Cubic critical, wide range

Average Error: 52.5 → 1.4

Time: 12.3s

Precision: binary64

Cost: 53888

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

↓

\[\mathsf{fma}\left(-0.16666666666666666, \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 6.328125\right) \cdot \frac{{a}^{3}}{b}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, \frac{-0.5625}{{b}^{5}} \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)\right)\right)\right) \]

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
  -0.16666666666666666
  (* (* (* (* c c) (* (* c c) (pow b -6.0))) 6.328125) (/ (pow a 3.0) b))
  (fma
   -0.5
   (/ c b)
   (fma
    -0.375
    (* a (/ c (/ (pow b 3.0) c)))
    (* (/ -0.5625 (pow b 5.0)) (* (pow c 3.0) (* a 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(-0.16666666666666666, ((((c * c) * ((c * c) * pow(b, -6.0))) * 6.328125) * (pow(a, 3.0) / b)), fma(-0.5, (c / b), fma(-0.375, (a * (c / (pow(b, 3.0) / c))), ((-0.5625 / pow(b, 5.0)) * (pow(c, 3.0) * (a * 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(-0.16666666666666666, Float64(Float64(Float64(Float64(c * c) * Float64(Float64(c * c) * (b ^ -6.0))) * 6.328125) * Float64((a ^ 3.0) / b)), fma(-0.5, Float64(c / b), fma(-0.375, Float64(a * Float64(c / Float64((b ^ 3.0) / c))), Float64(Float64(-0.5625 / (b ^ 5.0)) * Float64((c ^ 3.0) * Float64(a * 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[(-0.16666666666666666 * N[(N[(N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision] * N[(N[Power[a, 3.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5625 / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5

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

2. Simplified 52.5

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   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)): 4 points increase in error, 3 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)))): 13 points increase in error, 7 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)))): 5 points increase in error, 6 points decrease in error
   (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 -1)) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 3 a) -1))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (/.f64 (neg.f64 -1) -1))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (/.f64 (Rewrite=> metadata-eval 1) -1)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (Rewrite=> metadata-eval -1)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (*.f64 3 a))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in a around 0 1.4

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

4. Simplified 1.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot \frac{{a}^{3}}{b}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, \frac{-0.5625}{{b}^{5}} \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)\right)\right)\right)} \]
   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) (fma.f64 -3/8 (*.f64 a (/.f64 c (/.f64 (pow.f64 b 3) c))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.f64 a 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) (fma.f64 -3/8 (*.f64 a (/.f64 c (/.f64 (pow.f64 b 3) c))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.f64 a 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) (fma.f64 -3/8 (*.f64 a (/.f64 c (/.f64 (pow.f64 b 3) c))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.f64 a 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) (fma.f64 -3/8 (*.f64 a (/.f64 c (/.f64 (pow.f64 b 3) c))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.f64 a 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) (fma.f64 -3/8 (*.f64 a (/.f64 c (/.f64 (pow.f64 b 3) c))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.f64 a 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) (fma.f64 -3/8 (*.f64 a (/.f64 c (/.f64 (pow.f64 b 3) c))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.f64 a 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) (fma.f64 -3/8 (*.f64 a (/.f64 c (/.f64 (pow.f64 b 3) c))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.f64 a 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) (fma.f64 -3/8 (*.f64 a (/.f64 c (/.f64 (pow.f64 b 3) c))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.f64 a 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) (fma.f64 -3/8 (*.f64 a (/.f64 c (/.f64 (pow.f64 b 3) c))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.f64 a 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) (fma.f64 -3/8 (*.f64 a (/.f64 c (/.f64 (pow.f64 b 3) c))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.f64 a 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) (fma.f64 -3/8 (*.f64 a (/.f64 c (/.f64 (pow.f64 b 3) c))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.f64 a 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) (fma.f64 -3/8 (*.f64 a (/.f64 c (/.f64 (pow.f64 b 3) c))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.f64 a 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) (fma.f64 -3/8 (*.f64 a (/.f64 c (/.f64 (pow.f64 b 3) c))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.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) (fma.f64 -3/8 (*.f64 a (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c c) (pow.f64 b 3)))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.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) (fma.f64 -3/8 (*.f64 a (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (pow.f64 b 3))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.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) (fma.f64 -3/8 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (pow.f64 c 2) (pow.f64 b 3)) a)) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.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) (fma.f64 -3/8 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (*.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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (*.f64 (/.f64 -9/16 (pow.f64 b 5)) (*.f64 (pow.f64 c 3) (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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (Rewrite<= associate-/r/_binary64 (/.f64 -9/16 (/.f64 (pow.f64 b 5) (*.f64 (pow.f64 c 3) (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) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -9/16 (*.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) (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (Rewrite<= associate-*r/_binary64 (*.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) (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 -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<= 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)))))): 5 points increase in error, 5 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

5. Applied egg-rr 1.4

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \left(\color{blue}{\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right)} \cdot 6.328125\right) \cdot \frac{{a}^{3}}{b}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, \frac{-0.5625}{{b}^{5}} \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)\right)\right)\right) \]

6. Final simplification 1.4

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 6.328125\right) \cdot \frac{{a}^{3}}{b}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, \frac{-0.5625}{{b}^{5}} \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)\right)\right)\right) \]

Alternatives

Alternative 1
Error	1.4
Cost	53760

\[\mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(6.328125 \cdot \frac{{c}^{4}}{{b}^{6}}\right), \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \mathsf{fma}\left(-0.375, a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), \frac{c}{\frac{b}{-0.5}}\right)\right)\right) \]

Alternative 2
Error	1.9
Cost	33536

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

Alternative 3
Error	1.9
Cost	27328

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

Alternative 4
Error	5.7
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5
Error	5.7
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6
Error	2.9
Cost	13696

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

Alternative 7
Error	6.2
Cost	320

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

Reproduce

herbie shell --seed 2022308 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))