Quadratic roots, narrow range

Average Error: 28.7 → 5.0

Time: 25.7s

Precision: binary64

Cost: 60868

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

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2} \leq -10:\\ \;\;\;\;\sqrt[3]{{\left(\mathsf{fma}\left({\left(\mathsf{fma}\left(b, b, t_0\right)\right)}^{0.5}, \frac{0.5}{a}, b \cdot \frac{-0.5}{a}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a -4.0))))
   (if (<= (/ (- (sqrt (+ (* b b) t_0)) b) (* a 2.0)) -10.0)
     (cbrt (pow (fma (pow (fma b b t_0) 0.5) (/ 0.5 a) (* b (/ -0.5 a))) 3.0))
     (-
      (fma
       -0.25
       (* (/ (pow a 3.0) b) (* (/ (pow c 4.0) (pow b 6.0)) 20.0))
       (* -2.0 (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))))
      (fma (/ (* c c) (pow b 3.0)) 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) {
	double t_0 = c * (a * -4.0);
	double tmp;
	if (((sqrt(((b * b) + t_0)) - b) / (a * 2.0)) <= -10.0) {
		tmp = cbrt(pow(fma(pow(fma(b, b, t_0), 0.5), (0.5 / a), (b * (-0.5 / a))), 3.0));
	} else {
		tmp = fma(-0.25, ((pow(a, 3.0) / b) * ((pow(c, 4.0) / pow(b, 6.0)) * 20.0)), (-2.0 * (pow(c, 3.0) / (pow(b, 5.0) / (a * a))))) - fma(((c * c) / pow(b, 3.0)), a, (c / b));
	}
	return tmp;
}

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)
	t_0 = Float64(c * Float64(a * -4.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) + t_0)) - b) / Float64(a * 2.0)) <= -10.0)
		tmp = cbrt((fma((fma(b, b, t_0) ^ 0.5), Float64(0.5 / a), Float64(b * Float64(-0.5 / a))) ^ 3.0));
	else
		tmp = Float64(fma(-0.25, Float64(Float64((a ^ 3.0) / b) * Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 20.0)), Float64(-2.0 * Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))))) - fma(Float64(Float64(c * c) / (b ^ 3.0)), a, Float64(c / b)));
	end
	return tmp
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_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -10.0], N[Power[N[Power[N[(N[Power[N[(b * b + t$95$0), $MachinePrecision], 0.5], $MachinePrecision] * N[(0.5 / a), $MachinePrecision] + N[(b * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], N[(N[(-0.25 * N[(N[(N[Power[a, 3.0], $MachinePrecision] / b), $MachinePrecision] * N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * a + N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -10

   1. Initial program 9.8

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

   2. Simplified 9.8

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}} \]
      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)): 2 points increase in error, 1 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. Applied egg-rr 10.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}^{0.25}}{a}, \frac{{\left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}^{0.25}}{2}, -b \cdot \frac{0.5}{a}\right)} \]

   4. Simplified 10.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.25}}{a}, \frac{{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.25}}{2}, b \cdot \frac{-0.5}{a}\right)} \]
      Proof
      (fma.f64 (/.f64 (pow.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b)) 1/4) 2) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (pow.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (*.f64 c -4)) (*.f64 b b))) 1/4) a) (/.f64 (pow.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b)) 1/4) 2) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 1 points decrease in error
      (fma.f64 (/.f64 (pow.f64 (+.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 a c) -4)) (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b)) 1/4) 2) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (pow.f64 (Rewrite<= fma-udef_binary64 (fma.f64 (*.f64 a c) -4 (*.f64 b b))) 1/4) a) (/.f64 (pow.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b)) 1/4) 2) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (pow.f64 (fma.f64 (*.f64 a c) -4 (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (*.f64 c -4)) (*.f64 b b))) 1/4) 2) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 1 points decrease in error
      (fma.f64 (/.f64 (pow.f64 (fma.f64 (*.f64 a c) -4 (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (+.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 a c) -4)) (*.f64 b b)) 1/4) 2) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (pow.f64 (fma.f64 (*.f64 a c) -4 (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (Rewrite<= fma-udef_binary64 (fma.f64 (*.f64 a c) -4 (*.f64 b b))) 1/4) 2) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (pow.f64 (fma.f64 (*.f64 a c) -4 (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 (*.f64 a c) -4 (*.f64 b b)) 1/4) 2) (*.f64 b (/.f64 (Rewrite<= metadata-eval (neg.f64 1/2)) a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (pow.f64 (fma.f64 (*.f64 a c) -4 (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 (*.f64 a c) -4 (*.f64 b b)) 1/4) 2) (*.f64 b (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1/2 a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (pow.f64 (fma.f64 (*.f64 a c) -4 (*.f64 b b)) 1/4) a) (/.f64 (pow.f64 (fma.f64 (*.f64 a c) -4 (*.f64 b b)) 1/4) 2) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 b (/.f64 1/2 a))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 11.3

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{{\left(\frac{{\left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}^{0.25}}{a}\right)}^{3}}}, \frac{{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.25}}{2}, b \cdot \frac{-0.5}{a}\right) \]

   6. Applied egg-rr 10.4

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{b \cdot -0.5}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a \cdot 2}\right)}^{3}}} \]

