Cubic critical, narrow range

Average Error: 28.1 → 5.4

Time: 20.0s

Precision: binary64

Cost: 54724

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -600:\\ \;\;\;\;\left(b \cdot \frac{1}{a} - \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a}\right) \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{c \cdot \left(a \cdot c\right)}{b \cdot b} \cdot \frac{-0.375}{b}\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -600.0)
   (*
    (- (* b (/ 1.0 a)) (/ (sqrt (fma c (* a -3.0) (* b b))) a))
    -0.3333333333333333)
   (fma
    -0.5625
    (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
    (fma
     -0.16666666666666666
     (* (/ (pow (* a c) 4.0) (pow b 7.0)) (/ 6.328125 a))
     (fma -0.5 (/ c b) (* (/ (* c (* a c)) (* b b)) (/ -0.375 b)))))))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -600.0) {
		tmp = ((b * (1.0 / a)) - (sqrt(fma(c, (a * -3.0), (b * b))) / a)) * -0.3333333333333333;
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.16666666666666666, ((pow((a * c), 4.0) / pow(b, 7.0)) * (6.328125 / a)), fma(-0.5, (c / b), (((c * (a * c)) / (b * b)) * (-0.375 / b)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

↓

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -600.0)
		tmp = Float64(Float64(Float64(b * Float64(1.0 / a)) - Float64(sqrt(fma(c, Float64(a * -3.0), Float64(b * b))) / a)) * -0.3333333333333333);
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.16666666666666666, Float64(Float64((Float64(a * c) ^ 4.0) / (b ^ 7.0)) * Float64(6.328125 / a)), fma(-0.5, Float64(c / b), Float64(Float64(Float64(c * Float64(a * c)) / Float64(b * b)) * Float64(-0.375 / b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -600.0], N[(N[(N[(b * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(6.328125 / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(-0.375 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -600

   1. Initial program 6.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Simplified 6.4

      \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a} \cdot -0.3333333333333333} \]
      Proof
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c -3))))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 3))))))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a c) (neg.f64 3)))))) a) -1/3): 1 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 3)))))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 a c))))))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 3 a) c)))))) a) -1/3): 3 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) a) -1/3): 18 points increase in error, 12 points decrease in error
      (*.f64 (/.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) 1)) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite<= metadata-eval (/.f64 -1 -1))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) -1)) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 -1 a))) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite<= neg-mul-1_binary64 (neg.f64 a))) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 a)) (Rewrite<= metadata-eval (/.f64 -1 3))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (*.f64 (neg.f64 a) 3))): 47 points increase in error, 56 points decrease in error
      (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 a 3)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 a)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (*.f64 3 a)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 3 a) -1))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (/.f64 -1 -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (Rewrite=> metadata-eval 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (Rewrite<= metadata-eval (neg.f64 -1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (/.f64 (*.f64 3 a) (neg.f64 -1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (/.f64 (*.f64 3 a) (Rewrite=> metadata-eval 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite=> /-rgt-identity_binary64 (*.f64 3 a))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 9.4

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{{\left(\sqrt[3]{b}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{\sqrt[3]{b}}{\sqrt[3]{a}}, -\frac{{\left(\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)\right)}^{0.25}}{a} \cdot \frac{{\left(\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)\right)}^{0.25}}{1}\right) + \mathsf{fma}\left(-\frac{{\left(\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)\right)}^{0.25}}{a}, \frac{{\left(\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)\right)}^{0.25}}{1}, \frac{{\left(\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)\right)}^{0.25}}{a} \cdot \frac{{\left(\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)\right)}^{0.25}}{1}\right)\right)} \cdot -0.3333333333333333 \]

