Cubic critical, narrow range

Average Error: 28.9 → 5.0

Time: 19.8s

Precision: binary64

Cost: 60932

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -0.8:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}, \mathsf{fma}\left(-0.5625, a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}, \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{-0.16666666666666666}{\frac{a}{6.328125}}\right)\right)\right)\\ \end{array} \]

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) (* c (* a -3.0)))) b) (* 3.0 a)) -0.8)
   (* (- (sqrt (fma b b (* -3.0 (* a c)))) b) (/ 0.3333333333333333 a))
   (fma
    -0.5
    (/ c b)
    (fma
     -0.375
     (* (* a c) (/ c (pow b 3.0)))
     (fma
      -0.5625
      (* a (/ (pow c 3.0) (/ (pow b 5.0) a)))
      (*
       (/ (pow (* a c) 4.0) (pow b 7.0))
       (/ -0.16666666666666666 (/ a 6.328125))))))))

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) + (c * (a * -3.0)))) - b) / (3.0 * a)) <= -0.8) {
		tmp = (sqrt(fma(b, b, (-3.0 * (a * c)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = fma(-0.5, (c / b), fma(-0.375, ((a * c) * (c / pow(b, 3.0))), fma(-0.5625, (a * (pow(c, 3.0) / (pow(b, 5.0) / a))), ((pow((a * c), 4.0) / pow(b, 7.0)) * (-0.16666666666666666 / (a / 6.328125))))));
	}
	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(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a)) <= -0.8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-3.0 * Float64(a * c)))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = fma(-0.5, Float64(c / b), fma(-0.375, Float64(Float64(a * c) * Float64(c / (b ^ 3.0))), fma(-0.5625, Float64(a * Float64((c ^ 3.0) / Float64((b ^ 5.0) / a))), Float64(Float64((Float64(a * c) ^ 4.0) / (b ^ 7.0)) * Float64(-0.16666666666666666 / Float64(a / 6.328125))))));
	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[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[Sqrt[N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(a * c), $MachinePrecision] * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(a * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 / N[(a / 6.328125), $MachinePrecision]), $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)) < -0.80000000000000004

   1. Initial program 10.7

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

   2. Simplified 10.6

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
      Proof
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -3)))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a (Rewrite<= metadata-eval (neg.f64 3)))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 3)))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 a)))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 c (*.f64 3 a)))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 3 a) c))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) b) (/.f64 1/3 a)): 22 points increase in error, 10 points decrease in error
      (*.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))) (neg.f64 b))) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (+.f64 (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)))) (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 (neg.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 (/.f64 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 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 (/.f64 (Rewrite=> associate-+l-_binary64 (-.f64 0 (-.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 (/.f64 (Rewrite=> sub0-neg_binary64 (neg.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 (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.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 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1)) -1) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite=> associate-/l*_binary64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (/.f64 -1 -1))) -1) (/.f64 1/3 a)): 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 1)) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite=> /-rgt-identity_binary64 (-.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)))) -1) (/.f64 (Rewrite<= metadata-eval (/.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)))) -1) (/.f64 (/.f64 (Rewrite<= metadata-eval (neg.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)))) -1) (Rewrite<= associate-/r*_binary64 (/.f64 (neg.f64 -1) (*.f64 3 a)))): 42 points increase in error, 34 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 -1)) (*.f64 -1 (*.f64 3 a)))): 17 points increase in error, 22 points decrease in error
      (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 -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 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (/.f64 (neg.f64 -1) -1))): 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)))) (*.f64 3 a)) (/.f64 (Rewrite=> metadata-eval 1) -1)): 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)))) (*.f64 3 a)) (Rewrite=> metadata-eval -1)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (*.f64 3 a))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in c around 0 10.6

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 31.9

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

   2. Simplified 31.8

      \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a} \cdot -0.3333333333333333} \]
      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): 0 points increase in error, 3 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): 22 points increase in error, 9 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))): 53 points increase in error, 43 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. Taylor expanded in b around inf 4.4

      \[\leadsto \color{blue}{\left(1.6875 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)\right)} \cdot -0.3333333333333333 \]

