Cubic critical, wide range

Average Error: 52.4 → 1.5

Time: 17.0s

Precision: binary64

Cost: 41088

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

Math

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

↓

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

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
  (* (* (* c c) (* c (pow b -5.0))) (* a a))
  (fma
   -0.16666666666666666
   (/ (* (pow (* c a) 4.0) 6.328125) (* a (pow b 7.0)))
   (fma -0.375 (* (/ (* c c) (* b b)) (/ a b)) (* -0.5 (/ c b))))))

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, (((c * c) * (c * pow(b, -5.0))) * (a * a)), fma(-0.16666666666666666, ((pow((c * a), 4.0) * 6.328125) / (a * pow(b, 7.0))), fma(-0.375, (((c * c) / (b * b)) * (a / b)), (-0.5 * (c / b)))));
}

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

Derivation

1. Initial program 52.4

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

2. Simplified 52.4

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}} \]
   Proof
   (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -3)))) b) (*.f64 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 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 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 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 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 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 3 a)): 11 points increase in error, 2 points decrease in error
   (/.f64 (-.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))) 1)) b) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))) (Rewrite<= metadata-eval (/.f64 -1 -1))) b) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))) -1) -1)) b) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) -1) b) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) -1) b) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite<= /-rgt-identity_binary64 (/.f64 b 1))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (/.f64 b (Rewrite<= metadata-eval (/.f64 -1 -1)))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 b -1) -1))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 b)) -1)) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (/.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 b)) -1)) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 b)) -1)) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite=> sub-neg_binary64 (+.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 (neg.f64 b)))) -1) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite=> remove-double-neg_binary64 b)) -1) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 b (neg.f64 (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 (/.f64 (Rewrite<= sub-neg_binary64 (-.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
   (Rewrite<= associate-/r*_binary64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 -1 (*.f64 3 a)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 -1 (*.f64 3 a))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (*.f64 -1 (*.f64 3 a))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (neg.f64 -1) -1) (/.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 (/.f64 (Rewrite=> metadata-eval 1) -1) (/.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=> metadata-eval -1) (/.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
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (*.f64 3 a))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in b around inf 1.5

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

4. Simplified 1.5

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

5. Applied egg-rr 1.5

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

6. Applied egg-rr 1.5

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

7. Final simplification 1.5

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

Alternatives

Alternative 1
Error	4.0
Cost	34116

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 3} \leq -0.05:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b\right) \cdot \frac{\frac{0.3333333333333333}{\sqrt[3]{a \cdot a}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 2
Error	4.0
Cost	33924

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 3} \leq -0.05:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b\right) \cdot {\left(\sqrt{\frac{0.3333333333333333}{a}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 3
Error	2.3
Cost	33536

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

Alternative 4
Error	2.0
Cost	33536

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

Alternative 5
Error	5.9
Cost	21060

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 3} \leq -0.05:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6
Error	5.9
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3} \leq -0.05:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7
Error	5.9
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -0.05:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 8
Error	3.3
Cost	13696

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

Alternative 9
Error	3.1
Cost	13696

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

Alternative 10
Error	6.5
Cost	320

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

Alternative 11
Error	6.3
Cost	320

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

Reproduce

herbie shell --seed 2022338 
(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)))