Cubic critical, medium range

Average Error: 43.6 → 3.0

Time: 12.9s

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

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

↓

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

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
  -0.5
  (/ c b)
  (fma
   -0.375
   (* (/ a (pow b 3.0)) (* c c))
   (fma
    (/ -0.16666666666666666 a)
    (/ (pow (* c a) 4.0) (/ (pow b 7.0) 6.328125))
    (* a (/ (* -0.5625 (pow c 3.0)) (/ (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(-0.5, (c / b), fma(-0.375, ((a / pow(b, 3.0)) * (c * c)), fma((-0.16666666666666666 / a), (pow((c * a), 4.0) / (pow(b, 7.0) / 6.328125)), (a * ((-0.5625 * pow(c, 3.0)) / (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(-0.5, Float64(c / b), fma(-0.375, Float64(Float64(a / (b ^ 3.0)) * Float64(c * c)), fma(Float64(-0.16666666666666666 / a), Float64((Float64(c * a) ^ 4.0) / Float64((b ^ 7.0) / 6.328125)), Float64(a * Float64(Float64(-0.5625 * (c ^ 3.0)) / 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[(-0.5 * 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 / a), $MachinePrecision] * N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / 6.328125), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.6

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

2. Taylor expanded in b around inf 3.4

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

3. Simplified 3.4

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}}, \mathsf{fma}\left(-1.125, \frac{c \cdot c}{\frac{{b}^{3}}{a \cdot a}}, \mathsf{fma}\left(-1.5, \frac{c}{\frac{b}{a}}, -1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)\right)\right)}}{3 \cdot a} \]
   Proof
   (fma.f64 -1/2 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (*.f64 c c) (*.f64 a a))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (pow.f64 b 7)) (fma.f64 -9/8 (/.f64 (*.f64 c c) (/.f64 (pow.f64 b 3) (*.f64 a a))) (fma.f64 -3/2 (/.f64 c (/.f64 b a)) (*.f64 -27/16 (/.f64 (pow.f64 (*.f64 c a) 3) (pow.f64 b 5)))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/2 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (*.f64 a a))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (pow.f64 b 7)) (fma.f64 -9/8 (/.f64 (*.f64 c c) (/.f64 (pow.f64 b 3) (*.f64 a a))) (fma.f64 -3/2 (/.f64 c (/.f64 b a)) (*.f64 -27/16 (/.f64 (pow.f64 (*.f64 c a) 3) (pow.f64 b 5)))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/2 (/.f64 (+.f64 (pow.f64 (*.f64 -9/8 (*.f64 (pow.f64 c 2) (Rewrite<= unpow2_binary64 (pow.f64 a 2)))) 2) (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (pow.f64 b 7)) (fma.f64 -9/8 (/.f64 (*.f64 c c) (/.f64 (pow.f64 b 3) (*.f64 a a))) (fma.f64 -3/2 (/.f64 c (/.f64 b a)) (*.f64 -27/16 (/.f64 (pow.f64 (*.f64 c a) 3) (pow.f64 b 5)))))): 0 points increase in error, 0 points decrease in error