   4. Taylor expanded in c around 0 4.2

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

   5. Simplified 4.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{{b}^{3}} \cdot \left(c \cdot a\right), \mathsf{fma}\left(-0.5625, a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}, \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{-0.16666666666666666}{\frac{a}{6.328125}}\right)\right)\right)} \]
      Proof
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (*.f64 (/.f64 c (pow.f64 b 3)) (*.f64 c a)) (fma.f64 -9/16 (*.f64 a (/.f64 (pow.f64 c 3) (/.f64 (pow.f64 b 5) a))) (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (Rewrite<= associate-/r/_binary64 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a)))) (fma.f64 -9/16 (*.f64 a (/.f64 (pow.f64 c 3) (/.f64 (pow.f64 b 5) a))) (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 1 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c (*.f64 c a)) (pow.f64 b 3))) (fma.f64 -9/16 (*.f64 a (/.f64 (pow.f64 c 3) (/.f64 (pow.f64 b 5) a))) (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 1 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 c c) a)) (pow.f64 b 3)) (fma.f64 -9/16 (*.f64 a (/.f64 (pow.f64 c 3) (/.f64 (pow.f64 b 5) a))) (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) a) (pow.f64 b 3)) (fma.f64 -9/16 (*.f64 a (/.f64 (pow.f64 c 3) (/.f64 (pow.f64 b 5) a))) (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (pow.f64 c 3) (/.f64 (pow.f64 b 5) a)) a)) (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 c 3) (/.f64 (/.f64 (pow.f64 b 5) a) a))) (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (pow.f64 c 3) (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 b 5) (*.f64 a a)))) (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 c 3) (*.f64 a a)) (pow.f64 b 5))) (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (Rewrite<= unpow2_binary64 (pow.f64 a 2))) (pow.f64 b 5)) (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (pow.f64 (*.f64 c a) (Rewrite<= metadata-eval (*.f64 2 2))) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (*.f64 c a) 2) (pow.f64 (*.f64 c a) 2))) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (Rewrite=> unpow2_binary64 (*.f64 (*.f64 c a) (*.f64 c a))) (pow.f64 (*.f64 c a) 2)) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (Rewrite=> swap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 a a))) (pow.f64 (*.f64 c a) 2)) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (*.f64 (*.f64 c c) (*.f64 a a)) (Rewrite=> unpow2_binary64 (*.f64 (*.f64 c a) (*.f64 c a)))) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (*.f64 (*.f64 c c) (*.f64 a a)) (Rewrite=> swap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 a a)))) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.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 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (*.f64 c c) c) c)) (*.f64 (*.f64 a a) (*.f64 a a))) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (*.f64 (Rewrite<= unpow3_binary64 (pow.f64 c 3)) c) (*.f64 (*.f64 a a) (*.f64 a a))) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (Rewrite=> pow-plus_binary64 (pow.f64 c (+.f64 3 1))) (*.f64 (*.f64 a a) (*.f64 a a))) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 c (Rewrite=> metadata-eval 4)) (*.f64 (*.f64 a a) (*.f64 a a))) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (*.f64 a a) a) a))) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (*.f64 (Rewrite<= unpow3_binary64 (pow.f64 a 3)) a)) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (Rewrite=> pow-plus_binary64 (pow.f64 a (+.f64 3 1)))) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (pow.f64 a (Rewrite=> metadata-eval 4))) (pow.f64 b 7)) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (Rewrite=> associate-/l*_binary64 (/.f64 (pow.f64 c 4) (/.f64 (pow.f64 b 7) (pow.f64 a 4)))) (/.f64 -1/6 (/.f64 a 405/64)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (*.f64 (/.f64 (pow.f64 c 4) (/.f64 (pow.f64 b 7) (pow.f64 a 4))) (/.f64 -1/6 (/.f64 a (Rewrite<= metadata-eval (+.f64 81/16 81/64)))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 c 4) -1/6) (*.f64 (/.f64 (pow.f64 b 7) (pow.f64 a 4)) (/.f64 a (+.f64 81/16 81/64)))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/6 (pow.f64 c 4))) (*.f64 (/.f64 (pow.f64 b 7) (pow.f64 a 4)) (/.f64 a (+.f64 81/16 81/64))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (/.f64 (*.f64 -1/6 (pow.f64 c 4)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 b 7) a) (*.f64 (pow.f64 a 4) (+.f64 81/16 81/64)))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (/.f64 (*.f64 -1/6 (pow.f64 c 4)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 a (pow.f64 b 7))) (*.f64 (pow.f64 a 4) (+.f64 81/16 81/64))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (/.f64 (*.f64 -1/6 (pow.f64 c 4)) (/.f64 (*.f64 a (pow.f64 b 7)) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 81/16 (pow.f64 a 4)) (*.f64 81/64 (pow.f64 a 4))))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (/.f64 (*.f64 -1/6 (pow.f64 c 4)) (/.f64 (*.f64 a (pow.f64 b 7)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 81/64 (pow.f64 a 4)) (*.f64 81/16 (pow.f64 a 4))))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 -1/6 (pow.f64 c 4)) (+.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
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1/6 (*.f64 (pow.f64 c 4) (+.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
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (/.f64 (*.f64 -1/6 (*.f64 (pow.f64 c 4) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 81/16 (pow.f64 a 4)) (*.f64 81/64 (pow.f64 a 4)))))) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (/.f64 (*.f64 -1/6 (*.f64 (pow.f64 c 4) (Rewrite=> distribute-rgt-out_binary64 (*.f64 (pow.f64 a 4) (+.f64 81/16 81/64))))) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (/.f64 (*.f64 -1/6 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (+.f64 81/16 81/64)))) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (/.f64 (*.f64 -1/6 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))))) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (Rewrite<= associate-*r/_binary64 (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))))))): 1 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 c b) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))))) (*.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 -1/2 (/.f64 c b) (Rewrite<= associate-+r+_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 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))))): 1 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/2 (/.f64 c b)) (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (*.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<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)))) (+.f64 (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)))))): 20 points increase in error, 23 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (*.f64 -1/2 (/.f64 c b)))) (+.f64 (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (*.f64 -1/2 (/.f64 c b))) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-+r+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (*.f64 -1/2 (/.f64 c b))) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)))) (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))))): 12 points increase in error, 12 points decrease in error
      (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)))))) (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))))): 1 points increase in error, 4 points decrease in error
      (+.f64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1/6 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4))))) (*.f64 a (pow.f64 b 7))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))) (/.f64 (*.f64 -1/6 (Rewrite=> distribute-rgt-out_binary64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (+.f64 81/16 81/64)))) (*.f64 a (pow.f64 b 7)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))) (/.f64 (*.f64 -1/6 (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)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))) (/.f64 (*.f64 -1/6 (*.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)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))) (/.f64 (*.f64 -1/6 (*.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
      (+.f64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))) (Rewrite<= associate-*r/_binary64 (*.f64 -1/6 (/.f64 (*.f64 (pow.f64 c 4) (+.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
      (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)))) (*.f64 -1/6 (/.f64 (*.f64 (pow.f64 c 4) (+.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
      (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1/6 (/.f64 (*.f64 (pow.f64 c 4) (+.f64 (*.f64 81/64 (pow.f64 a 4)) (*.f64 81/16 (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
3. Recombined 2 regimes into one program.

