ENA, Section 1.4, Exercise 4b, n=5

Average Error: 7.3 → 0.8

Time: 14.4s

Precision: binary64

Cost: 40392

\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

\[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

↓

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-298}:\\ \;\;\;\;{\varepsilon}^{5} + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right) + x \cdot \left({\varepsilon}^{3} \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)\right)\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) + \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 5.0) (pow x 5.0)))

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-298)
     (+
      (pow eps 5.0)
      (+
       (* (* eps eps) (* 10.0 (pow x 3.0)))
       (* x (* (pow eps 3.0) (fma 10.0 x (* eps 5.0))))))
     (if (<= t_0 0.0)
       (+ (* eps (* 10.0 (* eps (pow x 3.0)))) (* eps (* 5.0 (pow x 4.0))))
       t_0))))

C

double code(double x, double eps) {
	return pow((x + eps), 5.0) - pow(x, 5.0);
}

↓

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -2e-298) {
		tmp = pow(eps, 5.0) + (((eps * eps) * (10.0 * pow(x, 3.0))) + (x * (pow(eps, 3.0) * fma(10.0, x, (eps * 5.0)))));
	} else if (t_0 <= 0.0) {
		tmp = (eps * (10.0 * (eps * pow(x, 3.0)))) + (eps * (5.0 * pow(x, 4.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
end

↓

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -2e-298)
		tmp = Float64((eps ^ 5.0) + Float64(Float64(Float64(eps * eps) * Float64(10.0 * (x ^ 3.0))) + Float64(x * Float64((eps ^ 3.0) * fma(10.0, x, Float64(eps * 5.0))))));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(eps * Float64(10.0 * Float64(eps * (x ^ 3.0)))) + Float64(eps * Float64(5.0 * (x ^ 4.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-298], N[(N[Power[eps, 5.0], $MachinePrecision] + N[(N[(N[(eps * eps), $MachinePrecision] * N[(10.0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Power[eps, 3.0], $MachinePrecision] * N[(10.0 * x + N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(eps * N[(10.0 * N[(eps * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -1.99999999999999982e-298

   1. Initial program 1.5

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Applied egg-rr 63.0

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{2.5} \cdot {\left(x + \varepsilon\right)}^{2.5}} - {x}^{5} \]

   3. Taylor expanded in x around 0 3.7

      \[\leadsto \color{blue}{{\varepsilon}^{5} + \left(\left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}\right) \cdot x + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)\right)} \]