   (fma.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)))) (pow.f64 b 7)) (fma.f64 -9/8 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (/.f64 (pow.f64 b 3) (*.f64 a a))) (fma.f64 -3/2 (/.f64 c (/.f64 b a)) (*.f64 -27/16 (/.f64 (pow.f64 (*.f64 c a) 3) (pow.f64 b 5)))))): 0 points increase in error, 0 points decrease in error
   (fma.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)))) (pow.f64 b 7)) (fma.f64 -9/8 (/.f64 (pow.f64 c 2) (/.f64 (pow.f64 b 3) (Rewrite<= unpow2_binary64 (pow.f64 a 2)))) (fma.f64 -3/2 (/.f64 c (/.f64 b a)) (*.f64 -27/16 (/.f64 (pow.f64 (*.f64 c a) 3) (pow.f64 b 5)))))): 0 points increase in error, 0 points decrease in error
   (fma.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)))) (pow.f64 b 7)) (fma.f64 -9/8 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 c 2) (pow.f64 a 2)) (pow.f64 b 3))) (fma.f64 -3/2 (/.f64 c (/.f64 b a)) (*.f64 -27/16 (/.f64 (pow.f64 (*.f64 c a) 3) (pow.f64 b 5)))))): 0 points increase in error, 0 points decrease in error
   (fma.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)))) (pow.f64 b 7)) (fma.f64 -9/8 (/.f64 (*.f64 (pow.f64 c 2) (pow.f64 a 2)) (pow.f64 b 3)) (fma.f64 -3/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c a) b)) (*.f64 -27/16 (/.f64 (pow.f64 (*.f64 c a) 3) (pow.f64 b 5)))))): 41 points increase in error, 37 points decrease in error
   (fma.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)))) (pow.f64 b 7)) (fma.f64 -9/8 (/.f64 (*.f64 (pow.f64 c 2) (pow.f64 a 2)) (pow.f64 b 3)) (fma.f64 -3/2 (/.f64 (*.f64 c a) b) (*.f64 -27/16 (/.f64 (Rewrite=> cube-prod_binary64 (*.f64 (pow.f64 c 3) (pow.f64 a 3))) (pow.f64 b 5)))))): 0 points increase in error, 0 points decrease in error
   (fma.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)))) (pow.f64 b 7)) (fma.f64 -9/8 (/.f64 (*.f64 (pow.f64 c 2) (pow.f64 a 2)) (pow.f64 b 3)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -3/2 (/.f64 (*.f64 c a) b)) (*.f64 -27/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 3)) (pow.f64 b 5))))))): 27 points increase in error, 26 points decrease in error
   (fma.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)))) (pow.f64 b 7)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -9/8 (/.f64 (*.f64 (pow.f64 c 2) (pow.f64 a 2)) (pow.f64 b 3))) (+.f64 (*.f64 -3/2 (/.f64 (*.f64 c a) b)) (*.f64 -27/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 3)) (pow.f64 b 5))))))): 0 points increase in error, 1 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.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)))) (pow.f64 b 7))) (+.f64 (*.f64 -9/8 (/.f64 (*.f64 (pow.f64 c 2) (pow.f64 a 2)) (pow.f64 b 3))) (+.f64 (*.f64 -3/2 (/.f64 (*.f64 c a) b)) (*.f64 -27/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 3)) (pow.f64 b 5))))))): 0 points increase in error, 0 points decrease in error

