Cubic critical, medium range

Average Error: 43.8 → 2.9

Time: 17.8s

Precision: binary64

Cost: 59968

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

Math

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

↓

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

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
  -0.5625
  (* (* a a) (/ (pow c 3.0) (pow b 5.0)))
  (fma
   -0.5
   (/ c b)
   (fma
    -0.375
    (* (/ a (pow b 3.0)) (* c c))
    (*
     (/ -0.16666666666666666 b)
     (* (pow a 3.0) (* (/ (pow c 4.0) (pow b 6.0)) 6.328125)))))))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

double code(double a, double b, double c) {
	return fma(-0.5625, ((a * a) * (pow(c, 3.0) / pow(b, 5.0))), fma(-0.5, (c / b), fma(-0.375, ((a / pow(b, 3.0)) * (c * c)), ((-0.16666666666666666 / b) * (pow(a, 3.0) * ((pow(c, 4.0) / pow(b, 6.0)) * 6.328125))))));
}

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(Float64(a / (b ^ 3.0)) * Float64(c * c)), Float64(Float64(-0.16666666666666666 / b) * Float64((a ^ 3.0) * Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 6.328125))))))
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[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 / b), $MachinePrecision] * N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.8

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

2. Simplified 43.7

   \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a} \cdot -0.3333333333333333} \]
   Proof
   (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c -3))))) a) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 3))))))) a) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a c) (neg.f64 3)))))) a) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 3)))))) a) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 a c))))))) a) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 3 a) c)))))) a) -1/3): 0 points increase in error, 1 points decrease in error
   (*.f64 (/.f64 (-.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) a) -1/3): 7 points increase in error, 7 points decrease in error
   (*.f64 (/.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) 1)) a) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite<= metadata-eval (/.f64 -1 -1))) a) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) -1)) a) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 -1 a))) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite<= neg-mul-1_binary64 (neg.f64 a))) -1/3): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 a)) (Rewrite<= metadata-eval (/.f64 -1 3))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (*.f64 (neg.f64 a) 3))): 21 points increase in error, 23 points decrease in error
   (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 a 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 a)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (*.f64 3 a)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 3 a) -1))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (/.f64 -1 -1))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (Rewrite=> metadata-eval 1)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (Rewrite<= metadata-eval (neg.f64 -1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r/_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (/.f64 (*.f64 3 a) (neg.f64 -1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (/.f64 (*.f64 3 a) (Rewrite=> metadata-eval 1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite=> /-rgt-identity_binary64 (*.f64 3 a))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 43.2

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(b, \frac{1}{a}, -\frac{{\left(\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)\right)}^{0.25}}{a} \cdot \frac{{\left(\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)\right)}^{0.25}}{1}\right) + \mathsf{fma}\left(-\frac{{\left(\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)\right)}^{0.25}}{a}, \frac{{\left(\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)\right)}^{0.25}}{1}, \frac{{\left(\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)\right)}^{0.25}}{a} \cdot \frac{{\left(\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)\right)}^{0.25}}{1}\right)\right)} \cdot -0.3333333333333333 \]

4. Simplified 44.1

   \[\leadsto \color{blue}{\left(0 \cdot \frac{{\left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}^{0.5}}{a} + \left(\frac{b}{a} - \frac{{\left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}^{0.5}}{a}\right)\right)} \cdot -0.3333333333333333 \]
   Proof
   (+.f64 (*.f64 0 (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a)) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a)) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (+.f64 -1 1) (/.f64 (pow.f64 (fma.f64 c (Rewrite<= *-commutative_binary64 (*.f64 -3 a)) (*.f64 b b)) 1/2) a)) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (+.f64 -1 1) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) (Rewrite<= metadata-eval (*.f64 2 1/4))) a)) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (+.f64 -1 1) (/.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4))) a)) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (+.f64 -1 1) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4)))) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (+.f64 -1 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (Rewrite<= /-rgt-identity_binary64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)))) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 -1 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)))) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)))) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (neg.f64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (Rewrite=> /-rgt-identity_binary64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4)))) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4))) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (Rewrite<= /-rgt-identity_binary64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= fma-udef_binary64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)))) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2) a))): 125 points increase in error, 128 points decrease in error
   (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (Rewrite<= *-commutative_binary64 (*.f64 -3 a)) (*.f64 b b)) 1/2) a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (/.f64 b a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) (Rewrite<= metadata-eval (*.f64 2 1/4))) a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (/.f64 b a) (/.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4))) a))): 86 points increase in error, 21 points decrease in error
   (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (/.f64 b a) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4))))): 35 points increase in error, 38 points decrease in error
   (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (/.f64 b a) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (Rewrite<= /-rgt-identity_binary64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 b 1)) a) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 b (/.f64 1 a))) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)))): 24 points increase in error, 33 points decrease in error
   (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 b (/.f64 1 a)) (neg.f64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))) (Rewrite<= fma-udef_binary64 (fma.f64 b (/.f64 1 a) (neg.f64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)))))): 117 points increase in error, 135 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (fma.f64 b (/.f64 1 a) (neg.f64 (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1)))) (fma.f64 (neg.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a)) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1) (*.f64 (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 c (*.f64 -3 a) (*.f64 b b)) 1/4) 1))))): 0 points increase in error, 0 points decrease in error

