Quadratic roots, narrow range

Average Error: 28.7 → 5.7

Time: 16.7s

Precision: binary64

Cost: 153220

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-c}{b}\\ t_1 := e^{t_0}\\ t_2 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.028:\\ \;\;\;\;\frac{\frac{b \cdot b - t_2}{\left(-b\right) - \sqrt{t_2}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{fma}\left(-1, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot t_1\right), t_1 \cdot \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \frac{{c}^{4}}{{b}^{6}}, -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + {a}^{3} \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{{c}^{6}}{{b}^{9}}, \mathsf{fma}\left(-0.25, \frac{{\left(\frac{c \cdot \left(c \cdot -2\right)}{{b}^{3}}\right)}^{2} + \frac{16}{\frac{{b}^{6}}{{c}^{4}}}}{b}, \frac{2}{\frac{{b}^{8}}{{c}^{5}}}\right)\right)\right)\right) + \mathsf{expm1}\left(t_0\right)\right)\\ \end{array} \]

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) b)) (t_1 (exp t_0)) (t_2 (fma c (* a -4.0) (* b b))))
   (if (<= (/ (- (sqrt (+ (* b b) (* c (* a -4.0)))) b) (* a 2.0)) -0.028)
     (/ (/ (- (* b b) t_2) (- (- b) (sqrt t_2))) (* a 2.0))
     (log1p
      (+
       (fma
        -1.0
        (* (/ c (/ (pow b 3.0) c)) (* a t_1))
        (*
         t_1
         (+
          (*
           (* a a)
           (fma
            0.5
            (/ (pow c 4.0) (pow b 6.0))
            (* -2.0 (/ (pow c 3.0) (pow b 5.0)))))
          (*
           (pow a 3.0)
           (fma
            -0.16666666666666666
            (/ (pow c 6.0) (pow b 9.0))
            (fma
             -0.25
             (/
              (+
               (pow (/ (* c (* c -2.0)) (pow b 3.0)) 2.0)
               (/ 16.0 (/ (pow b 6.0) (pow c 4.0))))
              b)
             (/ 2.0 (/ (pow b 8.0) (pow c 5.0)))))))))
       (expm1 t_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 t_0 = -c / b;
	double t_1 = exp(t_0);
	double t_2 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a * 2.0)) <= -0.028) {
		tmp = (((b * b) - t_2) / (-b - sqrt(t_2))) / (a * 2.0);
	} else {
		tmp = log1p((fma(-1.0, ((c / (pow(b, 3.0) / c)) * (a * t_1)), (t_1 * (((a * a) * fma(0.5, (pow(c, 4.0) / pow(b, 6.0)), (-2.0 * (pow(c, 3.0) / pow(b, 5.0))))) + (pow(a, 3.0) * fma(-0.16666666666666666, (pow(c, 6.0) / pow(b, 9.0)), fma(-0.25, ((pow(((c * (c * -2.0)) / pow(b, 3.0)), 2.0) + (16.0 / (pow(b, 6.0) / pow(c, 4.0)))) / b), (2.0 / (pow(b, 8.0) / pow(c, 5.0))))))))) + expm1(t_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)
	t_0 = Float64(Float64(-c) / b)
	t_1 = exp(t_0)
	t_2 = fma(c, Float64(a * -4.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0)) <= -0.028)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_2) / Float64(Float64(-b) - sqrt(t_2))) / Float64(a * 2.0));
	else
		tmp = log1p(Float64(fma(-1.0, Float64(Float64(c / Float64((b ^ 3.0) / c)) * Float64(a * t_1)), Float64(t_1 * Float64(Float64(Float64(a * a) * fma(0.5, Float64((c ^ 4.0) / (b ^ 6.0)), Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))))) + Float64((a ^ 3.0) * fma(-0.16666666666666666, Float64((c ^ 6.0) / (b ^ 9.0)), fma(-0.25, Float64(Float64((Float64(Float64(c * Float64(c * -2.0)) / (b ^ 3.0)) ^ 2.0) + Float64(16.0 / Float64((b ^ 6.0) / (c ^ 4.0)))) / b), Float64(2.0 / Float64((b ^ 8.0) / (c ^ 5.0))))))))) + expm1(t_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_] := Block[{t$95$0 = N[((-c) / b), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.028], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$2), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(N[(-1.0 * N[(N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * N[(a * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 3.0], $MachinePrecision] * N[(-0.16666666666666666 * N[(N[Power[c, 6.0], $MachinePrecision] / N[Power[b, 9.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(N[Power[N[(N[(c * N[(c * -2.0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(16.0 / N[(N[Power[b, 6.0], $MachinePrecision] / N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(2.0 / N[(N[Power[b, 8.0], $MachinePrecision] / N[Power[c, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Exp[t$95$0] - 1), $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)) < -0.0280000000000000006

   1. Initial program 13.6

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

   2. Simplified 13.6

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

      [Start] 13.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 13.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Applied egg-rr 12.9

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

   4. Simplified 12.6

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

      [Start] 12.9	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
      associate-/l/ [=>] 12.9	\[ \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}}{a \cdot 2} \]
      fma-def [<=] 12.6	\[ \frac{\frac{b \cdot b - \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
      +-commutative [=>] 12.6	\[ \frac{\frac{b \cdot b - \color{blue}{\left(c \cdot \left(a \cdot -4\right) + b \cdot b\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
      fma-def [=>] 12.6	\[ \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
      distribute-lft-neg-in [<=] 12.6	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{\color{blue}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}}{a \cdot 2} \]
      rem-square-sqrt [=>] 12.6	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}}{a \cdot 2} \]
      fma-def [<=] 12.6	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}\right)}}{a \cdot 2} \]
      +-commutative [=>] 12.6	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}\right)}}{a \cdot 2} \]
      fma-def [=>] 12.6	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\right)}}{a \cdot 2} \]

   if -0.0280000000000000006 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 33.7