4. Taylor expanded in c around 0 3.0

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

5. Simplified 3.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right), \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, \frac{{\left(c \cdot a\right)}^{4}}{\frac{{b}^{7}}{6.328125}}, \frac{-0.5625 \cdot {c}^{3}}{\frac{{b}^{5}}{a}} \cdot a\right)\right)\right)} \]
   Proof
   (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (*.f64 (/.f64 a (pow.f64 b 3)) (*.f64 c c)) (fma.f64 (/.f64 -1/6 a) (/.f64 (pow.f64 (*.f64 c a) 4) (/.f64 (pow.f64 b 7) 405/64)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (Rewrite<= associate-/r/_binary64 (/.f64 a (/.f64 (pow.f64 b 3) (*.f64 c c)))) (fma.f64 (/.f64 -1/6 a) (/.f64 (pow.f64 (*.f64 c a) 4) (/.f64 (pow.f64 b 7) 405/64)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a (*.f64 c c)) (pow.f64 b 3))) (fma.f64 (/.f64 -1/6 a) (/.f64 (pow.f64 (*.f64 c a) 4) (/.f64 (pow.f64 b 7) 405/64)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 c c) a)) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (pow.f64 (*.f64 c a) 4) (/.f64 (pow.f64 b 7) 405/64)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (pow.f64 (*.f64 c a) 4) (/.f64 (pow.f64 b 7) 405/64)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (pow.f64 (*.f64 c a) (Rewrite<= metadata-eval (*.f64 2 2))) (/.f64 (pow.f64 b 7) 405/64)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (*.f64 c a) 2) (pow.f64 (*.f64 c a) 2))) (/.f64 (pow.f64 b 7) 405/64)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (Rewrite=> unpow2_binary64 (*.f64 (*.f64 c a) (*.f64 c a))) (pow.f64 (*.f64 c a) 2)) (/.f64 (pow.f64 b 7) 405/64)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (*.f64 (*.f64 c a) (*.f64 c a)) (Rewrite=> unpow2_binary64 (*.f64 (*.f64 c a) (*.f64 c a)))) (/.f64 (pow.f64 b 7) 405/64)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (*.f64 (*.f64 c a) (*.f64 c a)) 2)) (/.f64 (pow.f64 b 7) 405/64)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (pow.f64 (*.f64 (*.f64 c a) (*.f64 c a)) 2) (/.f64 (pow.f64 b 7) (Rewrite<= metadata-eval (+.f64 81/64 81/16)))) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 (*.f64 (*.f64 c a) (*.f64 c a)) 2) (+.f64 81/64 81/16)) (pow.f64 b 7))) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (Rewrite=> unpow2_binary64 (*.f64 (*.f64 (*.f64 c a) (*.f64 c a)) (*.f64 (*.f64 c a) (*.f64 c a)))) (+.f64 81/64 81/16)) (pow.f64 b 7)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (*.f64 (Rewrite=> swap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 a a))) (*.f64 (*.f64 c a) (*.f64 c a))) (+.f64 81/64 81/16)) (pow.f64 b 7)) (*.f64 (/.f64 (*.f64 -9/16 (pow.f64 c 3)) (/.f64 (pow.f64 b 5) a)) a)))): 1 points increase in error, 0 points decrease in error
   (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (*.f64 (*.f64 (*.f64 c c) (*.f64 a a)) (Rewrite=> swap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 a a)))) (+.f64 81/64 81/16)) (pow.f64 b 7)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 (*.f64 c c) (*.f64 c c)) (*.f64 (*.f64 a a) (*.f64 a a)))) (+.f64 81/64 81/16)) (pow.f64 b 7)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (*.f64 c c)) (*.f64 (*.f64 a a) (*.f64 a a))) (+.f64 81/64 81/16)) (pow.f64 b 7)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (*.f64 (*.f64 (pow.f64 c 2) (Rewrite<= unpow2_binary64 (pow.f64 c 2))) (*.f64 (*.f64 a a) (*.f64 a a))) (+.f64 81/64 81/16)) (pow.f64 b 7)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (*.f64 (Rewrite=> pow-sqr_binary64 (pow.f64 c (*.f64 2 2))) (*.f64 (*.f64 a a) (*.f64 a a))) (+.f64 81/64 81/16)) (pow.f64 b 7)) (*.f64 (/.f64 (*.f64 -9/16 (pow.f64 c 3)) (/.f64 (pow.f64 b 5) a)) a)))): 0 points increase in error, 1 points decrease in error
   (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (*.f64 (pow.f64 c (Rewrite=> metadata-eval 4)) (*.f64 (*.f64 a a) (*.f64 a a))) (+.f64 81/64 81/16)) (pow.f64 b 7)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (*.f64 a a) a) a))) (+.f64 81/64 81/16)) (pow.f64 b 7)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (*.f64 (Rewrite<= unpow3_binary64 (pow.f64 a 3)) a)) (+.f64 81/64 81/16)) (pow.f64 b 7)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (Rewrite=> pow-plus_binary64 (pow.f64 a (+.f64 3 1)))) (+.f64 81/64 81/16)) (pow.f64 b 7)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a (Rewrite=> metadata-eval 4))) (+.f64 81/64 81/16)) (pow.f64 b 7)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (Rewrite=> metadata-eval 405/64)) (pow.f64 b 7)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (Rewrite<= metadata-eval (+.f64 81/16 81/64))) (pow.f64 b 7)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4))))) (pow.f64 b 7)) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (pow.f64 b 7)) (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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (pow.f64 b 7)) (/.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, 1 points decrease in error
   (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (pow.f64 b 7)) (Rewrite<= associate-*r/_binary64 (*.f64 -9/16 (/.f64 (pow.f64 c 3) (/.f64 (pow.f64 b 5) (*.f64 a a)))))))): 1 points increase in error, 0 points decrease in error