   4. Simplified 9.4

      \[\leadsto \color{blue}{\left(0 + \left({\left(\frac{\sqrt[3]{b}}{\sqrt[3]{a}}\right)}^{3} - \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a}\right)\right)} \cdot -0.3333333333333333 \]
      Proof
      (+.f64 0 (-.f64 (pow.f64 (/.f64 (cbrt.f64 b) (cbrt.f64 a)) 3) (/.f64 (sqrt.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b))) a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)))) (-.f64 (pow.f64 (/.f64 (cbrt.f64 b) (cbrt.f64 a)) 3) (/.f64 (sqrt.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b))) a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)) (neg.f64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))))) (-.f64 (pow.f64 (/.f64 (cbrt.f64 b) (cbrt.f64 a)) 3) (/.f64 (sqrt.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b))) a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)))) (-.f64 (pow.f64 (/.f64 (cbrt.f64 b) (cbrt.f64 a)) 3) (/.f64 (sqrt.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b))) a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (neg.f64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (Rewrite=> /-rgt-identity_binary64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4)))) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (pow.f64 (/.f64 (cbrt.f64 b) (cbrt.f64 a)) 3) (/.f64 (sqrt.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b))) a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4))) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (pow.f64 (/.f64 (cbrt.f64 b) (cbrt.f64 a)) 3) (/.f64 (sqrt.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b))) a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= fma-udef_binary64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)))) (-.f64 (pow.f64 (/.f64 (cbrt.f64 b) (cbrt.f64 a)) 3) (/.f64 (sqrt.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b))) a))): 135 points increase in error, 115 points decrease in error
      (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (Rewrite<= /-rgt-identity_binary64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (pow.f64 (/.f64 (cbrt.f64 b) (cbrt.f64 a)) 3) (/.f64 (sqrt.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b))) a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (Rewrite<= cube-unmult_binary64 (*.f64 (/.f64 (cbrt.f64 b) (cbrt.f64 a)) (*.f64 (/.f64 (cbrt.f64 b) (cbrt.f64 a)) (/.f64 (cbrt.f64 b) (cbrt.f64 a))))) (/.f64 (sqrt.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b))) a))): 34 points increase in error, 32 points decrease in error
      (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (*.f64 (/.f64 (cbrt.f64 b) (cbrt.f64 a)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (cbrt.f64 b) (cbrt.f64 b)) (*.f64 (cbrt.f64 a) (cbrt.f64 a))))) (/.f64 (sqrt.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b))) a))): 62 points increase in error, 57 points decrease in error
      (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (*.f64 (/.f64 (cbrt.f64 b) (cbrt.f64 a)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (cbrt.f64 b) 2)) (*.f64 (cbrt.f64 a) (cbrt.f64 a)))) (/.f64 (sqrt.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b))) a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (*.f64 (/.f64 (cbrt.f64 b) (cbrt.f64 a)) (/.f64 (pow.f64 (cbrt.f64 b) 2) (Rewrite<= unpow2_binary64 (pow.f64 (cbrt.f64 a) 2)))) (/.f64 (sqrt.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b))) a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (pow.f64 (cbrt.f64 b) 2) (pow.f64 (cbrt.f64 a) 2)) (/.f64 (cbrt.f64 b) (cbrt.f64 a)))) (/.f64 (sqrt.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b))) a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (*.f64 (/.f64 (pow.f64 (cbrt.f64 b) 2) (pow.f64 (cbrt.f64 a) 2)) (/.f64 (cbrt.f64 b) (cbrt.f64 a))) (/.f64 (sqrt.f64 (fma.f64 c (Rewrite<= *-commutative_binary64 (*.f64 -3 a)) (*.f64 b b))) a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (*.f64 (/.f64 (pow.f64 (cbrt.f64 b) 2) (pow.f64 (cbrt.f64 a) 2)) (/.f64 (cbrt.f64 b) (cbrt.f64 a))) (/.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/2)) a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (*.f64 (/.f64 (pow.f64 (cbrt.f64 b) 2) (pow.f64 (cbrt.f64 a) 2)) (/.f64 (cbrt.f64 b) (cbrt.f64 a))) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) (Rewrite<= metadata-eval (*.f64 2 1/4))) a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (*.f64 (/.f64 (pow.f64 (cbrt.f64 b) 2) (pow.f64 (cbrt.f64 a) 2)) (/.f64 (cbrt.f64 b) (cbrt.f64 a))) (/.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4))) a))): 50 points increase in error, 53 points decrease in error
      (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (*.f64 (/.f64 (pow.f64 (cbrt.f64 b) 2) (pow.f64 (cbrt.f64 a) 2)) (/.f64 (cbrt.f64 b) (cbrt.f64 a))) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4))))): 34 points increase in error, 45 points decrease in error
      (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (*.f64 (/.f64 (pow.f64 (cbrt.f64 b) 2) (pow.f64 (cbrt.f64 a) 2)) (/.f64 (cbrt.f64 b) (cbrt.f64 a))) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (Rewrite<= /-rgt-identity_binary64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (Rewrite=> fma-neg_binary64 (fma.f64 (/.f64 (pow.f64 (cbrt.f64 b) 2) (pow.f64 (cbrt.f64 a) 2)) (/.f64 (cbrt.f64 b) (cbrt.f64 a)) (neg.f64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)))))): 145 points increase in error, 107 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (fma.f64 (/.f64 (pow.f64 (cbrt.f64 b) 2) (pow.f64 (cbrt.f64 a) 2)) (/.f64 (cbrt.f64 b) (cbrt.f64 a)) (neg.f64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)))) (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 7.1

      \[\leadsto \left(0 + \left(\color{blue}{b \cdot \frac{1}{a}} - \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a}\right)\right) \cdot -0.3333333333333333 \]