   7. Applied egg-rr 10.9

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left({\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{0.25}, {\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{0.25} \cdot \frac{0.5}{a}, -0.5 \cdot \frac{b}{a}\right)\right)}}^{3}} \]

   8. Simplified 10.2

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.5}, \frac{0.5}{a}, b \cdot \frac{-0.5}{a}\right)\right)}}^{3}} \]
      Proof
      (fma.f64 (pow.f64 (fma.f64 b b (*.f64 c (*.f64 a -4))) 1/2) (/.f64 1/2 a) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 (fma.f64 b b (*.f64 c (*.f64 a (Rewrite<= metadata-eval (*.f64 -2 2))))) 1/2) (/.f64 1/2 a) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 (fma.f64 b b (*.f64 c (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a -2) 2)))) 1/2) (/.f64 1/2 a) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 (fma.f64 b b (*.f64 c (Rewrite=> *-commutative_binary64 (*.f64 2 (*.f64 a -2))))) 1/2) (/.f64 1/2 a) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 (fma.f64 b b (*.f64 c (*.f64 2 (Rewrite=> *-commutative_binary64 (*.f64 -2 a))))) 1/2) (/.f64 1/2 a) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 (fma.f64 b b (*.f64 c (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 2 -2) a)))) 1/2) (/.f64 1/2 a) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 (fma.f64 b b (*.f64 c (*.f64 (Rewrite=> metadata-eval -4) a))) 1/2) (/.f64 1/2 a) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 (fma.f64 b b (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 c -4) a))) 1/2) (/.f64 1/2 a) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 (fma.f64 b b (Rewrite<= *-commutative_binary64 (*.f64 a (*.f64 c -4)))) 1/2) (/.f64 1/2 a) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) (Rewrite<= metadata-eval (*.f64 2 1/4))) (/.f64 1/2 a) (*.f64 b (/.f64 -1/2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) 1/4) (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) 1/4))) (/.f64 1/2 a) (*.f64 b (/.f64 -1/2 a))): 91 points increase in error, 26 points decrease in error
      (fma.f64 (*.f64 (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) 1/4) (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) 1/4)) (/.f64 1/2 a) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 b -1/2) a))): 34 points increase in error, 34 points decrease in error
      (fma.f64 (*.f64 (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) 1/4) (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) 1/4)) (/.f64 1/2 a) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/2 b)) a)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) 1/4) (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) 1/4)) (/.f64 1/2 a) (Rewrite<= associate-*r/_binary64 (*.f64 -1/2 (/.f64 b a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) 1/4) (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) 1/4)) (/.f64 1/2 a)) (*.f64 -1/2 (/.f64 b a)))): 157 points increase in error, 92 points decrease in error
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) 1/4) (*.f64 (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) 1/4) (/.f64 1/2 a)))) (*.f64 -1/2 (/.f64 b a))): 53 points increase in error, 42 points decrease in error
      (Rewrite<= fma-udef_binary64 (fma.f64 (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) 1/4) (*.f64 (pow.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))) 1/4) (/.f64 1/2 a)) (*.f64 -1/2 (/.f64 b a)))): 95 points increase in error, 155 points decrease in error

   if -10 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 30.7

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

   2. Simplified 30.7

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}} \]
      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)): 2 points increase in error, 1 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 a around 0 4.4

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

   4. Simplified 4.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)} \]
      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 -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
      (-.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 -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
      (-.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 -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
      (-.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 -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
      (-.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 -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
      (-.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 -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
      (-.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 -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
      (-.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 -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
      (-.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 -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
      (-.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 -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
      (-.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 -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
      (-.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 -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
      (-.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 -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
      (-.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 -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
      (-.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 -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
      (-.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))))) (fma.f64 (/.f64 (*.f64 c c) (pow.f64 b 3)) a (/.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)))) (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
      (-.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<= 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
      (-.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 (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
      (-.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 (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
      (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))))))): 1 points increase in error, 3 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -10:\\ \;\;\;\;\sqrt[3]{{\left(\mathsf{fma}\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.5}, \frac{0.5}{a}, b \cdot \frac{-0.5}{a}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	5.0
Cost	48580

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2} \leq -10:\\ \;\;\;\;\sqrt[3]{{\left(\mathsf{fma}\left({\left(\mathsf{fma}\left(b, b, t_0\right)\right)}^{0.5}, \frac{0.5}{a}, b \cdot \frac{-0.5}{a}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(20 \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right)\right), -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \mathsf{fma}\left(\frac{c}{b} \cdot \frac{c}{b \cdot b}, a, \frac{c}{b}\right)\\ \end{array} \]

Alternative 2
Error	6.5
Cost	28164

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2} \leq -1.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 3
Error	9.5
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.007:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 4
Error	9.5
Cost	21060

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2} \leq -0.007:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 5
Error	9.5
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.007:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 6
Error	9.5
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 3.45:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 7
Error	11.6
Cost	7232

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

Alternative 8
Error	11.8
Cost	1728

\[\frac{-2 \cdot \left(a \cdot \left(c \cdot \frac{1}{b}\right) + \frac{c \cdot c}{\frac{b}{a} \cdot \frac{b \cdot b}{a}}\right)}{a \cdot 2} \]

Alternative 9
Error	11.8
Cost	1600

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

Alternative 10
Error	22.7
Cost	256

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

Alternative 11
Error	62.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022318 
(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)))