   4. Simplified 3.7

      \[\leadsto \color{blue}{{\varepsilon}^{5} + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right) + x \cdot \mathsf{fma}\left(x, {\varepsilon}^{3} \cdot 10, 5 \cdot {\varepsilon}^{4}\right)\right)} \]
      Proof
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (*.f64 eps eps) (*.f64 10 (pow.f64 x 3))) (*.f64 x (fma.f64 x (*.f64 (pow.f64 eps 3) 10) (*.f64 5 (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 eps 2)) (*.f64 10 (pow.f64 x 3))) (*.f64 x (fma.f64 x (*.f64 (pow.f64 eps 3) 10) (*.f64 5 (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (pow.f64 eps 2) (*.f64 (Rewrite<= metadata-eval (+.f64 2 8)) (pow.f64 x 3))) (*.f64 x (fma.f64 x (*.f64 (pow.f64 eps 3) 10) (*.f64 5 (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 eps 2) (+.f64 2 8)) (pow.f64 x 3))) (*.f64 x (fma.f64 x (*.f64 (pow.f64 eps 3) 10) (*.f64 5 (pow.f64 eps 4)))))): 2 points increase in error, 4 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2)))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (*.f64 (pow.f64 eps 3) 10) (*.f64 5 (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (*.f64 (pow.f64 eps 3) (Rewrite<= metadata-eval (+.f64 6 4))) (*.f64 5 (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (*.f64 (pow.f64 eps 3) (+.f64 (Rewrite<= metadata-eval (+.f64 2 4)) 4)) (*.f64 5 (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (pow.f64 eps 3) (+.f64 2 4)) (*.f64 (pow.f64 eps 3) 4))) (*.f64 5 (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (+.f64 (*.f64 (Rewrite=> cube-mult_binary64 (*.f64 eps (*.f64 eps eps))) (+.f64 2 4)) (*.f64 (pow.f64 eps 3) 4)) (*.f64 5 (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (+.f64 (*.f64 (*.f64 eps (Rewrite<= unpow2_binary64 (pow.f64 eps 2))) (+.f64 2 4)) (*.f64 (pow.f64 eps 3) 4)) (*.f64 5 (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 eps (*.f64 (pow.f64 eps 2) (+.f64 2 4)))) (*.f64 (pow.f64 eps 3) 4)) (*.f64 5 (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (+.f64 (*.f64 eps (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))))) (*.f64 (pow.f64 eps 3) 4)) (*.f64 5 (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps)) (*.f64 (pow.f64 eps 3) 4)) (*.f64 5 (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (Rewrite<= *-commutative_binary64 (*.f64 4 (pow.f64 eps 3)))) (*.f64 5 (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 eps 4) 5)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (*.f64 (pow.f64 eps 4) (Rewrite<= metadata-eval (neg.f64 -5))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (*.f64 (pow.f64 eps 4) (neg.f64 (Rewrite<= metadata-eval (+.f64 -4 -1)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (pow.f64 eps 4) (+.f64 -4 -1)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (fma.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (neg.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -4 (pow.f64 eps 4)) (*.f64 -1 (pow.f64 eps 4))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3)))) (+.f64 (*.f64 -4 (pow.f64 eps 4)) (*.f64 -1 (pow.f64 eps 4)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3)))) (neg.f64 (+.f64 (*.f64 -4 (pow.f64 eps 4)) (*.f64 -1 (pow.f64 eps 4))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3)))) (neg.f64 (Rewrite=> distribute-rgt-out_binary64 (*.f64 (pow.f64 eps 4) (+.f64 -4 -1)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3)))) (Rewrite=> distribute-rgt-neg-in_binary64 (*.f64 (pow.f64 eps 4) (neg.f64 (+.f64 -4 -1)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3)))) (*.f64 (pow.f64 eps 4) (neg.f64 (Rewrite=> metadata-eval -5))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3)))) (*.f64 (pow.f64 eps 4) (Rewrite=> metadata-eval 5)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3)))) (Rewrite<= *-commutative_binary64 (*.f64 5 (pow.f64 eps 4))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3)))) (*.f64 (Rewrite<= metadata-eval (+.f64 4 1)) (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 x (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3)))) (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 4 (pow.f64 eps 4)) (pow.f64 eps 4))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))))) (*.f64 x (+.f64 (*.f64 4 (pow.f64 eps 4)) (pow.f64 eps 4))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x x) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))))) (*.f64 x (+.f64 (*.f64 4 (pow.f64 eps 4)) (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3)))) (*.f64 x (+.f64 (*.f64 4 (pow.f64 eps 4)) (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2))) (*.f64 x (+.f64 (*.f64 4 (pow.f64 eps 4)) (pow.f64 eps 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 4 (pow.f64 eps 4)) (pow.f64 eps 4)) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2))) (*.f64 (+.f64 (*.f64 4 (pow.f64 eps 4)) (pow.f64 eps 4)) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 eps 5) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (+.f64 (*.f64 4 (pow.f64 eps 4)) (pow.f64 eps 4)) x) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in x around 0 3.7

      \[\leadsto {\varepsilon}^{5} + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right) + \color{blue}{\left(10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 5 \cdot \left({\varepsilon}^{4} \cdot x\right)\right)}\right) \]

