Cubic critical, wide range

Average Error: 52.8 → 1.3

Time: 12.3s

Precision: binary64

Cost: 41280

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

Math

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

↓

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

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

C

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

↓

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

Derivation

1. Initial program 52.8

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

2. Simplified 52.8

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   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)): 6 points increase in error, 4 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)))): 11 points increase in error, 8 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. Applied egg-rr 52.8

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

4. Taylor expanded in a around 0 1.3

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

5. Simplified 1.3

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

6. Applied egg-rr 1.3

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

7. Applied egg-rr 1.3

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

8. Final simplification 1.3

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

Alternatives

Alternative 1
Error	1.8
Cost	20928

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

Alternative 2
Error	2.7
Cost	7424

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

Alternative 3
Error	6.1
Cost	320

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

Alternative 4
Error	5.9
Cost	320

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

Reproduce

herbie shell --seed 2022285 
(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)))