Cubic critical, narrow range

Average Error: 28.8 → 4.7

Time: 23.7s

Precision: binary64

Cost: 88068

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\\ t_1 := \frac{c \cdot 0.375}{{b}^{3}}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -4.6:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(3, {a}^{3} \cdot \mathsf{fma}\left(0.2222222222222222, \frac{b}{c} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{c}, \mathsf{fma}\left(-0.75, \frac{c}{b \cdot b} \cdot t_1, \frac{c \cdot \left(c \cdot -0.5625\right)}{{b}^{5}}\right)\right), \mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, t_1 \cdot \left(a \cdot a\right), \frac{1.5}{\frac{b}{a}}\right)\right)\right)\right)}^{-1}\\ \end{array} \]

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 (* c (* a -3.0)))) (t_1 (/ (* c 0.375) (pow b 3.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -4.6)
     (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (* 3.0 a))
     (pow
      (fma
       3.0
       (*
        (pow a 3.0)
        (fma
         0.2222222222222222
         (* (/ b c) (/ (* (/ (pow c 4.0) (pow b 6.0)) 6.328125) c))
         (fma
          -0.75
          (* (/ c (* b b)) t_1)
          (/ (* c (* c -0.5625)) (pow b 5.0)))))
       (fma -2.0 (/ b c) (fma 3.0 (* t_1 (* a a)) (/ 1.5 (/ b a)))))
      -1.0))))

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, (c * (a * -3.0)));
	double t_1 = (c * 0.375) / pow(b, 3.0);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -4.6) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = pow(fma(3.0, (pow(a, 3.0) * fma(0.2222222222222222, ((b / c) * (((pow(c, 4.0) / pow(b, 6.0)) * 6.328125) / c)), fma(-0.75, ((c / (b * b)) * t_1), ((c * (c * -0.5625)) / pow(b, 5.0))))), fma(-2.0, (b / c), fma(3.0, (t_1 * (a * a)), (1.5 / (b / a))))), -1.0);
	}
	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(c * Float64(a * -3.0)))
	t_1 = Float64(Float64(c * 0.375) / (b ^ 3.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -4.6)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = fma(3.0, Float64((a ^ 3.0) * fma(0.2222222222222222, Float64(Float64(b / c) * Float64(Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 6.328125) / c)), fma(-0.75, Float64(Float64(c / Float64(b * b)) * t_1), Float64(Float64(c * Float64(c * -0.5625)) / (b ^ 5.0))))), fma(-2.0, Float64(b / c), fma(3.0, Float64(t_1 * Float64(a * a)), Float64(1.5 / Float64(b / a))))) ^ -1.0;
	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[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * 0.375), $MachinePrecision] / N[Power[b, 3.0], $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], -4.6], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[Power[N[(3.0 * N[(N[Power[a, 3.0], $MachinePrecision] * N[(0.2222222222222222 * N[(N[(b / c), $MachinePrecision] * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] + N[(-0.75 * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(c * N[(c * -0.5625), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b / c), $MachinePrecision] + N[(3.0 * N[(t$95$1 * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(1.5 / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $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)) < -4.5999999999999996

   1. Initial program 10.1

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

   2. Applied egg-rr 10.2

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{{b}^{4} - \left(c \cdot \left(a \cdot -3\right)\right) \cdot \left(c \cdot \left(a \cdot -3\right)\right)}{b \cdot b - c \cdot \left(a \cdot -3\right)}}}}{3 \cdot a} \]

   3. Applied egg-rr 9.3

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

   if -4.5999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 31.0

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

   2. Applied egg-rr 31.1

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{{b}^{4} - \left(c \cdot \left(a \cdot -3\right)\right) \cdot \left(c \cdot \left(a \cdot -3\right)\right)}{b \cdot b - c \cdot \left(a \cdot -3\right)}}}}{3 \cdot a} \]

   3. Applied egg-rr 30.9

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

   4. Taylor expanded in a around 0 4.2

      \[\leadsto {\color{blue}{\left(3 \cdot \left({a}^{3} \cdot \left(-0.5625 \cdot \frac{{c}^{2}}{{b}^{5}} + \left(0.2222222222222222 \cdot \frac{b \cdot \left({\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + -0.75 \cdot \frac{c \cdot \left(0.75 \cdot \frac{c}{{b}^{3}} + -0.375 \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}}\right)\right)\right) + \left(-2 \cdot \frac{b}{c} + \left(3 \cdot \left({a}^{2} \cdot \left(-0.375 \cdot \frac{c}{{b}^{3}} + 0.75 \cdot \frac{c}{{b}^{3}}\right)\right) + 1.5 \cdot \frac{a}{b}\right)\right)\right)}}^{-1} \]

