Quadratic roots, wide range

Average Error: 52.3 → 1.5

Time: 17.8s

Precision: binary64

Cost: 34496

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

Math

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

↓

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

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 (* c a) 4.0) (/ 20.0 (* a (pow b 7.0))))
   (- (/ -2.0 (/ (pow b 5.0) (* (pow c 3.0) (* a a)))) (/ c b)))
  (* (/ c (* b b)) (/ (* c a) 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((c * a), 4.0) * (20.0 / (a * pow(b, 7.0)))), ((-2.0 / (pow(b, 5.0) / (pow(c, 3.0) * (a * a)))) - (c / b))) - ((c / (b * b)) * ((c * a) / 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 Float64(fma(-0.25, Float64((Float64(c * a) ^ 4.0) * Float64(20.0 / Float64(a * (b ^ 7.0)))), Float64(Float64(-2.0 / Float64((b ^ 5.0) / Float64((c ^ 3.0) * Float64(a * a)))) - Float64(c / b))) - Float64(Float64(c / Float64(b * b)) * Float64(Float64(c * a) / 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[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 / N[(N[Power[b, 5.0], $MachinePrecision] / N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.3

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

2. Simplified 52.3

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   Proof
   (/.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -4)))) b) (*.f64 a 2)): 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 4)))))) b) (*.f64 a 2)): 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 4)))))) b) (*.f64 a 2)): 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 4 a)))))) b) (*.f64 a 2)): 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 4 a)))))) b) (*.f64 a 2)): 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 4 a) c))))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) b) (*.f64 a 2)): 2 points increase in error, 3 points decrease in error
   (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) (neg.f64 b))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite<= *-commutative_binary64 (*.f64 2 a))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in b around inf 1.5

