Cubic critical, medium range

Average Error: 43.6 → 2.9

Time: 17.1s

Precision: binary64

Cost: 53376

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

↓

\[\mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{-1.0546875}{a}\right)\right)\right) \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5625
  (* (* a a) (/ (pow c 3.0) (pow b 5.0)))
  (fma
   -0.5
   (/ c b)
   (fma
    -0.375
    (* a (/ (* c c) (pow b 3.0)))
    (* (/ (pow (* a c) 4.0) (pow b 7.0)) (/ -1.0546875 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) {
	return fma(-0.5625, ((a * a) * (pow(c, 3.0) / pow(b, 5.0))), fma(-0.5, (c / b), fma(-0.375, (a * ((c * c) / pow(b, 3.0))), ((pow((a * c), 4.0) / pow(b, 7.0)) * (-1.0546875 / a)))));
}

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)
	return fma(-0.5625, Float64(Float64(a * a) * Float64((c ^ 3.0) / (b ^ 5.0))), fma(-0.5, Float64(c / b), fma(-0.375, Float64(a * Float64(Float64(c * c) / (b ^ 3.0))), Float64(Float64((Float64(a * c) ^ 4.0) / (b ^ 7.0)) * Float64(-1.0546875 / a)))))
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_] := N[(-0.5625 * N[(N[(a * a), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0546875 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.6

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

2. Taylor expanded in b around inf 2.9

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

3. Simplified 2.9

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

4. Applied egg-rr 2.9

   \[\leadsto \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, \color{blue}{\frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{-1.0546875}{a}}\right)\right)\right) \]

5. Final simplification 2.9

   \[\leadsto \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{-1.0546875}{a}\right)\right)\right) \]

Alternatives

Alternative 1
Error	2.9
Cost	46976

\[\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(a \cdot a\right) \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1.0546875 \cdot {c}^{4}}{\frac{{b}^{7}}{a}}\right)\right)\right) \]

Alternative 2
Error	3.9
Cost	33536

\[\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, \frac{-0.5625}{{b}^{5}} \cdot \left(\left(a \cdot a\right) \cdot {c}^{3}\right)\right)\right) \]

Alternative 3
Error	3.9
Cost	27136

\[\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left(-0.375 \cdot \frac{c \cdot c}{{b}^{3}} + \frac{-0.5625 \cdot {c}^{3}}{\frac{{b}^{5}}{a}}\right)\right) \]

Alternative 4
Error	6.0
Cost	13696

\[\mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, -0.5 \cdot \frac{c}{b}\right) \]

Alternative 5
Error	6.0
Cost	7232

\[\frac{c}{b} \cdot \mathsf{fma}\left(\frac{c}{b}, -0.375 \cdot \frac{a}{b}, -0.5\right) \]

Alternative 6
Error	6.4
Cost	1728

\[\left(-1.125 \cdot \left(\frac{c \cdot c}{b \cdot b} \cdot \frac{a \cdot a}{b}\right) + -1.5 \cdot \frac{a \cdot c}{b}\right) \cdot \frac{0.3333333333333333}{a} \]

Alternative 7
Error	12.3
Cost	320

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

Alternative 8
Error	12.2
Cost	320

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

Reproduce

herbie shell --seed 2022295 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))