   5. Simplified 4.2

      \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(3, {a}^{3} \cdot \mathsf{fma}\left(0.2222222222222222, \frac{b}{c} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{c}, \mathsf{fma}\left(-0.75, \frac{c}{b \cdot b} \cdot \frac{c \cdot 0.375}{{b}^{3}}, \frac{\left(-0.5625 \cdot c\right) \cdot c}{{b}^{5}}\right)\right), \mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \left(a \cdot a\right) \cdot \frac{c \cdot 0.375}{{b}^{3}}, \frac{1.5}{\frac{b}{a}}\right)\right)\right)\right)}}^{-1} \]
      Proof
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (*.f64 (/.f64 b c) (/.f64 (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 405/64) c)) (fma.f64 -3/4 (*.f64 (/.f64 c (*.f64 b b)) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (*.f64 (/.f64 b c) (/.f64 (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) (Rewrite<= metadata-eval (+.f64 81/16 81/64))) c)) (fma.f64 -3/4 (*.f64 (/.f64 c (*.f64 b b)) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (*.f64 (/.f64 b c) (/.f64 (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) (+.f64 81/16 (Rewrite<= metadata-eval (*.f64 -9/8 -9/8)))) c)) (fma.f64 -3/4 (*.f64 (/.f64 c (*.f64 b b)) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (*.f64 (/.f64 b c) (/.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))))) c)) (fma.f64 -3/4 (*.f64 (/.f64 c (*.f64 b b)) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (*.f64 (/.f64 b c) (/.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)))) c)) (fma.f64 -3/4 (*.f64 (/.f64 c (*.f64 b b)) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (*.f64 (/.f64 b c) (/.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)))) c)) (fma.f64 -3/4 (*.f64 (/.f64 c (*.f64 b b)) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (*.f64 (/.f64 b c) (/.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)))))) c)) (fma.f64 -3/4 (*.f64 (/.f64 c (*.f64 b b)) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (*.f64 (/.f64 b c) (/.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)))))) c)) (fma.f64 -3/4 (*.f64 (/.f64 c (*.f64 b b)) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (*.f64 (/.f64 b c) (/.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)))))) c)) (fma.f64 -3/4 (*.f64 (/.f64 c (*.f64 b b)) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (*.f64 (/.f64 b c) (/.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)))))) c)) (fma.f64 -3/4 (*.f64 (/.f64 c (*.f64 b b)) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (*.f64 (/.f64 b c) (/.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))) c)) (fma.f64 -3/4 (*.f64 (/.f64 c (*.f64 b b)) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 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))) (*.f64 c c))) (fma.f64 -3/4 (*.f64 (/.f64 c (*.f64 b b)) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (Rewrite<= unpow2_binary64 (pow.f64 c 2))) (fma.f64 -3/4 (*.f64 (/.f64 c (*.f64 b b)) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)) (fma.f64 -3/4 (*.f64 (/.f64 c (Rewrite<= unpow2_binary64 (pow.f64 b 2))) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)) (fma.f64 -3/4 (*.f64 (/.f64 c (pow.f64 b 2)) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 c (pow.f64 b 3)) 3/8))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)) (fma.f64 -3/4 (*.f64 (/.f64 c (pow.f64 b 2)) (*.f64 (/.f64 c (pow.f64 b 3)) (Rewrite<= metadata-eval (+.f64 3/4 -3/8)))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)) (fma.f64 -3/4 (*.f64 (/.f64 c (pow.f64 b 2)) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3)))))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)) (fma.f64 -3/4 (Rewrite<= associate-/r/_binary64 (/.f64 c (/.f64 (pow.f64 b 2) (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)) (fma.f64 -3/4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2))) (/.f64 (*.f64 (*.f64 -9/16 c) c) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)) (fma.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2)) (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -9/16 (*.f64 c c))) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)) (fma.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2)) (/.f64 (*.f64 -9/16 (Rewrite<= unpow2_binary64 (pow.f64 c 2))) (pow.f64 b 5))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)) (fma.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2)) (Rewrite<= associate-*r/_binary64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2))) (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)) (+.f64 (*.f64 -3/4 (/.f64 (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) c) (*.f64 (*.f64 -3/8 (/.f64 c (pow.f64 b 3))) c))) (pow.f64 b 2))) (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5)))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)) (+.f64 (*.f64 -3/4 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (*.f64 -3/8 (/.f64 c (pow.f64 b 3))) c) (*.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) c))) (pow.f64 b 2))) (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5)))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)) (+.f64 (*.f64 -3/4 (/.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 c (+.f64 (*.f64 -3/8 (/.f64 c (pow.f64 b 3))) (*.f64 3/4 (/.f64 c (pow.f64 b 3)))))) (pow.f64 b 2))) (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5)))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (fma.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 -3/8 (/.f64 c (pow.f64 b 3))) (*.f64 3/4 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2))))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2))) (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 -3/8 (/.f64 c (pow.f64 b 3))) (*.f64 3/4 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2))))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 -3/8 (/.f64 c (pow.f64 b 3))) (*.f64 3/4 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2)))) (*.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2)))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (+.f64 (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (*.f64 -3/4 (/.f64 (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 (*.f64 -3/8 (/.f64 c (pow.f64 b 3))) c) (*.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) c))) (pow.f64 b 2)))) (*.