   (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (pow.f64 b 7)) (*.f64 -9/16 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 c 3) (*.f64 a a)) (pow.f64 b 5))))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (fma.f64 (/.f64 -1/6 a) (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (pow.f64 b 7)) (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (Rewrite<= unpow2_binary64 (pow.f64 a 2))) (pow.f64 b 5)))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 -1/6 a) (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (pow.f64 b 7))) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1/6 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4))))) (*.f64 a (pow.f64 b 7)))) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))))) (*.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 -1/2 (/.f64 c b) (fma.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.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 -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 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.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
   (fma.f64 -1/2 (/.f64 c b) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))))) (*.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 -1/2 (/.f64 c b) (Rewrite<= associate-+r+_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 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (*.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 -1/2 (/.f64 c b)) (+.f64 (*.f64 -9/16 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (*.f64 -3/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 -1/2 (/.f64 c b)) (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (*.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
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 -1/2 (/.f64 c b)) (+.f64 (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (*.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))))): 11 points increase in error, 6 points decrease in error
   (+.f64 (Rewrite=> associate-+r+_binary64 (+.f64 (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))))) (*.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)))): 6 points increase in error, 6 points decrease in error
   (Rewrite=> associate-+l+_binary64 (+.f64 (+.f64 (*.f64 -1/2 (/.f64 c b)) (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))))) (+.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)))))): 7 points increase in error, 10 points decrease in error
   (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (*.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
   (+.f64 (Rewrite=> fma-def_binary64 (fma.f64 -1/6 (/.f64 (+.f64 (*.f64 81/16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 81/64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))) (*.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
   (+.f64 (fma.f64 -1/6 (/.f64 (Rewrite=> distribute-rgt-out_binary64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (+.f64 81/16 81/64))) (*.f64 a (pow.f64 b 7))) (*.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
   (+.f64 (fma.f64 -1/6 (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (Rewrite=> metadata-eval 405/64)) (*.f64 a (pow.f64 b 7))) (*.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
   (+.f64 (fma.f64 -1/6 (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (Rewrite<= metadata-eval (+.f64 81/64 81/16))) (*.f64 a (pow.f64 b 7))) (*.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
   (+.f64 (fma.f64 -1/6 (/.f64 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 c 4) (*.f64 (pow.f64 a 4) (+.f64 81/64 81/16)))) (*.f64 a (pow.f64 b 7))) (*.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
   (+.f64 (fma.f64 -1/6 (/.f64 (*.f64 (pow.f64 c 4) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 81/64 (pow.f64 a 4)) (*.f64 81/16 (pow.f64 a 4))))) (*.f64 a (pow.f64 b 7))) (*.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
   (+.f64 (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 -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
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.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 -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))))): 10 points increase in error, 7 points decrease in error
   (+.f64 (Rewrite<= associate-+r+_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)))))) (*.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
   (Rewrite<= +-commutative_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

6. Final simplification 3.0

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

Alternatives

Alternative 1
Error	4.0
Cost	33536

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

Alternative 2
Error	4.0
Cost	27328

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

Alternative 3
Error	10.4
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4
Error	6.1
Cost	13696

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

Alternative 5
Error	6.4
Cost	8000

\[\frac{\mathsf{fma}\left(-1.125, \left(a \cdot a\right) \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), -1.5 \cdot \left(\frac{c}{b} \cdot a\right)\right)}{a \cdot 3} \]

Alternative 6
Error	12.2
Cost	320

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

Reproduce

herbie shell --seed 2022329 
(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)))