   6. Simplified 3.7

      \[\leadsto {\varepsilon}^{5} + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right) + \color{blue}{x \cdot \left({\varepsilon}^{3} \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)}\right) \]
      Proof
      (*.f64 x (*.f64 (pow.f64 eps 3) (fma.f64 10 x (*.f64 eps 5)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (pow.f64 eps 3) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 10 x) (*.f64 eps 5))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (pow.f64 eps 3) (*.f64 10 x)) (*.f64 (pow.f64 eps 3) (*.f64 eps 5))))): 2 points increase in error, 1 points decrease in error
      (*.f64 x (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 eps 3) 10) x)) (*.f64 (pow.f64 eps 3) (*.f64 eps 5)))): 2 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 10 (pow.f64 eps 3))) x) (*.f64 (pow.f64 eps 3) (*.f64 eps 5)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 (*.f64 (*.f64 10 (pow.f64 eps 3)) x) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 eps 3) eps) 5)))): 1 points increase in error, 3 points decrease in error
      (*.f64 x (+.f64 (*.f64 (*.f64 10 (pow.f64 eps 3)) x) (*.f64 (Rewrite=> pow-plus_binary64 (pow.f64 eps (+.f64 3 1))) 5))): 3 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 (*.f64 (*.f64 10 (pow.f64 eps 3)) x) (*.f64 (pow.f64 eps (Rewrite=> metadata-eval 4)) 5))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 (*.f64 (*.f64 10 (pow.f64 eps 3)) x) (Rewrite<= *-commutative_binary64 (*.f64 5 (pow.f64 eps 4))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (*.f64 10 (pow.f64 eps 3)) x) x) (*.f64 (*.f64 5 (pow.f64 eps 4)) x))): 1 points increase in error, 2 points decrease in error
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 10 (pow.f64 eps 3)) (*.f64 x x))) (*.f64 (*.f64 5 (pow.f64 eps 4)) x)): 3 points increase in error, 3 points decrease in error
      (+.f64 (*.f64 (*.f64 10 (pow.f64 eps 3)) (Rewrite<= unpow2_binary64 (pow.f64 x 2))) (*.f64 (*.f64 5 (pow.f64 eps 4)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 10 (*.f64 (pow.f64 eps 3) (pow.f64 x 2)))) (*.f64 (*.f64 5 (pow.f64 eps 4)) x)): 4 points increase in error, 3 points decrease in error
      (+.f64 (*.f64 10 (*.f64 (pow.f64 eps 3) (pow.f64 x 2))) (Rewrite<= associate-*r*_binary64 (*.f64 5 (*.f64 (pow.f64 eps 4) x)))): 4 points increase in error, 10 points decrease in error

   if -1.99999999999999982e-298 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 8.6

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around inf 0.3

      \[\leadsto \color{blue}{\left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)} \]