5. Taylor expanded in c around 0 44.1

   \[\leadsto \left(0 \cdot \color{blue}{\frac{b}{a}} + \left(\frac{b}{a} - \frac{{\left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}^{0.5}}{a}\right)\right) \cdot -0.3333333333333333 \]

6. Taylor expanded in a around 0 2.9

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

7. Simplified 2.9

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right), \frac{-0.16666666666666666}{b} \cdot \left({a}^{3} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)\right)\right)\right)} \]
   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 (/.f64 a (pow.f64 b 3)) (*.f64 c c)) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 405/64)))))): 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 (/.f64 a (pow.f64 b 3)) (*.f64 c c)) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 405/64)))))): 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 (/.f64 a (pow.f64 b 3)) (*.f64 c c)) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 405/64)))))): 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 (/.f64 a (pow.f64 b 3)) (*.f64 c c)) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 405/64)))))): 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 a (pow.f64 b 3)) (Rewrite<= unpow2_binary64 (pow.f64 c 2))) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 405/64)))))): 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-/r/_binary64 (/.f64 a (/.f64 (pow.f64 b 3) (pow.f64 c 2)))) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 405/64)))))): 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 a (pow.f64 c 2)) (pow.f64 b 3))) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 405/64)))))): 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 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 c 2) a)) (pow.f64 b 3)) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 405/64)))))): 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 -1/6 b) (*.f64 (pow.f64 a 3) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) (Rewrite<= metadata-eval (+.f64 81/64 81/16)))))))): 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 -1/6 b) (*.f64 (pow.f64 a 3) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) (+.f64 (Rewrite<= metadata-eval (*.f64 -9/8 -9/8)) 81/16))))))): 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 -1/6 b) (*.f64 (pow.f64 a 3) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 -9/8 -9/8) (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 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)) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (+.f64 (*.f64 (*.f64 -9/8 -9/8) (/.f64 (pow.f64 c (Rewrite<= metadata-eval (*.f64 2 2))) (pow.f64 b 6))) (*.f64 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 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)) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (+.f64 (*.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 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 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)) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (+.f64 (*.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 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 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)) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (+.f64 (*.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 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 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)) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (+.f64 (*.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 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 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)) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (+.f64 (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 81/16 (/.f64 (pow.f64 c 4) (pow.f64 b 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)) (*.f64 (/.f64 -1/6 b) (*.f64 (pow.f64 a 3) (+.f64 (Rewrite<= unpow2_binary64 (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))))))))): 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-/r/_binary64 (/.f64 -1/6 (/.f64 b (*.f64 (pow.f64 a 3) (+.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))))))))))): 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 -1/6 (*.f64 (pow.f64 a 3) (+.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)))))) b))))): 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-*r/_binary64 (*.f64 -1/6 (/.f64 (*.f64 (pow.f64 a 3) (+.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))))) b)))))): 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 a 3) (+.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))))) b)))))): 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 a 3) (+.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))))) b)))))): 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 a 3) (+.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))))) b))))): 5 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 a 3) (+.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))))) b)) (+.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 a 3) (+.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))))) b)) (+.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

8. Final simplification 2.9

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

Alternatives

Alternative 1
Error	2.9
Cost	47680

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

Alternative 2
Error	3.9
Cost	33536

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

Alternative 3
Error	5.9
Cost	7424

\[-0.375 \cdot \left({\left(\frac{c}{b}\right)}^{2} \cdot \frac{a}{b}\right) + -0.5 \cdot \frac{c}{b} \]

Alternative 4
Error	12.0
Cost	320

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

Reproduce

herbie shell --seed 2022325 
(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)))