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -0.8:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}, \mathsf{fma}\left(-0.5625, a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}, \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{-0.16666666666666666}{\frac{a}{6.328125}}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	5.1
Cost	60932

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -0.8:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{a \cdot c}}, \frac{-0.16666666666666666}{{b}^{7}} \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{a}{6.328125}}\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	5.3
Cost	54916

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -0.8:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(1.6875 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \left(\frac{c}{b} \cdot 1.5 + 0.5 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)\right)\right) \cdot -0.3333333333333333\\ \end{array} \]

Alternative 3
Error	6.4
Cost	40964

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -0.8:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(c \cdot \left(a \cdot c\right)\right)}{{b}^{3}}\right)\right)\\ \end{array} \]

Alternative 4
Error	9.1
Cost	21124

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -0.095:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 5
Error	9.2
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -0.095:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \mathsf{fma}\left(1.125, \frac{c \cdot c}{\left(b \cdot b\right) \cdot \frac{b}{a}}, \frac{c}{b} \cdot 1.5\right)\\ \end{array} \]

Alternative 6
Error	9.3
Cost	15108

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -0.095:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot c\right) \cdot \left(\frac{1.5}{b} + 1.125 \cdot \left(a \cdot \left(c \cdot {b}^{-3}\right)\right)\right)\right) \cdot \frac{-0.3333333333333333}{a}\\ \end{array} \]

Alternative 7
Error	9.2
Cost	15108

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -0.095:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(1.125 \cdot \left(c \cdot \left(c \cdot \frac{a}{{b}^{3}}\right)\right)\right) + \left(\frac{c}{b} \cdot 1.5\right) \cdot -0.3333333333333333\\ \end{array} \]

Alternative 8
Error	9.2
Cost	15044

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -0.095:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \mathsf{fma}\left(1.125, \frac{c \cdot c}{\left(b \cdot b\right) \cdot \frac{b}{a}}, \frac{c}{b} \cdot 1.5\right)\\ \end{array} \]

Alternative 9
Error	15.8
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -9 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10
Error	22.5
Cost	320

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

Reproduce

herbie shell --seed 2022325 
(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)))