   3. Simplified 0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot 10\right) + {x}^{3} \cdot 10\right)\right)} \]
      Proof
      (fma.f64 (*.f64 eps 5) (pow.f64 x 4) (*.f64 (*.f64 eps eps) (+.f64 (*.f64 (*.f64 x x) (*.f64 eps 10)) (*.f64 (pow.f64 x 3) 10)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= *-commutative_binary64 (*.f64 5 eps)) (pow.f64 x 4) (*.f64 (*.f64 eps eps) (+.f64 (*.f64 (*.f64 x x) (*.f64 eps 10)) (*.f64 (pow.f64 x 3) 10)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 4 1)) eps) (pow.f64 x 4) (*.f64 (*.f64 eps eps) (+.f64 (*.f64 (*.f64 x x) (*.f64 eps 10)) (*.f64 (pow.f64 x 3) 10)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 4 eps) eps)) (pow.f64 x 4) (*.f64 (*.f64 eps eps) (+.f64 (*.f64 (*.f64 x x) (*.f64 eps 10)) (*.f64 (pow.f64 x 3) 10)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 eps 2)) (+.f64 (*.f64 (*.f64 x x) (*.f64 eps 10)) (*.f64 (pow.f64 x 3) 10)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) (*.f64 eps 10)) (*.f64 (pow.f64 x 3) 10)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 (pow.f64 x 2) (*.f64 eps (Rewrite<= metadata-eval (+.f64 2 8)))) (*.f64 (pow.f64 x 3) 10)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 (pow.f64 x 2) (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 2 8) eps))) (*.f64 (pow.f64 x 3) 10)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (pow.f64 eps 2) (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 x 2) (+.f64 2 8)) eps)) (*.f64 (pow.f64 x 3) 10)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2)))) eps) (*.f64 (pow.f64 x 3) 10)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2))) eps) (*.f64 (pow.f64 x 3) (Rewrite<= metadata-eval (+.f64 4 6)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2))) eps) (*.f64 (pow.f64 x 3) (+.f64 4 (Rewrite<= metadata-eval (+.f64 2 4))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2))) eps) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (pow.f64 x 3) 4) (*.f64 (pow.f64 x 3) (+.f64 2 4))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2))) eps) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (pow.f64 x 3))) (*.f64 (pow.f64 x 3) (+.f64 2 4)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2))) eps) (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (Rewrite=> cube-mult_binary64 (*.f64 x (*.f64 x x))) (+.f64 2 4)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2))) eps) (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (*.f64 x (Rewrite<= unpow2_binary64 (pow.f64 x 2))) (+.f64 2 4)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2))) eps) (+.f64 (*.f64 4 (pow.f64 x 3)) (Rewrite<= associate-*r*_binary64 (*.f64 x (*.f64 (pow.f64 x 2) (+.f64 2 4)))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2))) eps) (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 x (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2))) eps) (+.f64 (*.f64 4 (pow.f64 x 3)) (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2))) eps) (pow.f64 eps 2)) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2))))): 1 points increase in error, 1 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2))) (*.f64 eps (pow.f64 eps 2)))) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 4 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2))) (*.f64 eps (Rewrite=> unpow2_binary64 (*.f64 eps eps)))) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2))) (Rewrite<= cube-mult_binary64 (pow.f64 eps 3))) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 2 points increase in error, 1 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 eps 3) (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 8 (pow.f64 x 2))))) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (pow.f64 eps 3) (Rewrite=> distribute-rgt-out_binary64 (*.f64 (pow.f64 x 2) (+.f64 2 8)))) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (pow.f64 eps 3) (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 2 8) (pow.f64 x 2)))) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (pow.f64 eps 3) (+.f64 2 8)) (pow.f64 x 2))) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 4 points increase in error, 1 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (*.f64 (pow.f64 eps 3) (Rewrite=> metadata-eval 10)) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (*.f64 (pow.f64 eps 3) (Rewrite<= metadata-eval (+.f64 6 4))) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (*.f64 (pow.f64 eps 3) (+.f64 (Rewrite<= metadata-eval (+.f64 2 4)) 4)) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (pow.f64 eps 3) (+.f64 2 4)) (*.f64 (pow.f64 eps 3) 4))) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 2 points increase in error, 1 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (Rewrite=> cube-mult_binary64 (*.f64 eps (*.f64 eps eps))) (+.f64 2 4)) (*.f64 (pow.f64 eps 3) 4)) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 2 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 eps (Rewrite<= unpow2_binary64 (pow.f64 eps 2))) (+.f64 2 4)) (*.f64 (pow.f64 eps 3) 4)) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 eps (*.f64 (pow.f64 eps 2) (+.f64 2 4)))) (*.f64 (pow.f64 eps 3) 4)) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 1 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 eps (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))))) (*.f64 (pow.f64 eps 3) 4)) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps)) (*.f64 (pow.f64 eps 3) 4)) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (Rewrite<= *-commutative_binary64 (*.f64 4 (pow.f64 eps 3)))) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (*.f64 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 x 3) 4)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 (pow.f64 x 3) 4) (Rewrite=> *-commutative_binary64 (*.f64 x (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2)))))) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 (pow.f64 x 3) 4) (*.f64 x (Rewrite=> distribute-rgt-out_binary64 (*.f64 (pow.f64 x 2) (+.f64 2 4))))) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 (pow.f64 x 3) 4) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 x (pow.f64 x 2)) (+.f64 2 4)))) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 (pow.f64 x 3) 4) (*.f64 (*.f64 x (Rewrite=> unpow2_binary64 (*.f64 x x))) (+.f64 2 4))) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (*.f64 (+.f64 (*.f64 (pow.f64 x 3) 4) (*.f64 (Rewrite<= cube-mult_binary64 (pow.f64 x 3)) (+.f64 2 4))) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (*.f64 (Rewrite=> distribute-lft-out_binary64 (*.f64 (pow.f64 x 3) (+.f64 4 (+.f64 2 4)))) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (*.f64 (*.f64 (pow.f64 x 3) (+.f64 4 (Rewrite=> metadata-eval 6))) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (*.f64 (*.f64 (pow.f64 x 3) (Rewrite=> metadata-eval 10)) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (*.f64 (*.f64 (pow.f64 x 3) (Rewrite<= metadata-eval (+.f64 2 8))) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 x 3) (*.f64 (+.f64 2 8) (pow.f64 eps 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (*.f64 (pow.f64 x 3) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 eps 2) (+.f64 2 8)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (*.f64 (pow.f64 x 3) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2)) (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 8 (pow.f64 eps 2))) (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 2 (pow.f64 eps 2)) (*.f64 4 (pow.f64 eps 2))) eps) (*.f64 4 (pow.f64 eps 3))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in eps around 0 0.3

