Quadratic roots, narrow range

Average Error: 28.4 → 5.9

Time: 12.9s

Precision: binary64

Cost: 34880

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

↓

\[\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 20\right), \frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot {c}^{3}}{{b}^{5}}\right) - \left(\frac{c}{b} + a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right)\right) \]

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 a 3.0) b) (* (* (* c c) (* (* c c) (pow b -6.0))) 20.0))
   (/ (* (* a (* a -2.0)) (pow c 3.0)) (pow b 5.0)))
  (+ (/ c b) (* a (* (/ c b) (/ c (* b 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(a, 3.0) / b) * (((c * c) * ((c * c) * pow(b, -6.0))) * 20.0)), (((a * (a * -2.0)) * pow(c, 3.0)) / pow(b, 5.0))) - ((c / b) + (a * ((c / b) * (c / (b * 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((a ^ 3.0) / b) * Float64(Float64(Float64(c * c) * Float64(Float64(c * c) * (b ^ -6.0))) * 20.0)), Float64(Float64(Float64(a * Float64(a * -2.0)) * (c ^ 3.0)) / (b ^ 5.0))) - Float64(Float64(c / b) + Float64(a * Float64(Float64(c / b) * Float64(c / Float64(b * 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[(N[Power[a, 3.0], $MachinePrecision] / b), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] + N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.4

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

2. Simplified 28.4

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}} \]
   Proof
   (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b))) b) (/.f64 1/2 a)): 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 1/2 a)): 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 1/2 a)): 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 1/2 a)): 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 1/2 a)): 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 1/2 a)): 0 points increase in error, 2 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 1/2 a)): 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 1/2 a)): 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 1/2 a)): 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 1/2 a)): 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 1/2 a)): 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 1/2 a)): 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 1/2 a)): 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)))) (/.f64 (Rewrite<= metadata-eval (/.f64 1 2)) a)): 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)))) (/.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 -1)) 2) a)): 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<= associate-/r*_binary64 (/.f64 (neg.f64 -1) (*.f64 2 a)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (neg.f64 -1)) (*.f64 2 a))): 24 points increase in error, 31 points decrease in error
   (Rewrite=> associate-/l*_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (/.f64 (*.f64 2 a) (neg.f64 -1)))): 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)))) (/.f64 (*.f64 2 a) (Rewrite=> metadata-eval 1))): 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=> /-rgt-identity_binary64 (*.f64 2 a))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 29.6

   \[\leadsto \left(\sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}\right)}^{3}}} - b\right) \cdot \frac{0.5}{a} \]

