Quadratic roots, narrow range

Average Error: 28.2 → 5.6

Time: 14.9s

Precision: binary64

Cost: 40836

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq 1.95:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, 4 \cdot \left(c \cdot a\right)\right) + \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}, \frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.95)
   (/
    (-
     (sqrt (+ (fma c (* a -4.0) (* 4.0 (* c a))) (fma b b (* -4.0 (* c a)))))
     b)
    (* a 2.0))
   (-
    (fma
     -0.25
     (* (/ (pow (* c a) 4.0) a) (/ 20.0 (pow b 7.0)))
     (- (/ (* -2.0 (* (pow c 3.0) (* a a))) (pow b 5.0)) (/ c b)))
    (* a (/ (* c c) (pow b 3.0))))))

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 tmp;
	if (b <= 1.95) {
		tmp = (sqrt((fma(c, (a * -4.0), (4.0 * (c * a))) + fma(b, b, (-4.0 * (c * a))))) - b) / (a * 2.0);
	} else {
		tmp = fma(-0.25, ((pow((c * a), 4.0) / a) * (20.0 / pow(b, 7.0))), (((-2.0 * (pow(c, 3.0) * (a * a))) / pow(b, 5.0)) - (c / b))) - (a * ((c * c) / pow(b, 3.0)));
	}
	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)
	tmp = 0.0
	if (b <= 1.95)
		tmp = Float64(Float64(sqrt(Float64(fma(c, Float64(a * -4.0), Float64(4.0 * Float64(c * a))) + fma(b, b, Float64(-4.0 * Float64(c * a))))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-0.25, Float64(Float64((Float64(c * a) ^ 4.0) / a) * Float64(20.0 / (b ^ 7.0))), Float64(Float64(Float64(-2.0 * Float64((c ^ 3.0) * Float64(a * a))) / (b ^ 5.0)) - Float64(c / b))) - Float64(a * Float64(Float64(c * c) / (b ^ 3.0))));
	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_] := If[LessEqual[b, 1.95], N[(N[(N[Sqrt[N[(N[(c * N[(a * -4.0), $MachinePrecision] + N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.94999999999999996

   1. Initial program 11.9

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

   2. Applied egg-rr 11.8

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

   if 1.94999999999999996 < b

   1. Initial program 31.7

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

   2. Taylor expanded in b around inf 4.2

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

   3. Simplified 4.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{\mathsf{fma}\left(16, {c}^{4} \cdot {a}^{4}, 4 \cdot \left({c}^{4} \cdot {a}^{4}\right)\right)}{a \cdot {b}^{7}}, \frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot c}{{b}^{3}} \cdot a} \]
      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 (*.f64 -2 (*.f64 (pow.f64 c 3) (*.f64 a a))) (pow.f64 b 5)) (/.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 (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 (*.f64 -2 (*.f64 (pow.f64 c 3) (*.f64 a a))) (pow.f64 b 5)) (/.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 (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 (*.f64 -2 (*.f64 (pow.f64 c 3) (*.f64 a a))) (pow.f64 b 5)) (/.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 (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 (*.f64 -2 (*.f64 (pow.f64 c 3) (*.f64 a a))) (pow.f64 b 5)) (/.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 (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 (*.f64 -2 (*.f64 (pow.f64 c 3) (*.f64 a a))) (pow.f64 b 5)) (/.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 (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 (*.f64 -2 (*.f64 (pow.f64 c 3) (*.f64 a a))) (pow.f64 b 5)) (/.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 (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 (*.f64 -2 (*.f64 (pow.f64 c 3) (*.f64 a a))) (pow.f64 b 5)) (/.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 (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 (*.f64 -2 (*.f64 (pow.f64 c 3) (*.f64 a a))) (pow.f64 b 5)) (/.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 (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 (*.f64 -2 (*.f64 (pow.f64 c 3) (*.f64 a a))) (pow.f64 b 5)) (/.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 (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 (*.f64 -2 (*.f64 (pow.f64 c 3) (*.f64 a a))) (pow.f64 b 5)) (/.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 (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 (*.f64 -2 (*.f64 (pow.f64 c 3) (*.f64 a a))) (pow.f64 b 5)) (/.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 (*.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 (*.f64 -2 (*.f64 (pow.f64 c 3) (Rewrite<= unpow2_binary64 (pow.f64 a 2)))) (pow.f64 b 5)) (/.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 (*.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 (*.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 (*.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 (*.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 (*.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 (*.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 (*.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 (*.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 (*.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 (*.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 (*.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)) (pow.f64 b 3)) a)): 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<= 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 (*.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<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b 3)))): 0 points increase in error, 1 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