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)}\right) \]

   5. Simplified 0.3

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left(\varepsilon \cdot \left(x \cdot \left(\left(\varepsilon \cdot x\right) \cdot \left(\varepsilon + x\right)\right)\right)\right)}\right) \]
      Proof
      (*.f64 10 (*.f64 eps (*.f64 x (*.f64 (*.f64 eps x) (+.f64 eps x))))): 0 points increase in error, 0 points decrease in error
      (*.f64 10 (*.f64 eps (*.f64 x (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 eps (*.f64 eps x)) (*.f64 x (*.f64 eps x))))))): 0 points increase in error, 1 points decrease in error
      (*.f64 10 (*.f64 eps (*.f64 x (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 eps eps) x)) (*.f64 x (*.f64 eps x)))))): 0 points increase in error, 3 points decrease in error
      (*.f64 10 (*.f64 eps (*.f64 x (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 eps 2)) x) (*.f64 x (*.f64 eps x)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 10 (*.f64 eps (*.f64 x (+.f64 (*.f64 (pow.f64 eps 2) x) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 eps x) x)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 10 (*.f64 eps (*.f64 x (+.f64 (*.f64 (pow.f64 eps 2) x) (Rewrite<= associate-*r*_binary64 (*.f64 eps (*.f64 x x))))))): 3 points increase in error, 4 points decrease in error
      (*.f64 10 (*.f64 eps (*.f64 x (+.f64 (*.f64 (pow.f64 eps 2) x) (*.f64 eps (Rewrite<= unpow2_binary64 (pow.f64 x 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 10 (*.f64 eps (*.f64 x (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 eps (pow.f64 x 2)) (*.f64 (pow.f64 eps 2) x)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 10 (*.f64 eps (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 eps (pow.f64 x 2)) x) (*.f64 (*.f64 (pow.f64 eps 2) x) x))))): 1 points increase in error, 2 points decrease in error
      (*.f64 10 (*.f64 eps (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 eps (*.f64 (pow.f64 x 2) x))) (*.f64 (*.f64 (pow.f64 eps 2) x) x)))): 3 points increase in error, 3 points decrease in error
      (*.f64 10 (*.f64 eps (+.f64 (*.f64 eps (*.f64 (Rewrite=> unpow2_binary64 (*.f64 x x)) x)) (*.f64 (*.f64 (pow.f64 eps 2) x) x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 10 (*.f64 eps (+.f64 (*.f64 eps (Rewrite<= unpow3_binary64 (pow.f64 x 3))) (*.f64 (*.f64 (pow.f64 eps 2) x) x)))): 3 points increase in error, 3 points decrease in error
      (*.f64 10 (*.f64 eps (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 x 3) eps)) (*.f64 (*.f64 (pow.f64 eps 2) x) x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 10 (*.f64 eps (+.f64 (*.f64 (pow.f64 x 3) eps) (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 eps 2) (*.f64 x x)))))): 3 points increase in error, 0 points decrease in error
      (*.f64 10 (*.f64 eps (+.f64 (*.f64 (pow.f64 x 3) eps) (*.f64 (pow.f64 eps 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 10 (*.f64 eps (+.f64 (*.f64 (pow.f64 x 3) eps) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 x 2) (pow.f64 eps 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 10 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (pow.f64 x 3) eps) eps) (*.f64 (*.f64 (pow.f64 x 2) (pow.f64 eps 2)) eps)))): 2 points increase in error, 1 points decrease in error
      (*.f64 10 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 x 3) (*.f64 eps eps))) (*.f64 (*.f64 (pow.f64 x 2) (pow.f64 eps 2)) eps))): 5 points increase in error, 3 points decrease in error
      (*.f64 10 (+.f64 (*.f64 (pow.f64 x 3) (Rewrite<= unpow2_binary64 (pow.f64 eps 2))) (*.f64 (*.f64 (pow.f64 x 2) (pow.f64 eps 2)) eps))): 0 points increase in error, 0 points decrease in error
      (*.f64 10 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 eps 2) (pow.f64 x 3))) (*.f64 (*.f64 (pow.f64 x 2) (pow.f64 eps 2)) eps))): 0 points increase in error, 0 points decrease in error
      (*.f64 10 (+.f64 (*.f64 (pow.f64 eps 2) (pow.f64 x 3)) (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 x 2) (*.f64 (pow.f64 eps 2) eps))))): 1 points increase in error, 4 points decrease in error
      (*.f64 10 (+.f64 (*.f64 (pow.f64 eps 2) (pow.f64 x 3)) (*.f64 (pow.f64 x 2) (*.f64 (Rewrite=> unpow2_binary64 (*.f64 eps eps)) eps)))): 0 points increase in error, 0 points decrease in error
      (*.f64 10 (+.f64 (*.f64 (pow.f64 eps 2) (pow.f64 x 3)) (*.f64 (pow.f64 x 2) (Rewrite<= unpow3_binary64 (pow.f64 eps 3))))): 2 points increase in error, 1 points decrease in error
      (*.f64 10 (+.f64 (*.f64 (pow.f64 eps 2) (pow.f64 x 3)) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 eps 3) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 10 (*.f64 (pow.f64 eps 2) (pow.f64 x 3))) (*.f64 10 (*.f64 (pow.f64 eps 3) (pow.f64 x 2))))): 1 points increase in error, 0 points decrease in error

