Jmat.Real.erf

Average Error: 13.3 → 0.8

Time: 15.3s

Precision: binary64

Cost: 26116

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-9}:\\ \;\;\;\;10^{-9} + \sqrt{{\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{6}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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
 (if (<= (fabs x) 2e-9)
   (+ 1e-9 (sqrt (pow (cbrt (* x 1.128386358070218)) 6.0)))
   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 tmp;
	if (fabs(x) <= 2e-9) {
		tmp = 1e-9 + sqrt(pow(cbrt((x * 1.128386358070218)), 6.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

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

↓

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2e-9) {
		tmp = 1e-9 + Math.sqrt(Math.pow(Math.cbrt((x * 1.128386358070218)), 6.0));
	} else {
		tmp = 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)
	tmp = 0.0
	if (abs(x) <= 2e-9)
		tmp = Float64(1e-9 + sqrt((cbrt(Float64(x * 1.128386358070218)) ^ 6.0)));
	else
		tmp = 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_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-9], N[(1e-9 + N[Sqrt[N[Power[N[Power[N[(x * 1.128386358070218), $MachinePrecision], 1/3], $MachinePrecision], 6.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000012e-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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, -0.254829592 + \frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)} \]
      Proof
      (fma.f64 (/.f64 (pow.f64 (exp.f64 x) (neg.f64 x)) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (+.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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (pow.f64 (exp.f64 x) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (+.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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 x (*.f64 -1 x)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (+.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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 x (Rewrite=> mul-1-neg_binary64 (neg.f64 x)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (+.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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 x x)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (+.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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (neg.f64 (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (+.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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (+.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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.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))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (Rewrite<= /-rgt-identity_binary64 (/.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 1))) (+.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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) 1) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.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))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) (+.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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (Rewrite<= metadata-eval (neg.f64 31853699/125000000)) (/.f64 (-.f64 8890523/31250000 (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 1061405429/1000000000 (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (Rewrite<= metadata-eval (-.f64 0 -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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 0 (+.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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (neg.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (neg.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (neg.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (neg.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) 1): 39 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) 1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))))) 1): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))))) 1): 0 points increase in error, 39 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))) 1): 39 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 0 points increase in error, 39 points decrease in error

   3. Applied egg-rr 27.2

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

   4. Taylor expanded in x around 0 0.7

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

   5. Applied egg-rr 0.0

      \[\leadsto 10^{-9} + \color{blue}{\sqrt{1.2732557730789702 \cdot \left(x \cdot x\right)}} \]

   6. Simplified 0.0

      \[\leadsto 10^{-9} + \color{blue}{\sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}} \]
      Proof
      (+.f64 1/1000000000 (sqrt.f64 (*.f64 x (*.f64 x 318313943269742557644254641881/250000000000000000000000000000)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/1000000000 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x x) 318313943269742557644254641881/250000000000000000000000000000)))): 0 points increase in error, 3 points decrease in error
      (+.f64 1/1000000000 (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 318313943269742557644254641881/250000000000000000000000000000 (*.f64 x x))))): 0 points increase in error, 3 points decrease in error

   7. Applied egg-rr 0.0

      \[\leadsto 10^{-9} + \sqrt{\color{blue}{{\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{6}}} \]

   if 2.00000000000000012e-9 < (fabs.f64 x)

   1. Initial program 0.4

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

   2. Simplified 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(e^{x}\right)}^{\left(-x\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, -0.254829592 + \frac{0.284496736 - \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 1\right)} \]
      Proof
      (fma.f64 (/.f64 (pow.f64 (exp.f64 x) (neg.f64 x)) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (+.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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (pow.f64 (exp.f64 x) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (+.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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 x (*.f64 -1 x)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (+.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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (*.f64 x (Rewrite=> mul-1-neg_binary64 (neg.f64 x)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (+.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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 x x)))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (+.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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (neg.f64 (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1)) (+.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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1))) (+.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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.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))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (/.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (Rewrite<= /-rgt-identity_binary64 (/.f64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))) 1))) (+.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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) 1) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.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))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) (+.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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (Rewrite<= metadata-eval (neg.f64 31853699/125000000)) (/.f64 (-.f64 8890523/31250000 (/.f64 (+.f64 1421413741/1000000000 (/.f64 (+.f64 -1453152027/1000000000 (/.f64 1061405429/1000000000 (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (Rewrite<= metadata-eval (-.f64 0 -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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (-.f64 (-.f64 0 -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))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 0 (+.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))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))) (fma.f64 3275911/10000000 (fabs.f64 x) 1))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (neg.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3275911/10000000 (fabs.f64 x)) 1)))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (/.f64 (neg.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) 1): 39 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (neg.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))) (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))))) 1): 0 points increase in error, 39 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (+.f64 (neg.f64 31853699/125000000) (neg.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) 1): 39 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x))))) (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) 1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (neg.f64 (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))))) 1): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))))) 1): 0 points increase in error, 39 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))) 1): 39 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 (neg.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000)))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 31853699/125000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -8890523/31250000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 1421413741/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) (+.f64 -1453152027/1000000000 (*.f64 (/.f64 1 (+.f64 1 (*.f64 3275911/10000000 (fabs.f64 x)))) 1061405429/1000000000))))))))) (exp.f64 (neg.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 0 points increase in error, 39 points decrease in error

   3. Applied egg-rr 63.6

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

   4. Taylor expanded in x around inf 1.5

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-9}:\\ \;\;\;\;10^{-9} + \sqrt{{\left(\sqrt[3]{x \cdot 1.128386358070218}\right)}^{6}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	0.9
Cost	584

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

Alternative 3
Error	1.4
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 4
Error	29.9
Cost	64

\[10^{-9} \]

Reproduce

herbie shell --seed 2022343 
(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)))))))