Quadratic roots, medium range

Average Error: 43.8 → 2.9

Time: 11.3s

Precision: binary64

Cost: 40768

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

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

↓

\[\mathsf{fma}\left(\frac{-0.25}{{b}^{7}}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}, \frac{-2 \cdot {c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right) - \mathsf{fma}\left(\frac{c}{b \cdot b} \cdot \frac{c}{b}, a, \frac{c}{b}\right)\right) \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (fma
  (/ -0.25 (pow b 7.0))
  (/ (* (pow (* c a) 4.0) 20.0) a)
  (-
   (* (/ (* -2.0 (pow c 3.0)) (pow b 5.0)) (* a a))
   (fma (* (/ c (* b b)) (/ c b)) a (/ c b)))))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

↓

double code(double a, double b, double c) {
	return fma((-0.25 / pow(b, 7.0)), ((pow((c * a), 4.0) * 20.0) / a), ((((-2.0 * pow(c, 3.0)) / pow(b, 5.0)) * (a * a)) - fma(((c / (b * b)) * (c / b)), a, (c / b))));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	return fma(Float64(-0.25 / (b ^ 7.0)), Float64(Float64((Float64(c * a) ^ 4.0) * 20.0) / a), Float64(Float64(Float64(Float64(-2.0 * (c ^ 3.0)) / (b ^ 5.0)) * Float64(a * a)) - fma(Float64(Float64(c / Float64(b * b)) * Float64(c / b)), a, Float64(c / b))))
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := N[(N[(-0.25 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / a), $MachinePrecision] + N[(N[(N[(N[(-2.0 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision] * a + N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.8

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

2. Taylor expanded in b around inf 2.9

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

3. Simplified 2.9

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.25}{{b}^{7}}, \frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot 20}{a}, \frac{-2 \cdot {c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\right)} \]
   Proof
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) 20) a) (-.f64 (*.f64 (/.f64 (*.f64 -2 (pow.f64 c 3)) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (Rewrite<= metadata-eval (+.f64 4 16))) a) (-.f64 (*.f64 (/.f64 (*.f64 -2 (pow.f64 c 3)) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 4 (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))))) a) (-.f64 (*.f64 (/.f64 (*.f64 -2 (pow.f64 c 3)) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (*.f64 (Rewrite<= metadata-eval (*.f64 -2 -2)) (*.f64 (pow.f64 c 4) (pow.f64 a 4))) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (*.f64 (/.f64 (*.f64 -2 (pow.f64 c 3)) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (*.f64 (*.f64 -2 -2) (*.f64 (pow.f64 c (Rewrite<= metadata-eval (*.f64 2 2))) (pow.f64 a 4))) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (*.f64 (/.f64 (*.f64 -2 (pow.f64 c 3)) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (*.f64 (*.f64 -2 -2) (*.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 c 2) (pow.f64 c 2))) (pow.f64 a 4))) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (*.f64 (/.f64 (*.f64 -2 (pow.f64 c 3)) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (*.f64 (*.f64 -2 -2) (*.f64 (*.f64 (pow.f64 c 2) (pow.f64 c 2)) (pow.f64 a (Rewrite<= metadata-eval (*.f64 2 2))))) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (*.f64 (/.f64 (*.f64 -2 (pow.f64 c 3)) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (*.f64 (*.f64 -2 -2) (*.f64 (*.f64 (pow.f64 c 2) (pow.f64 c 2)) (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 a 2) (pow.f64 a 2))))) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (*.f64 (/.f64 (*.f64 -2 (pow.f64 c 3)) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (*.f64 (*.f64 -2 -2) (Rewrite=> unswap-sqr_binary64 (*.f64 (*.f64 (pow.f64 c 2) (pow.f64 a 2)) (*.f64 (pow.f64 c 2) (pow.f64 a 2))))) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (*.f64 (/.f64 (*.f64 -2 (pow.f64 c 3)) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))))) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (*.f64 (/.f64 (*.f64 -2 (pow.f64 c 3)) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2)) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (*.f64 (/.f64 (*.f64 -2 (pow.f64 c 3)) (pow.f64 b 5)) (*.f64 a a)) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (*.f64 (/.f64 (*.f64 -2 (pow.f64 c 3)) (pow.f64 b 5)) (Rewrite<= unpow2_binary64 (pow.f64 a 2))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 -2 (pow.f64 c 3)) (/.f64 (pow.f64 b 5) (pow.f64 a 2)))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -2 (/.f64 (pow.f64 c 3) (/.f64 (pow.f64 b 5) (pow.f64 a 2))))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (*.f64 -2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (fma.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (pow.f64 b 3)) a (/.f64 c b)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 (pow.f64 c 2) (pow.f64 b 3)) a) (/.f64 c b))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (-.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (/.f64 c b)))): 1 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (neg.f64 (+.f64 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 c b)))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (+.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 c b)))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (+.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (*.f64 -1 (/.f64 c b)))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (*.f64 -1 (/.f64 c b))) (*.f64 -2 (/.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 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a) (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (+.f64 (*.f64 -1 (/.f64 c b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))))))): 0 points increase in error, 1 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 -1/4 (pow.f64 b 7)) (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) a)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (+.f64 (*.f64 -1 (/.f64 c b)) (*.f64 -2 (/.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<= times-frac_binary64 (/.f64 (*.f64 -1/4 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))))) (*.f64 (pow.f64 b 7) a))) (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (+.f64 (*.f64 -1 (/.f64 c b)) (*.f64 -2 (/.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 (/.f64 (*.f64 -1/4 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4))))) (Rewrite<= *-commutative_binary64 (*.f64 a (pow.f64 b 7)))) (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (+.f64 (*.f64 -1 (/.f64 c b)) (*.f64 -2 (/.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<= associate-*r/_binary64 (*.f64 -1/4 (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7))))) (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (+.f64 (*.f64 -1 (/.f64 c b)) (*.f64 -2 (/.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 (*.f64 -1/4 (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 (*.f64 -1 (/.f64 c b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)))) (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 -1/4 (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (+.f64 (*.f64 -1 (/.f64 c b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))))) (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))))): 1 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (+.f64 (*.f64 -1/4 (/.f64 (+.f64 (pow.f64 (*.f64 -2 (*.f64 (pow.f64 c 2) (pow.f64 a 2))) 2) (*.f64 16 (*.f64 (pow.f64 c 4) (pow.f64 a 4)))) (*.f64 a (pow.f64 b 7)))) (+.f64 (*.f64 -1 (/.f64 c b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))))))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 2.9