   4. Taylor expanded in c around 0 4.2

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

   5. Simplified 4.2

      \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}}, \frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot c}{{b}^{3}} \cdot a \]
      Proof
      (*.f64 (/.f64 (pow.f64 (*.f64 c a) 4) a) (/.f64 20 (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (*.f64 c a) (Rewrite<= metadata-eval (*.f64 2 2))) a) (/.f64 20 (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (*.f64 c a) 2) (pow.f64 (*.f64 c a) 2))) a) (/.f64 20 (pow.f64 b 7))): 50 points increase in error, 49 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 (*.f64 c a) 2) (Rewrite=> unpow2_binary64 (*.f64 (*.f64 c a) (*.f64 c a)))) a) (/.f64 20 (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (Rewrite=> unpow2_binary64 (*.f64 (*.f64 c a) (*.f64 c a))) (*.f64 (*.f64 c a) (*.f64 c a))) a) (/.f64 20 (pow.f64 b 7))): 0 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (*.f64 (Rewrite=> swap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 a a))) (*.f64 (*.f64 c a) (*.f64 c a))) a) (/.f64 20 (pow.f64 b 7))): 64 points increase in error, 49 points decrease in error
      (*.f64 (/.f64 (*.f64 (*.f64 (*.f64 c c) (*.f64 a a)) (Rewrite=> swap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 a a)))) a) (/.f64 20 (pow.f64 b 7))): 56 points increase in error, 61 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 (*.f64 c c) (*.f64 c c)) (*.f64 (*.f64 a a) (*.f64 a a)))) a) (/.f64 20 (pow.f64 b 7))): 54 points increase in error, 52 points decrease in error
      (*.f64 (/.f64 (*.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (*.f64 c c) c) c)) (*.f64 (*.f64 a a) (*.f64 a a))) a) (/.f64 20 (pow.f64 b 7))): 37 points increase in error, 28 points decrease in error
      (*.f64 (/.f64 (*.f64 (*.f64 (Rewrite<= unpow3_binary64 (pow.f64 c 3)) c) (*.f64 (*.f64 a a) (*.f64 a a))) a) (/.f64 20 (pow.f64 b 7))): 22 points increase in error, 17 points decrease in error
      (*.f64 (/.f64 (*.f64 (Rewrite=> pow-plus_binary64 (pow.f64 c (+.f64 3 1))) (*.f64 (*.f64 a a) (*.f64 a a))) a) (/.f64 20 (pow.f64 b 7))): 28 points increase in error, 32 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 c (Rewrite=> metadata-eval 4)) (*.f64 (*.f64 a a) (*.f64 a a))) a) (/.f64 20 (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (*.f64 a a) a) a))) a) (/.f64 20 (pow.f64 b 7))): 29 points increase in error, 32 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (*.f64 (Rewrite<= unpow3_binary64 (pow.f64 a 3)) a)) a) (/.f64 20 (pow.f64 b 7))): 29 points increase in error, 32 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (Rewrite=> pow-plus_binary64 (pow.f64 a (+.f64 3 1)))) a) (/.f64 20 (pow.f64 b 7))): 25 points increase in error, 26 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 c 4) (pow.f64 a (Rewrite=> metadata-eval 4))) a) (/.f64 20 (pow.f64 b 7))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 c 4) (pow.f64 a 4)) 20) (*.f64 a (pow.f64 b 7)))): 71 points increase in error, 62 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))): 43 points increase in error, 50 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
3. Recombined 2 regimes into one program.

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.95:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, 4 \cdot \left(c \cdot a\right)\right) + \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}, \frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.9
Cost	20868

\[\begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 2
Error	6.9
Cost	20868

\[\begin{array}{l} \mathbf{if}\;b \leq 2.65:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, 4 \cdot \left(c \cdot a\right)\right) + \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 3
Error	9.7
Cost	13764

\[\begin{array}{l} \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c}{b} \cdot \frac{c \cdot a}{b \cdot b}\\ \end{array} \]

Alternative 4
Error	9.7
Cost	13764

\[\begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c}{b} \cdot \frac{c \cdot a}{b \cdot b}\\ \end{array} \]

Alternative 5
Error	9.7
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c}{b} \cdot \frac{c \cdot a}{b \cdot b}\\ \end{array} \]

Alternative 6
Error	12.1
Cost	1024

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

Alternative 7
Error	23.1
Cost	256

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

Alternative 8
Error	63.0
Cost	192

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

Reproduce

herbie shell --seed 2022330 
(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)))