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

4. Simplified 1.5

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

5. Taylor expanded in c around 0 1.5

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

6. Simplified 1.5

   \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{{\left(c \cdot a\right)}^{4} \cdot \frac{20}{a \cdot {b}^{7}}}, \frac{-2}{\frac{{b}^{5}}{{c}^{3} \cdot \left(a \cdot a\right)}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]
   Proof
   (*.f64 (pow.f64 (*.f64 c a) 4) (/.f64 20 (*.f64 a (pow.f64 b 7)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (pow.f64 (*.f64 c a) 4) (/.f64 (Rewrite<= metadata-eval (*.f64 20 1)) (*.f64 a (pow.f64 b 7)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (pow.f64 (*.f64 c a) 4) (Rewrite<= associate-*r/_binary64 (*.f64 20 (/.f64 1 (*.f64 a (pow.f64 b 7)))))): 28 points increase in error, 36 points decrease in error
   (*.f64 (pow.f64 (*.f64 c a) 4) (*.f64 20 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 a) (pow.f64 b 7))))): 31 points increase in error, 27 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (*.f64 c a) 4) 20) (/.f64 (/.f64 1 a) (pow.f64 b 7)))): 46 points increase in error, 34 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 (*.f64 c a) 4) 20) (/.f64 1 a)) (pow.f64 b 7))): 32 points increase in error, 43 points decrease in error
   (/.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 (*.f64 c a) 4) 20) 1) a)) (pow.f64 b 7)): 32 points increase in error, 26 points decrease in error
   (/.f64 (/.f64 (Rewrite=> *-rgt-identity_binary64 (*.f64 (pow.f64 (*.f64 c a) 4) 20)) a) (pow.f64 b 7)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (pow.f64 (*.f64 c a) 4) 20) (*.f64 a (pow.f64 b 7)))): 39 points increase in error, 48 points decrease in error
   (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 20 (pow.f64 (*.f64 c a) 4))) (*.f64 a (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 20 (pow.f64 (*.f64 c a) (Rewrite<= metadata-eval (*.f64 2 2)))) (*.f64 a (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 20 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (*.f64 c a) 2) (pow.f64 (*.f64 c a) 2)))) (*.f64 a (pow.f64 b 7))): 65 points increase in error, 48 points decrease in error
   (/.f64 (*.f64 20 (*.f64 (Rewrite=> unpow2_binary64 (*.f64 (*.f64 c a) (*.f64 c a))) (pow.f64 (*.f64 c a) 2))) (*.f64 a (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 20 (*.f64 (*.f64 (*.f64 c a) (*.f64 c a)) (Rewrite=> unpow2_binary64 (*.f64 (*.f64 c a) (*.f64 c a))))) (*.f64 a (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 20 (*.f64 (Rewrite=> swap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 a a))) (*.f64 (*.f64 c a) (*.f64 c a)))) (*.f64 a (pow.f64 b 7))): 54 points increase in error, 50 points decrease in error
   (/.f64 (*.f64 20 (*.f64 (*.f64 (*.f64 c c) (*.f64 a a)) (Rewrite=> swap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 a a))))) (*.f64 a (pow.f64 b 7))): 54 points increase in error, 47 points decrease in error
   (/.f64 (*.f64 20 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 (*.f64 c c) (*.f64 c c)) (*.f64 (*.f64 a a) (*.f64 a a))))) (*.f64 a (pow.f64 b 7))): 60 points increase in error, 56 points decrease in error
   (/.f64 (*.f64 20 (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (*.f64 c c)) (*.f64 (*.f64 a a) (*.f64 a a)))) (*.f64 a (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 20 (*.f64 (*.f64 (pow.f64 c 2) (Rewrite<= unpow2_binary64 (pow.f64 c 2))) (*.f64 (*.f64 a a) (*.f64 a a)))) (*.f64 a (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 20 (*.f64 (Rewrite=> pow-sqr_binary64 (pow.f64 c (*.f64 2 2))) (*.f64 (*.f64 a a) (*.f64 a a)))) (*.f64 a (pow.f64 b 7))): 46 points increase in error, 50 points decrease in error
   (/.f64 (*.f64 20 (*.f64 (pow.f64 c (Rewrite=> metadata-eval 4)) (*.f64 (*.f64 a a) (*.f64 a a)))) (*.f64 a (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 20 (*.f64 (pow.f64 c 4) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (*.f64 a a) a) a)))) (*.f64 a (pow.f64 b 7))): 22 points increase in error, 41 points decrease in error
   (/.f64 (*.f64 20 (*.f64 (pow.f64 c 4) (*.f64 (Rewrite<= unpow3_binary64 (pow.f64 a 3)) a))) (*.f64 a (pow.f64 b 7))): 21 points increase in error, 23 points decrease in error
   (/.f64 (*.f64 20 (*.f64 (pow.f64 c 4) (Rewrite=> pow-plus_binary64 (pow.f64 a (+.f64 3 1))))) (*.f64 a (pow.f64 b 7))): 27 points increase in error, 17 points decrease in error
   (/.f64 (*.f64 20 (*.f64 (pow.f64 c 4) (pow.f64 a (Rewrite=> metadata-eval 4)))) (*.f64 a (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) 20)) (*.f64 a (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 c 4) (*.f64 (pow.f64 a 4) 20))) (*.f64 a (pow.f64 b 7))): 45 points increase in error, 37 points decrease in error
   (/.f64 (*.f64 (pow.f64 c 4) (*.f64 (pow.f64 a 4) (Rewrite<= metadata-eval (+.f64 4 16)))) (*.f64 a (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (pow.f64 c 4) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 4 (pow.f64 a 4)) (*.f64 16 (pow.f64 a 4))))) (*.f64 a (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error

7. Applied egg-rr 1.5

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

8. Final simplification 1.5

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

Alternatives

Alternative 1
Error	2.0
Cost	14528

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

Alternative 2
Error	3.1
Cost	1024

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

Alternative 3
Error	3.1
Cost	832

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

Alternative 4
Error	6.3
Cost	256

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

Alternative 5
Error	63.0
Cost	192

\[\frac{b}{a} \]

Reproduce

herbie shell --seed 2022318 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))