   6. Taylor expanded in eps around 0 0.3

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, 10 \cdot \left(\varepsilon \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot {x}^{2}\right)}\right)\right)\right) \]

   7. Simplified 0.3

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, 10 \cdot \left(\varepsilon \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]
      Proof
      (*.f64 eps (*.f64 x x)): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (Rewrite<= unpow2_binary64 (pow.f64 x 2))): 0 points increase in error, 0 points decrease in error

   8. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot {x}^{3}\right) \cdot 10\right) + \varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 1.7

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-298}:\\ \;\;\;\;{\varepsilon}^{5} + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot {x}^{3}\right) + x \cdot \left({\varepsilon}^{3} \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) + \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	40392

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) + \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.8
Cost	40392

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-298}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(x \cdot \left(10 \cdot {\varepsilon}^{3}\right) + 5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) + \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.6
Cost	40264

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.6
Cost	39880

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.3
Cost	7432

\[\begin{array}{l} t_0 := \varepsilon \cdot \left({x}^{3} \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \mathbf{if}\;x \leq -1.3383859603913675 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.237969128891156 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.4
Cost	7048

\[\begin{array}{l} t_0 := \left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{if}\;x \leq -1.3383859603913675 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.237969128891156 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.4
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3383859603913675 \cdot 10^{-58}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 1.237969128891156 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 8
Error	1.5
Cost	6792

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{if}\;x \leq -1.3383859603913675 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.237969128891156 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	11.3
Cost	704

\[\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right) \]

Alternative 10
Error	18.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022312 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=5"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 5.0) (pow x 5.0)))