Cubic critical, medium range

Average Error: 43.7 → 3.0

Time: 15.2s

Precision: binary64

Cost: 53504

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

Math

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

↓

\[\mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c}{\frac{{b}^{3}}{a \cdot c}}\right)\right)\right) \]

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
   -0.16666666666666666
   (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0)))
   (fma -0.5 (/ c b) (* -0.375 (/ c (/ (pow b 3.0) (* a c))))))))

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(-0.16666666666666666, ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))), fma(-0.5, (c / b), (-0.375 * (c / (pow(b, 3.0) / (a * c)))))));
}

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(-0.16666666666666666, Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0))), fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(c / Float64((b ^ 3.0) / Float64(a * c)))))))
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[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.7

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

2. Simplified 43.6

   \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a} \cdot -0.3333333333333333} \]
   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, 2 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): 12 points increase in error, 9 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))): 27 points increase in error, 30 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. Taylor expanded in b around inf 3.2

   \[\leadsto \color{blue}{\left(1.6875 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)\right)} \cdot -0.3333333333333333 \]

4. Simplified 3.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(1.125, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{0.5 \cdot \left({\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)\right)}{a \cdot {b}^{7}}\right)\right)\right)} \cdot -0.3333333333333333 \]
   Proof
   (fma.f64 27/16 (/.f64 (pow.f64 c 3) (/.f64 (pow.f64 b 5) (*.f64 a a))) (fma.f64 9/8 (/.f64 (*.f64 c c) (/.f64 (pow.f64 b 3) a)) (fma.f64 3/2 (/.f64 c b) (/.f64 (*.f64 1/2 (+.f64 (pow.f64 (*.f64 (*.f64 (*.f64 c c) (*.f64 a a)) -9/8) 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 27/16 (/.f64 (pow.f64 c 3) (/.f64 (pow.f64 b 5) (Rewrite<= unpow2_binary64 (pow.f64 a 2)))) (fma.f64 9/8 (/.f64 (*.f64 c c) (/.f64 (pow.f64 b 3) a)) (fma.f64 3/2 (/.f64 c b) (/.f64 (*.f64 1/2 (+.f64 (pow.f64 (*.f64 (*.f64 (*.f64 c c) (*.f64 a a)) -9/8) 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 27/16 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (fma.f64 9/8 (/.f64 (*.f64 c c) (/.f64 (pow.f64 b 3) a)) (fma.f64 3/2 (/.f64 c b) (/.f64 (*.f64 1/2 (+.f64 (pow.f64 (*.f64 (*.f64 (*.f64 c c) (*.f64 a a)) -9/8) 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 27/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 9/8 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (/.f64 (pow.f64 b 3) a)) (fma.f64 3/2 (/.f64 c b) (/.f64 (*.f64 1/2 (+.f64 (pow.f64 (*.f64 (*.f64 (*.f64 c c) (*.f64 a a)) -9/8) 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 27/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 9/8 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (fma.f64 3/2 (/.f64 c b) (/.f64 (*.f64 1/2 (+.f64 (pow.f64 (*.f64 (*.f64 (*.f64 c c) (*.f64 a a)) -9/8) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))))) (*.f64 a (pow.f64 b 7)))))): 0 points increase in error, 1 points decrease in error
   (fma.f64 27/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 9/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 3/2 (/.f64 c b) (/.f64 (*.f64 1/2 (+.f64 (pow.f64 (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (*.f64 a a)) -9/8) 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 27/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 9/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 3/2 (/.f64 c b) (/.f64 (*.f64 1/2 (+.f64 (pow.f64 (*.f64 (*.f64 (pow.f64 c 2) (Rewrite<= unpow2_binary64 (pow.f64 a 2))) -9/8) 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 27/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 9/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 3/2 (/.f64 c b) (/.f64 (*.f64 1/2 (+.f64 (pow.f64 (Rewrite<= *-commutative_binary64 (*.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 27/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 9/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 3/2 (/.f64 c b) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.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 27/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 9/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3/2 (/.f64 c b)) (*.f64 1/2 (/.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)))))))): 6 points increase in error, 54 points decrease in error
   (fma.f64 27/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 9/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (+.f64 (*.f64 3/2 (/.f64 c b)) (*.f64 1/2 (/.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 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 27/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (*.f64 9/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (+.f64 (*.f64 3/2 (/.f64 c b)) (*.f64 1/2 (/.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