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (+.f64 (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (*.f64 -3/4 (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) c) (*.f64 (*.f64 -3/8 (/.f64 c (pow.f64 b 3))) c))) (pow.f64 b 2)))) (*.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (+.f64 (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (*.f64 -3/4 (/.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3)))))) (pow.f64 b 2)))) (*.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (Rewrite=> associate-+l+_binary64 (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (+.f64 (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2))) (*.f64 2/9 (/.f64 (*.f64 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))) (pow.f64 c 2))))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (+.f64 (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2))) (*.f64 2/9 (/.f64 (*.f64 b (Rewrite=> +-commutative_binary64 (+.f64 (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2) (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6)))))) (pow.f64 c 2)))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2/9 (/.f64 (*.f64 b (+.f64 (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2) (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))))) (pow.f64 c 2))) (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2))))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (*.f64 a a) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (+.f64 (*.f64 2/9 (/.f64 (*.f64 b (+.f64 (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2) (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))))) (pow.f64 c 2))) (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2)))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (/.f64 (*.f64 c 3/8) (pow.f64 b 3))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (+.f64 (*.f64 2/9 (/.f64 (*.f64 b (+.f64 (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2) (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))))) (pow.f64 c 2))) (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2)))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (pow.f64 a 2) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 c (pow.f64 b 3)) 3/8))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (+.f64 (*.f64 2/9 (/.f64 (*.f64 b (+.f64 (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2) (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))))) (pow.f64 c 2))) (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2)))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (pow.f64 a 2) (*.f64 (/.f64 c (pow.f64 b 3)) (Rewrite<= metadata-eval (+.f64 -3/8 3/4)))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (+.f64 (*.f64 2/9 (/.f64 (*.f64 b (+.f64 (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2) (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))))) (pow.f64 c 2))) (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2)))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (pow.f64 a 2) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -3/8 (/.f64 c (pow.f64 b 3))) (*.f64 3/4 (/.f64 c (pow.f64 b 3)))))) (/.f64 3/2 (/.f64 b a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (+.f64 (*.f64 2/9 (/.f64 (*.f64 b (+.f64 (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2) (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))))) (pow.f64 c 2))) (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2)))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (pow.f64 a 2) (+.f64 (*.f64 -3/8 (/.f64 c (pow.f64 b 3))) (*.f64 3/4 (/.f64 c (pow.f64 b 3))))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 3/2 a) b))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (+.f64 (*.f64 2/9 (/.f64 (*.f64 b (+.f64 (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2) (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))))) (pow.f64 c 2))) (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2)))))) (fma.f64 -2 (/.f64 b c) (fma.f64 3 (*.f64 (pow.f64 a 2) (+.f64 (*.f64 -3/8 (/.f64 c (pow.f64 b 3))) (*.f64 3/4 (/.f64 c (pow.f64 b 3))))) (Rewrite<= associate-*r/_binary64 (*.f64 3/2 (/.f64 a b)))))): 2 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (+.f64 (*.f64 2/9 (/.f64 (*.f64 b (+.f64 (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2) (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))))) (pow.f64 c 2))) (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2)))))) (fma.f64 -2 (/.f64 b c) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3 (*.f64 (pow.f64 a 2) (+.f64 (*.f64 -3/8 (/.f64 c (pow.f64 b 3))) (*.f64 3/4 (/.f64 c (pow.f64 b 3)))))) (*.f64 3/2 (/.f64 a b)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 3 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (+.f64 (*.f64 2/9 (/.f64 (*.f64 b (+.f64 (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2) (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))))) (pow.f64 c 2))) (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2)))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -2 (/.f64 b c)) (+.f64 (*.f64 3 (*.f64 (pow.f64 a 2) (+.f64 (*.f64 -3/8 (/.f64 c (pow.f64 b 3))) (*.f64 3/4 (/.f64 c (pow.f64 b 3)))))) (*.f64 3/2 (/.f64 a b)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 -9/16 (/.f64 (pow.f64 c 2) (pow.f64 b 5))) (+.f64 (*.f64 2/9 (/.f64 (*.f64 b (+.f64 (pow.f64 (*.f64 -9/8 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2) (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))))) (pow.f64 c 2))) (*.f64 -3/4 (/.f64 (*.f64 c (+.f64 (*.f64 3/4 (/.f64 c (pow.f64 b 3))) (*.f64 -3/8 (/.f64 c (pow.f64 b 3))))) (pow.f64 b 2))))))) (+.f64 (*.f64 -2 (/.f64 b c)) (+.f64 (*.f64 3 (*.f64 (pow.f64 a 2) (+.f64 (*.f64 -3/8 (/.f64 c (pow.f64 b 3))) (*.f64 3/4 (/.f64 c (pow.f64 b 3)))))) (*.f64 3/2 (/.f64 a b)))))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -4.6:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(3, {a}^{3} \cdot \mathsf{fma}\left(0.2222222222222222, \frac{b}{c} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{c}, \mathsf{fma}\left(-0.75, \frac{c}{b \cdot b} \cdot \frac{c \cdot 0.375}{{b}^{3}}, \frac{c \cdot \left(c \cdot -0.5625\right)}{{b}^{5}}\right)\right), \mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{c \cdot 0.375}{{b}^{3}} \cdot \left(a \cdot a\right), \frac{1.5}{\frac{b}{a}}\right)\right)\right)\right)}^{-1}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.7
Cost	75268