   \[\leadsto \mathsf{fma}\left(\frac{-0.25}{{b}^{7}}, \frac{\color{blue}{{\left(c \cdot a\right)}^{4}} \cdot 20}{a}, \frac{-2 \cdot {c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\right) \]

5. Applied egg-rr 2.9

   \[\leadsto \mathsf{fma}\left(\frac{-0.25}{{b}^{7}}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}, \frac{-2 \cdot {c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right) - \mathsf{fma}\left(\color{blue}{\frac{c}{b \cdot b} \cdot \frac{c}{b}}, a, \frac{c}{b}\right)\right) \]

6. Final simplification 2.9

   \[\leadsto \mathsf{fma}\left(\frac{-0.25}{{b}^{7}}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a}, \frac{-2 \cdot {c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right) - \mathsf{fma}\left(\frac{c}{b \cdot b} \cdot \frac{c}{b}, a, \frac{c}{b}\right)\right) \]

Alternatives

Alternative 1
Error	4.0
Cost	20736

\[\frac{{c}^{3} \cdot \left(-2 \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \left(\frac{c}{b} + a \cdot \left(\left(c \cdot c\right) \cdot {b}^{-3}\right)\right) \]

Alternative 2
Error	4.0
Cost	20608

\[a \cdot \left(\frac{-2 \cdot {c}^{3}}{\frac{{b}^{5}}{a}} - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b} \]

Alternative 3
Error	6.0
Cost	1472

\[0.5 \cdot \frac{-2}{\frac{b \cdot b}{c \cdot c} \cdot \frac{b \cdot b}{b \cdot a}} - \frac{c}{b} \]

Alternative 4
Error	6.1
Cost	832

\[\frac{c}{b} \cdot \left(-1 - \frac{c \cdot a}{b \cdot b}\right) \]

Alternative 5
Error	12.0
Cost	256

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

Reproduce

herbie shell --seed 2022313 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))