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

   2. Simplified 33.7

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

      [Start] 33.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 33.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Applied egg-rr 33.7

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\right)\right)} \]

   4. Taylor expanded in a around 0 21.7

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(e^{-1 \cdot \frac{c}{b}} + \left(-1 \cdot \frac{{c}^{2} \cdot \left(e^{-1 \cdot \frac{c}{b}} \cdot a\right)}{{b}^{3}} + \left(e^{-1 \cdot \frac{c}{b}} \cdot \left({a}^{3} \cdot \left(-0.16666666666666666 \cdot \frac{{c}^{6}}{{b}^{9}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}}{b} + 2 \cdot \frac{{c}^{5}}{{b}^{8}}\right)\right)\right) + e^{-1 \cdot \frac{c}{b}} \cdot \left({a}^{2} \cdot \left(0.5 \cdot \frac{{c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right)\right)\right) - 1}\right) \]

   5. Simplified 3.4

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(-1, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot e^{-\frac{c}{b}}\right), e^{-\frac{c}{b}} \cdot \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \frac{{c}^{4}}{{b}^{6}}, -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + {a}^{3} \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{{c}^{6}}{{b}^{9}}, \mathsf{fma}\left(-0.25, \frac{{\left(\frac{\left(-2 \cdot c\right) \cdot c}{{b}^{3}}\right)}^{2} + \frac{16}{\frac{{b}^{6}}{{c}^{4}}}}{b}, \frac{2}{\frac{{b}^{8}}{{c}^{5}}}\right)\right)\right)\right) + \mathsf{expm1}\left(-\frac{c}{b}\right)}\right) \]
      Proof

      [Start] 21.7

      \[ \mathsf{log1p}\left(\left(e^{-1 \cdot \frac{c}{b}} + \left(-1 \cdot \frac{{c}^{2} \cdot \left(e^{-1 \cdot \frac{c}{b}} \cdot a\right)}{{b}^{3}} + \left(e^{-1 \cdot \frac{c}{b}} \cdot \left({a}^{3} \cdot \left(-0.16666666666666666 \cdot \frac{{c}^{6}}{{b}^{9}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}}{b} + 2 \cdot \frac{{c}^{5}}{{b}^{8}}\right)\right)\right) + e^{-1 \cdot \frac{c}{b}} \cdot \left({a}^{2} \cdot \left(0.5 \cdot \frac{{c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right)\right)\right) - 1\right) \]

      +-commutative [=>] 21.7

      \[ \mathsf{log1p}\left(\color{blue}{\left(\left(-1 \cdot \frac{{c}^{2} \cdot \left(e^{-1 \cdot \frac{c}{b}} \cdot a\right)}{{b}^{3}} + \left(e^{-1 \cdot \frac{c}{b}} \cdot \left({a}^{3} \cdot \left(-0.16666666666666666 \cdot \frac{{c}^{6}}{{b}^{9}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}}{b} + 2 \cdot \frac{{c}^{5}}{{b}^{8}}\right)\right)\right) + e^{-1 \cdot \frac{c}{b}} \cdot \left({a}^{2} \cdot \left(0.5 \cdot \frac{{c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right)\right) + e^{-1 \cdot \frac{c}{b}}\right)} - 1\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.028:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{fma}\left(-1, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot e^{\frac{-c}{b}}\right), e^{\frac{-c}{b}} \cdot \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \frac{{c}^{4}}{{b}^{6}}, -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + {a}^{3} \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{{c}^{6}}{{b}^{9}}, \mathsf{fma}\left(-0.25, \frac{{\left(\frac{c \cdot \left(c \cdot -2\right)}{{b}^{3}}\right)}^{2} + \frac{16}{\frac{{b}^{6}}{{c}^{4}}}}{b}, \frac{2}{\frac{{b}^{8}}{{c}^{5}}}\right)\right)\right)\right) + \mathsf{expm1}\left(\frac{-c}{b}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	5.6
Cost	48260

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.028:\\ \;\;\;\;\frac{\frac{b \cdot b - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, \frac{{c}^{3} \cdot \left(a \cdot \left(a \cdot -2\right)\right)}{{b}^{5}}\right) - \frac{c}{b}\right) - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right)\\ \end{array} \]

Alternative 2
Error	6.7
Cost	28292

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.025:\\ \;\;\;\;\frac{\frac{b \cdot b - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{c}^{3} \cdot \left(a \cdot \left(a \cdot -2\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{\frac{\frac{a}{b}}{b}}{\frac{\frac{b}{c}}{c}}\\ \end{array} \]

Alternative 3
Error	6.8
Cost	28228

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2} \leq -0.025:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{\mathsf{fma}\left(b, b, t_0\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{c}^{3} \cdot \left(a \cdot \left(a \cdot -2\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{\frac{\frac{a}{b}}{b}}{\frac{\frac{b}{c}}{c}}\\ \end{array} \]

Alternative 4
Error	6.9
Cost	21956

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.028:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{c}^{3} \cdot \left(a \cdot \left(a \cdot -2\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{\frac{\frac{a}{b}}{b}}{\frac{\frac{b}{c}}{c}}\\ \end{array} \]

Alternative 5
Error	9.3
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.025:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}\\ \end{array} \]

Alternative 6
Error	9.6
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 60:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}\\ \end{array} \]

Alternative 7
Error	11.7
Cost	7232

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

Alternative 8
Error	11.8
Cost	1600

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

Alternative 9
Error	11.8
Cost	1600

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

Alternative 10
Error	22.6
Cost	256

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

Alternative 11
Error	63.0
Cost	192

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

Reproduce

herbie shell --seed 2023034 
(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)))