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\\ t_1 := 0.375 \cdot \frac{a \cdot a}{{b}^{3}}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -4.6:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(3 \cdot \left(c \cdot \left(t_1 + c \cdot \mathsf{fma}\left(0.2222222222222222, b \cdot \frac{6.328125 \cdot \frac{{a}^{4}}{{b}^{6}}}{a}, \mathsf{fma}\left(-0.75, t_1 \cdot \frac{a}{b \cdot b}, -0.5625 \cdot \frac{{a}^{3}}{{b}^{5}}\right)\right)\right)\right) + \mathsf{fma}\left(-2, \frac{b}{c}, \frac{1.5}{\frac{b}{a}}\right)\right)}^{-1}\\ \end{array} \]

Alternative 2
Error	4.8
Cost	54532

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -4.6:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \frac{c \cdot \left(6.328125 \cdot {\left(a \cdot c\right)}^{3}\right)}{{b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left(\frac{c \cdot \left(c \cdot -0.375\right)}{{b}^{3}} + \frac{-0.5625 \cdot {c}^{3}}{\frac{{b}^{5}}{a}}\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	6.5
Cost	27076

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\\ \mathbf{if}\;b \leq 3.45:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(3, \frac{c \cdot \left(a \cdot a\right)}{\frac{{b}^{3}}{0.375}}, \frac{1.5}{\frac{b}{a}}\right)\right)\right)}^{-1}\\ \end{array} \]

Alternative 4
Error	9.1
Cost	20932

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\\ \mathbf{if}\;b \leq 3.45:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \]

Alternative 5
Error	9.2
Cost	13892

\[\begin{array}{l} \mathbf{if}\;b \leq 3.45:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right) \cdot \frac{\frac{1}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \]

Alternative 6
Error	9.2
Cost	13764

\[\begin{array}{l} \mathbf{if}\;b \leq 3.45:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \]

Alternative 7
Error	9.2
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 3.45:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \]

Alternative 8
Error	11.2
Cost	7168

\[{\left(\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}\right)}^{-1} \]

Alternative 9
Error	22.5
Cost	320

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

Reproduce

herbie shell --seed 2022318 
(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)))