   if -600 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Simplified 28.9

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{-0.3333333333333333}{a}} \]
      Proof
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c -3))))) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 3))))))) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a c) (neg.f64 3)))))) (/.f64 -1/3 a)): 1 points increase in error, 1 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 3)))))) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 a c))))))) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 3 a) c)))))) (/.f64 -1/3 a)): 2 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 17 points increase in error, 12 points decrease in error
      (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) 1)) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite<= metadata-eval (/.f64 -1 -1))) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) -1)) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) -1) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (/.f64 (Rewrite<= metadata-eval (/.f64 -1 3)) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 3 a)))): 37 points increase in error, 51 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (*.f64 -1 (*.f64 3 a)))): 28 points increase in error, 29 points decrease in error
      (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 3 a) -1))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (/.f64 -1 -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (Rewrite=> metadata-eval 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (Rewrite<= metadata-eval (neg.f64 -1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (/.f64 (*.f64 3 a) (neg.f64 -1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (/.f64 (*.f64 3 a) (Rewrite=> metadata-eval 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite=> /-rgt-identity_binary64 (*.f64 3 a))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in b around inf 5.3

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

   4. Simplified 5.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(-1.125 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)\right)}^{2} + \left(5.0625 \cdot {c}^{4}\right) \cdot {a}^{4}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right)} \]
      Proof
      (fma.f64 -9/16 (/.f64 (pow.f64 c 3) (/.f64 (pow.f64 b 5) (*.f64 a a))) (fma.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 (*.f64 -9/8 (*.f64 c c)) (*.f64 a a)) 2) (*.f64 (*.f64 81/16 (pow.f64 c 4)) (pow.f64 a 4))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 -3/8 (*.f64 a (*.f64 c c))) (pow.f64 b 3))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -9/16 (/.f64 (pow.f64 c 3) (/.f64 (pow.f64 b 5) (Rewrite<= unpow2_binary64 (pow.f64 a 2)))) (fma.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 (*.f64 -9/8 (*.f64 c c)) (*.f64 a a)) 2) (*.f64 (*.f64 81/16 (pow.f64 c 4)) (pow.f64 a 4))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 -3/8 (*.f64 a (*.f64 c c))) (pow.f64 b 3))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -9/16 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (fma.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 (*.f64 -9/8 (*.f64 c c)) (*.f64 a a)) 2) (*.f64 (*.f64 81/16 (pow.f64 c 4)) (pow.f64 a 4))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 -3/8 (*.f64 a (*.f64 c c))) (pow.f64 b 3))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 (*.f64 -9/8 (Rewrite<= unpow2_binary64 (pow.f64 c 2))) (*.f64 a a)) 2) (*.f64 (*.f64 81/16 (pow.f64 c 4)) (pow.f64 a 4))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 -3/8 (*.f64 a (*.f64 c c))) (pow.f64 b 3))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 (*.f64 -9/8 (pow.f64 c 2)) (Rewrite<= unpow2_binary64 (pow.f64 a 2))) 2) (*.f64 (*.f64 81/16 (pow.f64 c 4)) (pow.f64 a 4))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 -3/8 (*.f64 a (*.f64 c c))) (pow.f64 b 3))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 -1/6 (/.f64 (+.f64 (pow.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2)))) 2) (*.f64 (*.f64 81/16 (pow.f64 c 4)) (pow.f64 a 4))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 -3/8 (*.f64 a (*.f64 c c))) (pow.f64 b 3))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (Rewrite<= associate-*r*_binary64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 -3/8 (*.f64 a (*.f64 c c))) (pow.f64 b 3))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 -3/8 (*.f64 a (Rewrite<= unpow2_binary64 (pow.f64 c 2)))) (pow.f64 b 3))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (/.f64 (*.f64 -3/8 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 c 2) a))) (pow.f64 b 3))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (Rewrite<= associate-*r/_binary64 (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))))): 0 points increase in error, 1 points decrease in error
      (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (*.f64 -1/6 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in c around 0 5.3

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right) \]

