fma.f64 (fabs.f64 x) 3275911/10000000 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2))))) (+.f64 (*.f64 1453152027/1000000000 (/.f64 1 (*.f64 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2) (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (*.f64 8890523/31250000 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (fabs.f64 x) 3275911/10000000) 1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2))))) (+.f64 (*.f64 1453152027/1000000000 (/.f64 1 (*.f64 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2) (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (*.f64 8890523/31250000 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))))) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 3275911/10000000 (fabs.f64 x))) 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (/.f64 (*.f64 (exp.f64 (neg.f64 (pow.f64 (fabs.f64 x) 2))) (-.f64 (+.f64 31853699/125000000 (+.f64 (*.f64 1061405429/1000000000 (/.f64 1 (pow.f64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1) 4))) (*.f64 1421413741/1000000000 (/.f64 1 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2))))) (+.f64 (*.f64 1453152027/1000000000 (/.f64 1 (*.f64 (pow.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 2) (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) (*.f64 8890523/31250000 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))): 0 points increase in error, 0 points decrease in error

   if -1.99999999999999991e-6 < x < 1.5e-6

   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

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]
      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, 1 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. Applied egg-rr 27.5

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

   4. Simplified 27.5

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\frac{\mathsf{fma}\left(0.3275911, x, 1\right)}{{\left(e^{x}\right)}^{x}}}\right)}} \]
      Proof
      (exp.f64 (log1p.f64 (/.f64 (-.f64 -31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 1061405429/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))) (/.f64 (fma.f64 3275911/10000000 x 1) (pow.f64 (exp.f64 x) x))))): 0 points increase in error, 0 points decrease in error
      (exp.f64 (log1p.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 x 1))) (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))) (/.f64 (fma.f64 3275911/10000000 x 1) (pow.f64 (exp.f64 x) x))))): 0 points increase in error, 0 points decrease in error
      (exp.f64 (log1p.f64 (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 31853699/125000000) (neg.f64 (/.f64 (+.f64 -8890523/31250000 (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 1061405429/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))))) (/.f64 (fma.f64 3275911/10000000 x 1) (pow.f64 (exp.f64 x) x))))): 0 points increase in error, 0 points decrease in error
      (exp.f64 (log1p.f64 (/.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 1061405429/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))))) (/.f64 (fma.f64 3275911/10000000 x 1) (pow.f64 (exp.f64 x) x))))): 0 points increase in error, 0 points decrease in error
      (exp.f64 (log1p.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (neg.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 1061405429/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1)))) (pow.f64 (exp.f64 x) x)) (fma.f64 3275911/10000000 x 1))))): 0 points increase in error, 1 points decrease in error
      (exp.f64 (log1p.f64 (/.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (+.f64 31853699/125000000 (/.f64 (+.f64 -8890523/31250000 (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 1061405429/1000000000 (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))) (fma.f64 3275911/10000000 x 1))) (pow.f64 (exp.f64 x) x)))) (fma.f64 3275911/10000000 x 1)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in x around 0 1.2

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

   6. Simplified 1.2

      \[\leadsto \color{blue}{10^{-9} + x \cdot 1.128386358070218} \]
      Proof
      (+.f64 1/1000000000 (*.f64 x 564193179035109/500000000000000)): 0 points increase in error, 0 points decrease in error
      (+.f64 1/1000000000 (Rewrite<= *-commutative_binary64 (*.f64 564193179035109/500000000000000 x))): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 0.2

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

   if 1.5e-6 < x

   1. Initial program 0.2

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

   2. Simplified 0.2

      \[\leadsto \color{blue}{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]
      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, 1 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, 1 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, 1 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, 1 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))))))): 1 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around inf 0.2

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

   4. Applied egg-rr 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-6}:\\ \;\;\;\;1 - \frac{\frac{\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{4}} + \frac{1.421413741}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{2}}\right) + \left(0.254829592 + \left(\frac{-1.453152027}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)}^{3}} - \frac{0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\right)\right)}{{\left(e^{x}\right)}^{x}}}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{1 - \frac{0.254829592 + \frac{\mathsf{fma}\left(1.061405429, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-3}, \frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \mathsf{fma}\left(1.453152027, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2}, 0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	85512

\[\begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{-7}:\\ \;\;\;\;1 + t_1 \cdot \left(e^{-x \cdot x} \cdot \left(-0.254829592 + t_1 \cdot \left(0.284496736 + t_1 \cdot \left(-1.421413741 + \left(1.061405429 \cdot t_1 + -1.453152027\right) \cdot \frac{-1}{t_0}\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{1 - \frac{0.254829592 + \frac{\mathsf{fma}\left(1.061405429, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-3}, \frac{1.421413741}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right) - \mathsf{fma}\left(1.453152027, {\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{-2}, 0.284496736\right)}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left(0.3275911, x, 1\right)}}\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	42116

\[\begin{array}{l} t_0 := \frac{1}{1 + \left|x\right| \cdot 0.3275911}\\ t_1 := e^{-x \cdot x}\\ t_2 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{-7}:\\ \;\;\;\;1 - t_0 \cdot \left(\left(0.254829592 + t_0 \cdot \left(-0.284496736 + t_0 \cdot \left(1.421413741 + t_0 \cdot \left(1.061405429 \cdot t_0 + -1.453152027\right)\right)\right)\right) \cdot t_1\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{t_2} \cdot \left(t_1 \cdot \left(-0.254829592 + \left(-0.284496736 + t_0 \cdot \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}, 1.421413741\right)\right) \cdot \frac{-1}{t_2}\right)\right)\\ \end{array} \]

Alternative 3
Error	0.2
Cost	41988

\[\begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := e^{-x \cdot x}\\ t_2 := 1 + x \cdot 0.3275911\\ t_3 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{-7}:\\ \;\;\;\;1 + \left(t_1 \cdot \left(0.254829592 + t_3 \cdot \left(-0.284496736 + t_3 \cdot \left(1.421413741 + t_3 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right)\right) \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{t_2} \cdot \left(t_1 \cdot \left(-0.254829592 + \left(-0.284496736 + t_3 \cdot \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}, 1.421413741\right)\right) \cdot \frac{-1}{t_2}\right)\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	35336

\[\begin{array}{l} t_0 := \frac{1}{1 + x \cdot 0.3275911}\\ \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1 + t_0 \cdot \left(e^{-x \cdot x} \cdot \left(-0.254829592 + t_0 \cdot \left(0.284496736 + \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)}, -1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}, 1.421413741\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	0.5
Cost	19716

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;10^{-9} + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	0.3
Cost	16520

\[\begin{array}{l} t_0 := 1 + x \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-6}:\\ \;\;\;\;10^{-9} + \sqrt{{\left(x \cdot 1.128386358070218\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1 + t_1 \cdot \left(e^{-x \cdot x} \cdot \left(-0.254829592 + t_1 \cdot \left(0.284496736 + t_1 \cdot \left(-1.421413741 + \left(-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}\right) \cdot \frac{-1}{t_0}\right)\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	1.0
Cost	584

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

Alternative 8
Error	1.6
Cost	328

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

Alternative 9
Error	29.8
Cost	64

\[10^{-9} \]

Reproduce

herbie shell --seed 2022328 
(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)))))))