Quadratic roots, wide range

Average Error: 52.6 → 1.4

Time: 12.2s

Precision: binary64

Cost: 34560

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

Math

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

↓

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

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 a)
  (/ (* (pow (* a c) 4.0) 20.0) (pow b 7.0))
  (-
   (* -2.0 (* (* c c) (* c (/ (* a a) (pow b 5.0)))))
   (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 / a), ((pow((a * c), 4.0) * 20.0) / pow(b, 7.0)), ((-2.0 * ((c * c) * (c * ((a * a) / pow(b, 5.0))))) - 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 / a), Float64(Float64((Float64(a * c) ^ 4.0) * 20.0) / (b ^ 7.0)), Float64(Float64(-2.0 * Float64(Float64(c * c) * Float64(c * Float64(Float64(a * a) / (b ^ 5.0))))) - 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 / a), $MachinePrecision] * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[(c * c), $MachinePrecision] * N[(c * N[(N[(a * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 52.6

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

2. Simplified 52.6

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}} \]
   Proof
   (/.f64 (-.f64 (sqrt.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (sqrt.f64 (fma.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 b b))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (sqrt.f64 (fma.f64 a (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 c 4))) (*.f64 b b))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (sqrt.f64 (fma.f64 a (neg.f64 (Rewrite=> *-commutative_binary64 (*.f64 4 c))) (*.f64 b b))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (sqrt.f64 (fma.f64 a (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 4) c)) (*.f64 b b))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (*.f64 (neg.f64 4) c)) (*.f64 b b)))) b) (*.f64 a 2)): 1 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (sqrt.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a (neg.f64 4)) c)) (*.f64 b b))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (sqrt.f64 (+.f64 (*.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 4))) c) (*.f64 b b))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (sqrt.f64 (+.f64 (*.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 a))) c) (*.f64 b b))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (*.f64 (neg.f64 (*.f64 4 a)) c)))) b) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (sqrt.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) b) (*.f64 a 2)): 0 points increase in error, 0 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 1.4

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

4. Simplified 1.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.25}{a}, \frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot 20}{{b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\right)} \]
   Proof
   (fma.f64 (/.f64 -1/4 a) (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) 20) (pow.f64 b 7)) (-.f64 (*.f64 -2 (/.f64 (pow.f64 c 3) (/.f64 (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 a) (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) (Rewrite<= metadata-eval (+.f64 4 16))) (pow.f64 b 7)) (-.f64 (*.f64 -2 (/.f64 (pow.f64 c 3) (/.f64 (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 a) (/.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))))) (pow.f64 b 7)) (-.f64 (*.f64 -2 (/.f64 (pow.f64 c 3) (/.f64 (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 a) (/.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)))) (pow.f64 b 7)) (-.f64 (*.f64 -2 (/.f64 (pow.f64 c 3) (/.f64 (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 a) (/.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)))) (pow.f64 b 7)) (-.f64 (*.f64 -2 (/.f64 (pow.f64 c 3) (/.f64 (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 a) (/.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)))) (pow.f64 b 7)) (-.f64 (*.f64 -2 (/.f64 (pow.f64 c 3) (/.f64 (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 a) (/.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)))) (pow.f64 b 7)) (-.f64 (*.f64 -2 (/.f64 (pow.f64 c 3) (/.f64 (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 a) (/.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)))) (pow.f64 b 7)) (-.f64 (*.f64 -2 (/.f64 (pow.f64 c 3) (/.f64 (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 a) (/.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)))) (pow.f64 b 7)) (-.f64 (*.f64 -2 (/.f64 (pow.f64 c 3) (/.f64 (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 a) (/.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)))) (pow.f64 b 7)) (-.f64 (*.f64 -2 (/.f64 (pow.f64 c 3) (/.f64 (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 a) (/.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)))) (pow.f64 b 7)) (-.f64 (*.f64 -2 (/.f64 (pow.f64 c 3) (/.f64 (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 a) (/.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)))) (pow.f64 b 7)) (-.f64 (*.f64 -2 (/.f64 (pow.f64 c 3) (/.f64 (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 a) (/.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)))) (pow.f64 b 7)) (-.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 a) (/.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)))) (pow.f64 b 7)) (-.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 a) (/.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)))) (pow.f64 b 7)) (-.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, 1 points decrease in error
   (fma.f64 (/.f64 -1/4 a) (/.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)))) (pow.f64 b 7)) (-.f64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))) (+.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 c 2) (/.f64 (pow.f64 b 3) a))) (/.f64 c b)))): 1 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 a) (/.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)))) (pow.f64 b 7)) (-.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)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 -1/4 a) (/.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)))) (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 (/.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 a) (/.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)))) (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 (/.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 a) (/.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)))) (pow.f64 b 7)) (+.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 a) (/.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)))) (pow.f64 b 7)) (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 a) (/.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)))) (pow.f64 b 7)) (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, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 -1/4 a) (/.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)))) (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<= 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 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))))): 0 points increase in error, 2 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. Applied egg-rr 1.7

   \[\leadsto \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\color{blue}{\left(e^{\mathsf{log1p}\left({\left(c \cdot a\right)}^{4}\right)} - 1\right)} \cdot 20}{{b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\right) \]

6. Simplified 1.4

   \[\leadsto \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\color{blue}{{\left(c \cdot a\right)}^{4}} \cdot 20}{{b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\right) \]
   Proof
   (pow.f64 (*.f64 c a) 4): 0 points increase in error, 0 points decrease in error
   (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (pow.f64 (*.f64 c a) 4)))): 42 points increase in error, 40 points decrease in error
   (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (pow.f64 (*.f64 c a) 4))) 1)): 13 points increase in error, 168 points decrease in error

7. Applied egg-rr 1.4

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

8. Applied egg-rr 1.4

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

9. Final simplification 1.4

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

Alternatives

Alternative 1
Error	1.9
Cost	20736

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

Alternative 2
Error	3.0
Cost	7232

\[\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}} \]

Alternative 3
Error	3.3
Cost	1728

\[\left(-2 \cdot \left(\frac{c \cdot c}{b \cdot b} \cdot \frac{a \cdot a}{b}\right) + -2 \cdot \frac{a \cdot c}{b}\right) \cdot \frac{0.5}{a} \]

Alternative 4
Error	6.1
Cost	256

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

Alternative 5
Error	61.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022332 
(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)))