   6. Simplified 5.3

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right) \]
      Proof
      (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) (pow.f64 b 7)) (/.f64 405/64 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (*.f64 c a) (Rewrite<= metadata-eval (*.f64 2 2))) (pow.f64 b 7)) (/.f64 405/64 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (*.f64 c a) 2) (pow.f64 (*.f64 c a) 2))) (pow.f64 b 7)) (/.f64 405/64 a)): 57 points increase in error, 52 points decrease in error
      (*.f64 (/.f64 (*.f64 (Rewrite=> unpow2_binary64 (*.f64 (*.f64 c a) (*.f64 c a))) (pow.f64 (*.f64 c a) 2)) (pow.f64 b 7)) (/.f64 405/64 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (*.f64 (*.f64 c a) (*.f64 c a)) (Rewrite=> unpow2_binary64 (*.f64 (*.f64 c a) (*.f64 c a)))) (pow.f64 b 7)) (/.f64 405/64 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 a a))) (*.f64 (*.f64 c a) (*.f64 c a))) (pow.f64 b 7)) (/.f64 405/64 a)): 62 points increase in error, 59 points decrease in error
      (*.f64 (/.f64 (*.f64 (*.f64 (*.f64 c c) (*.f64 a a)) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 a a)))) (pow.f64 b 7)) (/.f64 405/64 a)): 55 points increase in error, 54 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 (*.f64 c c) (*.f64 c c)) (*.f64 (*.f64 a a) (*.f64 a a)))) (pow.f64 b 7)) (/.f64 405/64 a)): 53 points increase in error, 59 points decrease in error
      (*.f64 (/.f64 (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (*.f64 c c)) (*.f64 (*.f64 a a) (*.f64 a a))) (pow.f64 b 7)) (/.f64 405/64 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (*.f64 (pow.f64 c 2) (Rewrite<= unpow2_binary64 (pow.f64 c 2))) (*.f64 (*.f64 a a) (*.f64 a a))) (pow.f64 b 7)) (/.f64 405/64 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (Rewrite=> pow-sqr_binary64 (pow.f64 c (*.f64 2 2))) (*.f64 (*.f64 a a) (*.f64 a a))) (pow.f64 b 7)) (/.f64 405/64 a)): 49 points increase in error, 45 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 c (Rewrite=> metadata-eval 4)) (*.f64 (*.f64 a a) (*.f64 a a))) (pow.f64 b 7)) (/.f64 405/64 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (*.f64 a a) a) a))) (pow.f64 b 7)) (/.f64 405/64 a)): 33 points increase in error, 30 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (*.f64 (Rewrite<= unpow3_binary64 (pow.f64 a 3)) a)) (pow.f64 b 7)) (/.f64 405/64 a)): 23 points increase in error, 22 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (Rewrite=> pow-plus_binary64 (pow.f64 a (+.f64 3 1)))) (pow.f64 b 7)) (/.f64 405/64 a)): 24 points increase in error, 31 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (pow.f64 a (Rewrite=> metadata-eval 4))) (pow.f64 b 7)) (/.f64 405/64 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (pow.f64 b 7)) (/.f64 (Rewrite<= metadata-eval (+.f64 81/16 81/64)) a)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (+.f64 81/16 81/64)) (*.f64 (pow.f64 b 7) a))): 60 points increase in error, 66 points decrease in error
      (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (+.f64 81/16 81/64)) (Rewrite<= *-commutative_binary64 (*.f64 a (pow.f64 b 7)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 c 4) (*.f64 (pow.f64 a 4) (+.f64 81/16 81/64)))) (*.f64 a (pow.f64 b 7))): 44 points increase in error, 38 points decrease in error
      (/.f64 (*.f64 (pow.f64 c 4) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 81/16 (pow.f64 a 4)) (*.f64 81/64 (pow.f64 a 4))))) (*.f64 a (pow.f64 b 7))): 29 points increase in error, 23 points decrease in error
      (/.f64 (*.f64 (pow.f64 c 4) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 81/64 (pow.f64 a 4)) (*.f64 81/16 (pow.f64 a 4))))) (*.f64 a (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 5.3

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{\frac{c \cdot \left(c \cdot a\right)}{b \cdot b} \cdot \frac{-0.375}{b}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -600:\\ \;\;\;\;\left(b \cdot \frac{1}{a} - \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a}\right) \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{c \cdot \left(a \cdot c\right)}{b \cdot b} \cdot \frac{-0.375}{b}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.2
Cost	40964

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -600:\\ \;\;\;\;\left(b \cdot \frac{1}{a} - \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a}\right) \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(c \cdot \left(a \cdot c\right)\right) \cdot -0.375}{{b}^{3}}\right)\right)\\ \end{array} \]

Alternative 2
Error	9.4
Cost	21124

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.085:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 3
Error	15.5
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00016:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-0.5 \cdot \frac{c}{b}\right)\right)\\ \end{array} \]

Alternative 4
Error	15.5
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00016:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-0.5 \cdot \frac{c}{b}\right)\right)\\ \end{array} \]

Alternative 5
Error	15.5
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00016:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-0.5 \cdot \frac{c}{b}\right)\right)\\ \end{array} \]

Alternative 6
Error	15.6
Cost	20548

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00016:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-0.5 \cdot \frac{c}{b}\right)\right)\\ \end{array} \]

Alternative 7
Error	15.6
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00016:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8
Error	23.2
Cost	320

\[-0.5 \cdot \frac{c}{b} \]

Alternative 9
Error	62.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022326 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))