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

Average Error: 7.3 → 1.5

Time: 12.0s

Precision: binary64

Cost: 20164

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -5.86545167239441 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, 10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.555613563475806 \cdot 10^{-29}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.86545167239441e-50)
   (fma (* eps 5.0) (pow x 4.0) (* 10.0 (* eps (* eps (pow x 3.0)))))
   (if (<= x 4.555613563475806e-29)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (* eps (+ (* 5.0 (pow x 4.0)) (* eps (* 10.0 (pow x 3.0))))))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if (x <= -5.86545167239441e-50) {
		tmp = fma((eps * 5.0), pow(x, 4.0), (10.0 * (eps * (eps * pow(x, 3.0)))));
	} else if (x <= 4.555613563475806e-29) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = eps * ((5.0 * pow(x, 4.0)) + (eps * (10.0 * pow(x, 3.0))));
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	tmp = 0.0
	if (x <= -5.86545167239441e-50)
		tmp = fma(Float64(eps * 5.0), (x ^ 4.0), Float64(10.0 * Float64(eps * Float64(eps * (x ^ 3.0)))));
	elseif (x <= 4.555613563475806e-29)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64(eps * Float64(Float64(5.0 * (x ^ 4.0)) + Float64(eps * Float64(10.0 * (x ^ 3.0)))));
	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_] := If[LessEqual[x, -5.86545167239441e-50], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision] + N[(10.0 * N[(eps * N[(eps * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.555613563475806e-29], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(10.0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8654516723944096e-50

   1. Initial program 38.7

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

   2. Taylor expanded in x around inf 4.9

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

   3. Simplified 4.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot 10\right) + {x}^{3} \cdot 10\right)\right)} \]
      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)))): 3 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))))): 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 (*.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)))): 1 points increase in error, 1 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)))): 1 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)))): 1 points increase in error, 2 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)))): 1 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)))): 1 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 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)))): 1 points increase in error, 2 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, 1 points decrease in error

   4. Taylor expanded in eps around 0 5.3

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

   5. Simplified 5.3

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)}\right) \]
      Proof
      (*.f64 10 (*.f64 eps (*.f64 eps (pow.f64 x 3)))): 0 points increase in error, 0 points decrease in error
      (*.f64 10 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 eps eps) (pow.f64 x 3)))): 7 points increase in error, 4 points decrease in error
      (*.f64 10 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 eps 2)) (pow.f64 x 3))): 0 points increase in error, 0 points decrease in error