4. Taylor expanded in a around 0 5.9

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

5. Simplified 5.9

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), \frac{\left(\left(-2 \cdot a\right) \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right) - \left(\frac{c}{b} + \frac{c \cdot c}{{b}^{3}} \cdot a\right)} \]
   Proof
   (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) b) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) 20)) (/.f64 (*.f64 (*.f64 (*.f64 -2 a) a) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) b) (*.f64 (/.f64 (pow.f64 c 4) (pow.f64 b 6)) (Rewrite<= metadata-eval (+.f64 16 4)))) (/.f64 (*.f64 (*.f64 (*.f64 -2 a) a) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) b) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 4 (/.f64 (pow.f64 c 4) (pow.f64 b 6)))))) (/.f64 (*.f64 (*.f64 (*.f64 -2 a) a) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (Rewrite<= metadata-eval (*.f64 -2 -2)) (/.f64 (pow.f64 c 4) (pow.f64 b 6))))) (/.f64 (*.f64 (*.f64 (*.f64 -2 a) a) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -2 -2) (/.f64 (pow.f64 c (Rewrite<= metadata-eval (*.f64 2 2))) (pow.f64 b 6))))) (/.f64 (*.f64 (*.f64 (*.f64 -2 a) a) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -2 -2) (/.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 c 2) (pow.f64 c 2))) (pow.f64 b 6))))) (/.f64 (*.f64 (*.f64 (*.f64 -2 a) a) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -2 -2) (/.f64 (*.f64 (pow.f64 c 2) (pow.f64 c 2)) (pow.f64 b (Rewrite<= metadata-eval (*.f64 2 3))))))) (/.f64 (*.f64 (*.f64 (*.f64 -2 a) a) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -2 -2) (/.f64 (*.f64 (pow.f64 c 2) (pow.f64 c 2)) (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 b 3) (pow.f64 b 3))))))) (/.f64 (*.f64 (*.f64 (*.f64 -2 a) a) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (*.f64 (*.f64 -2 -2) (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (pow.f64 c 2) (pow.f64 b 3)) (/.f64 (pow.f64 c 2) (pow.f64 b 3))))))) (/.f64 (*.f64 (*.f64 (*.f64 -2 a) a) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))))))) (/.f64 (*.f64 (*.f64 (*.f64 -2 a) a) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (*.f64 (/.f64 (pow.f64 a 3) b) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (Rewrite<= unpow2_binary64 (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2)))) (/.f64 (*.f64 (*.f64 (*.f64 -2 a) a) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 a 3) (/.f64 b (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))))) (/.f64 (*.f64 (*.f64 (*.f64 -2 a) a) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (/.f64 (*.f64 (*.f64 (*.f64 -2 a) a) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b) (/.f64 (*.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -2 (*.f64 a a))) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b) (/.f64 (*.f64 (*.f64 -2 (Rewrite<= unpow2_binary64 (pow.f64 a 2))) (pow.f64 c 3)) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b) (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -2 (*.f64 (pow.f64 a 2) (pow.f64 c 3)))) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b) (/.f64 (*.f64 -2 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)))) (pow.f64 b 5))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (fma.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b) (Rewrite<= associate-*r/_binary64 (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)))) (+.f64 (/.f64 c b) (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (pow.f64 b 3)) a))): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)))) (+.f64 (/.f64 c b) (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 c 2) (/.f64 (pow.f64 b 3) a))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)))) (+.f64 (/.f64 c b) (Rewrite<= associate-/l*_binary64 (/.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 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5)))) (Rewrite<= +-commutative_binary64 (+.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
   (Rewrite<= unsub-neg_binary64 (+.f64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.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
   (+.f64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) 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 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)) (/.f64 c b))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.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
   (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (*.f64 -1 (/.f64 c b))) (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) 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<= associate-+r+_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3))) (+.f64 (*.f64 -1 (/.f64 c b)) (+.f64 (*.f64 -1/4 (/.f64 (*.f64 (pow.f64 a 3) (+.f64 (*.f64 16 (/.f64 (pow.f64 c 4) (pow.f64 b 6))) (pow.f64 (*.f64 -2 (/.f64 (pow.f64 c 2) (pow.f64 b 3))) 2))) b)) (*.f64 -2 (/.f64 (*.f64 (pow.f64 c 3) (pow.f64 a 2)) (pow.f64 b 5))))))): 2 points increase in error, 1 points decrease in error

6. Applied egg-rr 5.9

   \[\leadsto \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), \frac{\left(\left(-2 \cdot a\right) \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right) - \left(\frac{c}{b} + \color{blue}{\left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right)} \cdot a\right) \]

7. Applied egg-rr 5.9

   \[\leadsto \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right)} \cdot 20\right), \frac{\left(\left(-2 \cdot a\right) \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right) - \left(\frac{c}{b} + \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a\right) \]

8. Final simplification 5.9

   \[\leadsto \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 20\right), \frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot {c}^{3}}{{b}^{5}}\right) - \left(\frac{c}{b} + a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right)\right) \]

Alternatives

Alternative 1
Error	6.6
Cost	28036

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -3:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\frac{-2 \cdot {c}^{3}}{\frac{{b}^{5}}{a}} - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]

Alternative 2
Error	10.2
Cost	13764

\[\begin{array}{l} \mathbf{if}\;b \leq 230:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c \cdot \left(-c\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]

Alternative 3
Error	10.2
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 230:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c \cdot \left(-c\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]

Alternative 4
Error	11.8
Cost	7232

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

Alternative 5
Error	11.9
Cost	832

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

Alternative 6
Error	22.9
Cost	512

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

Alternative 7
Error	22.9
Cost	256

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

Reproduce

herbie shell --seed 2022298 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))