5. Taylor expanded in c around 0 3.0

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

6. Simplified 3.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c}{\frac{{b}^{3}}{c \cdot a}}\right)\right)\right)} \]
   Proof
   (fma.f64 -9/16 (*.f64 (*.f64 a a) (/.f64 (pow.f64 c 3) (pow.f64 b 5))) (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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)) (*.f64 a a))) (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -9/16 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (pow.f64 c 3) (*.f64 a a)) (pow.f64 b 5))) (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (Rewrite<= unpow2_binary64 (pow.f64 a 2))) (pow.f64 b 5)) (fma.f64 -1/6 (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (pow.f64 (*.f64 c a) (Rewrite<= metadata-eval (*.f64 2 2))) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (*.f64 c a) 2) (pow.f64 (*.f64 c a) 2))) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (*.f64 (Rewrite=> unpow2_binary64 (*.f64 (*.f64 c a) (*.f64 c a))) (pow.f64 (*.f64 c a) 2)) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (*.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 a a))) (pow.f64 (*.f64 c a) 2)) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (*.f64 (*.f64 (*.f64 c c) (*.f64 a a)) (Rewrite=> unpow2_binary64 (*.f64 (*.f64 c a) (*.f64 c a)))) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (*.f64 (*.f64 (*.f64 c c) (*.f64 a a)) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 a a)))) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 (*.f64 c c) (*.f64 c c)) (*.f64 (*.f64 a a) (*.f64 a a)))) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (*.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (*.f64 c c) c) c)) (*.f64 (*.f64 a a) (*.f64 a a))) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (*.f64 (*.f64 (Rewrite<= unpow3_binary64 (pow.f64 c 3)) c) (*.f64 (*.f64 a a) (*.f64 a a))) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (*.f64 (Rewrite=> pow-plus_binary64 (pow.f64 c (+.f64 3 1))) (*.f64 (*.f64 a a) (*.f64 a a))) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (*.f64 (pow.f64 c (Rewrite=> metadata-eval 4)) (*.f64 (*.f64 a a) (*.f64 a a))) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 a a))) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (*.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 a 2)))) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (Rewrite=> pow-sqr_binary64 (pow.f64 a (*.f64 2 2)))) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (pow.f64 a (Rewrite=> metadata-eval 4))) a) (/.f64 405/64 (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) a) (/.f64 (Rewrite<= metadata-eval (+.f64 81/16 81/64)) (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (+.f64 81/16 81/64)) (*.f64 a (pow.f64 b 7)))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 0 points increase in error, 1 points decrease in error
   (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 -1/6 (/.f64 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 c 4) (*.f64 (pow.f64 a 4) (+.f64 81/16 81/64)))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 1 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 -1/6 (/.f64 (*.f64 (pow.f64 c 4) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 81/16 (pow.f64 a 4)) (*.f64 81/64 (pow.f64 a 4))))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 0 points increase in error, 1 points decrease in error
   (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (fma.f64 -1/6 (/.f64 (*.f64 (pow.f64 c 4) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 81/64 (pow.f64 a 4)) (*.f64 81/16 (pow.f64 a 4))))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 c (/.f64 (pow.f64 b 3) (*.f64 c a))))))): 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 -1/6 (/.f64 (*.f64 (pow.f64 c 4) (+.f64 (*.f64 81/64 (pow.f64 a 4)) (*.f64 81/16 (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c (*.f64 c a)) (pow.f64 b 3))))))): 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 -1/6 (/.f64 (*.f64 (pow.f64 c 4) (+.f64 (*.f64 81/64 (pow.f64 a 4)) (*.f64 81/16 (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 c c) a)) (pow.f64 b 3)))))): 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 -1/6 (/.f64 (*.f64 (pow.f64 c 4) (+.f64 (*.f64 81/64 (pow.f64 a 4)) (*.f64 81/16 (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))) (fma.f64 -1/2 (/.f64 c b) (*.f64 -3/8 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) a) (pow.f64 b 3)))))): 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 -1/6 (/.f64 (*.f64 (pow.f64 c 4) (+.f64 (*.f64 81/64 (pow.f64 a 4)) (*.f64 81/16 (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))) (Rewrite<= fma-def_binary64 (+.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
   (fma.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/6 (/.f64 (*.f64 (pow.f64 c 4) (+.f64 (*.f64 81/64 (pow.f64 a 4)) (*.f64 81/16 (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 c 4) (+.f64 (*.f64 81/64 (pow.f64 a 4)) (*.f64 81/16 (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

7. Final simplification 3.0

   \[\leadsto \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c}{\frac{{b}^{3}}{a \cdot c}}\right)\right)\right) \]

Alternatives

Alternative 1
Error	3.0
Cost	47104

\[\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}, a \cdot \left(-0.375 \cdot \left(c \cdot \frac{c}{{b}^{3}}\right) + -0.5625 \cdot \left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right)\right) \]

Alternative 2
Error	4.0
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	4.0
Cost	27136

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

Alternative 4
Error	10.6
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -0.1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5
Error	6.0
Cost	13696

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

Alternative 6
Error	6.4
Cost	7680

\[\left(\left(a \cdot c\right) \cdot \left(\frac{1.5}{b} + 1.125 \cdot \left(a \cdot \left(c \cdot {b}^{-3}\right)\right)\right)\right) \cdot \frac{-0.3333333333333333}{a} \]

Alternative 7
Error	6.3
Cost	7680

\[-0.3333333333333333 \cdot \left(1.125 \cdot \left(c \cdot \left(c \cdot \frac{a}{{b}^{3}}\right)\right)\right) + \left(\frac{c}{b} \cdot 1.5\right) \cdot -0.3333333333333333 \]

Alternative 8
Error	6.3
Cost	7616

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

Alternative 9
Error	12.2
Cost	320

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

Reproduce

herbie shell --seed 2022330 
(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)))