   if -5.8654516723944096e-50 < x < 4.55561356347580599e-29

   1. Initial program 0.9

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

   if 4.55561356347580599e-29 < x

   1. Initial program 48.7

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

   2. Taylor expanded in x around inf 2.8

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

   3. Simplified 2.8

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10\right)\right)} \]
      Proof
      (*.f64 eps (+.f64 (*.f64 5 (pow.f64 x 4)) (*.f64 eps (*.f64 (pow.f64 x 3) 10)))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (+.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 4 1)) (pow.f64 x 4)) (*.f64 eps (*.f64 (pow.f64 x 3) 10)))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (+.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 4 (pow.f64 x 4)) (pow.f64 x 4))) (*.f64 eps (*.f64 (pow.f64 x 3) 10)))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (+.f64 (+.f64 (*.f64 4 (pow.f64 x 4)) (pow.f64 x 4)) (*.f64 eps (*.f64 (pow.f64 x 3) (Rewrite<= metadata-eval (+.f64 4 6)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (+.f64 (+.f64 (*.f64 4 (pow.f64 x 4)) (pow.f64 x 4)) (*.f64 eps (*.f64 (pow.f64 x 3) (+.f64 4 (Rewrite<= metadata-eval (+.f64 2 4))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (+.f64 (+.f64 (*.f64 4 (pow.f64 x 4)) (pow.f64 x 4)) (*.f64 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
      (*.f64 eps (+.f64 (+.f64 (*.f64 4 (pow.f64 x 4)) (pow.f64 x 4)) (*.f64 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
      (*.f64 eps (+.f64 (+.f64 (*.f64 4 (pow.f64 x 4)) (pow.f64 x 4)) (*.f64 eps (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (Rewrite=> cube-mult_binary64 (*.f64 x (*.f64 x x))) (+.f64 2 4)))))): 1 points increase in error, 0 points decrease in error
      (*.f64 eps (+.f64 (+.f64 (*.f64 4 (pow.f64 x 4)) (pow.f64 x 4)) (*.f64 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
      (*.f64 eps (+.f64 (+.f64 (*.f64 4 (pow.f64 x 4)) (pow.f64 x 4)) (*.f64 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
      (*.f64 eps (+.f64 (+.f64 (*.f64 4 (pow.f64 x 4)) (pow.f64 x 4)) (*.f64 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
      (*.f64 eps (+.f64 (+.f64 (*.f64 4 (pow.f64 x 4)) (pow.f64 x 4)) (*.f64 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
      (*.f64 eps (+.f64 (+.f64 (*.f64 4 (pow.f64 x 4)) (pow.f64 x 4)) (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) eps)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (+.f64 (*.f64 4 (pow.f64 x 4)) (pow.f64 x 4)) eps) (*.f64 (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) eps) eps))): 1 points increase in error, 2 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 eps (+.f64 (*.f64 4 (pow.f64 x 4)) (pow.f64 x 4)))) (*.f64 (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) eps) eps)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 eps (*.f64 4 (pow.f64 x 4))) (*.f64 eps (pow.f64 x 4)))) (*.f64 (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) eps) eps)): 11 points increase in error, 9 points decrease in error
      (+.f64 (+.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 eps 4) (pow.f64 x 4))) (*.f64 eps (pow.f64 x 4))) (*.f64 (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) eps) eps)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 eps)) (pow.f64 x 4)) (*.f64 eps (pow.f64 x 4))) (*.f64 (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) eps) eps)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 (pow.f64 x 4) (+.f64 (*.f64 4 eps) eps))) (*.f64 (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) eps) eps)): 8 points increase in error, 9 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4))) (*.f64 (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) eps) eps)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (Rewrite<= associate-*r*_binary64 (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (*.f64 eps eps)))): 0 points increase in error, 3 points decrease in error
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (*.f64 (+.f64 (*.f64 4 (pow.f64 x 3)) (*.f64 (+.f64 (*.f64 2 (pow.f64 x 2)) (*.f64 4 (pow.f64 x 2))) x)) (Rewrite<= unpow2_binary64 (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (*.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
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (*.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
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (*.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
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (*.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))): 1 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (*.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
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (*.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, 1 points decrease in error
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (*.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
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (*.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
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (*.f64 (*.f64 (pow.f64 x 3) (Rewrite=> metadata-eval 10)) (pow.f64 eps 2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (*.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
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 x 3) (*.f64 (+.f64 2 8) (pow.f64 eps 2))))): 3 points increase in error, 1 points decrease in error
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (*.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
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (*.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
      (+.f64 (*.f64 (+.f64 (*.f64 4 eps) eps) (pow.f64 x 4)) (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
3. Recombined 3 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.86545167239441 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, 10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot {x}^{3}\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.555613563475806 \cdot 10^{-29}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.5
Cost	13896

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right)\\ \mathbf{if}\;x \leq -5.86545167239441 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.555613563475806 \cdot 10^{-29}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.5
Cost	13572

\[\begin{array}{l} \mathbf{if}\;x \leq -5.86545167239441 \cdot 10^{-50}:\\ \;\;\;\;\left(\varepsilon \cdot {x}^{3}\right) \cdot \mathsf{fma}\left(x, 5, \varepsilon \cdot 10\right)\\ \mathbf{elif}\;x \leq 4.555613563475806 \cdot 10^{-29}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{3} \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \end{array} \]

Alternative 3
Error	1.5
Cost	13512

\[\begin{array}{l} t_0 := \varepsilon \cdot \left({x}^{3} \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \mathbf{if}\;x \leq -5.86545167239441 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.555613563475806 \cdot 10^{-29}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.7
Cost	7432

\[\begin{array}{l} t_0 := \varepsilon \cdot \left({x}^{3} \cdot \left(\varepsilon \cdot 10 + x \cdot 5\right)\right)\\ \mathbf{if}\;x \leq -5.86545167239441 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.555613563475806 \cdot 10^{-29}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.8
Cost	7176

\[\begin{array}{l} \mathbf{if}\;x \leq -5.86545167239441 \cdot 10^{-50}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 4.555613563475806 \cdot 10^{-29}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + x \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 6
Error	1.8
Cost	7048

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{if}\;x \leq -5.86545167239441 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.555613563475806 \cdot 10^{-29}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.8
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -5.86545167239441 \cdot 10^{-50}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 4.555613563475806 \cdot 10^{-29}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 8
Error	1.8
Cost	6792

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{if}\;x \leq -5.86545167239441 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.555613563475806 \cdot 10^{-29}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	11.1
Cost	704

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

Reproduce

herbie shell --seed 2022297 
(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)))