Cubic critical, narrow range

Average Error: 28.5 → 5.2

Time: 26.7s

Precision: binary64

Cost: 60996

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + \sqrt{t_0}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot \frac{{a}^{3}}{b}, \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left(\frac{c \cdot \left(c \cdot -0.375\right)}{{b}^{3}} + \frac{{c}^{3} \cdot -0.5625}{\frac{{b}^{5}}{a}}\right)\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* -3.0 (* a c)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -14.0)
     (* (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (/ 0.3333333333333333 a))
     (fma
      -0.16666666666666666
      (* (* (/ (pow c 4.0) (pow b 6.0)) 6.328125) (/ (pow a 3.0) b))
      (fma
       -0.5
       (/ c b)
       (*
        a
        (+
         (/ (* c (* c -0.375)) (pow b 3.0))
         (/ (* (pow c 3.0) -0.5625) (/ (pow b 5.0) a)))))))))

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 t_0 = fma(b, b, (-3.0 * (a * c)));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -14.0) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) * (0.3333333333333333 / a);
	} else {
		tmp = fma(-0.16666666666666666, (((pow(c, 4.0) / pow(b, 6.0)) * 6.328125) * (pow(a, 3.0) / b)), fma(-0.5, (c / b), (a * (((c * (c * -0.375)) / pow(b, 3.0)) + ((pow(c, 3.0) * -0.5625) / (pow(b, 5.0) / a))))));
	}
	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)
	t_0 = fma(b, b, Float64(-3.0 * Float64(a * c)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -14.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) * Float64(0.3333333333333333 / a));
	else
		tmp = fma(-0.16666666666666666, Float64(Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 6.328125) * Float64((a ^ 3.0) / b)), fma(-0.5, Float64(c / b), Float64(a * Float64(Float64(Float64(c * Float64(c * -0.375)) / (b ^ 3.0)) + Float64(Float64((c ^ 3.0) * -0.5625) / Float64((b ^ 5.0) / a))))));
	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_] := Block[{t$95$0 = N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, 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], -14.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.16666666666666666 * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision] * N[(N[Power[a, 3.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(a * N[(N[(N[(c * N[(c * -0.375), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[c, 3.0], $MachinePrecision] * -0.5625), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / a), $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)) < -14

   1. Initial program 9.3

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

   2. Simplified 9.3

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
      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)): 20 points increase in error, 11 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)))): 50 points increase in error, 37 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)))): 32 points increase in error, 35 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. Applied egg-rr 10.3

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

   4. Applied egg-rr 8.7

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

   if -14 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 30.4

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

   2. Simplified 30.3

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
      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)): 20 points increase in error, 11 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)))): 50 points increase in error, 37 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)))): 32 points increase in error, 35 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. Applied egg-rr 30.3

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

   4. Applied egg-rr 30.3

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

   5. Taylor expanded in a around 0 4.9

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

   6. Simplified 4.8

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

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot \frac{{a}^{3}}{b}, \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left(\frac{c \cdot \left(c \cdot -0.375\right)}{{b}^{3}} + \frac{{c}^{3} \cdot -0.5625}{\frac{{b}^{5}}{a}}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	5.2
Cost	60804

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + \sqrt{t_0}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, {c}^{3} \cdot \frac{a \cdot a}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, \frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} \cdot -1.0546875\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	6.6
Cost	34564

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.3:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left(\frac{c \cdot \left(c \cdot -0.375\right)}{{b}^{3}} + \frac{{c}^{3} \cdot -0.5625}{\frac{{b}^{5}}{a}}\right)\right)\\ \end{array} \]

Alternative 3
Error	9.1
Cost	28228

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0035:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + \sqrt{t_0}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 4
Error	9.1
Cost	28228

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0035:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 5
Error	9.3
Cost	27524

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0045:\\ \;\;\;\;{\left(\frac{a}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}{-3}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 6
Error	9.3
Cost	21188

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0045:\\ \;\;\;\;\frac{1}{\frac{a}{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 7
Error	9.3
Cost	21188

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0045:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}{-3} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 8
Error	9.3
Cost	21124

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0045:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, c \cdot \frac{-0.5}{b}\right)\\ \end{array} \]

Alternative 9
Error	9.3
Cost	21124

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0045:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 10
Error	15.1
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11
Error	15.1
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12
Error	15.1
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 13
Error	16.6
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 310:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 14
Error	22.9
Cost	320

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

Alternative 15
Error	22.9
Cost	320

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

Reproduce

herbie shell --seed 2022293 
(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)))