Quadratic roots, narrow range

Average Accuracy: 55.5% → 92.0%

Time: 20.7s

Precision: binary64

Cost: 69124

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -4.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left({c}^{3} \cdot \left(-4 \cdot \frac{a \cdot a}{{b}^{5}}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\left(b \cdot \frac{a \cdot a}{{b}^{4}}\right) \cdot -1.5}{a}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b}\\ \end{array} \]

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 (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -4.5)
   (* (- (sqrt (fma b b (* (* a c) -4.0))) b) (/ 0.5 a))
   (-
    (*
     0.5
     (+
      (+
       (* (pow c 3.0) (* -4.0 (/ (* a a) (pow b 5.0))))
       (*
        (* c c)
        (+
         (/ a (pow b 3.0))
         (* 2.0 (/ (* (* b (/ (* a a) (pow b 4.0))) -1.5) a)))))
      (*
       (pow c 4.0)
       (+
        (/ (* (/ (pow a 4.0) (pow b 8.0)) 2.25) (/ a b))
        (* -12.25 (/ (pow a 3.0) (pow b 7.0)))))))
    (/ 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 tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -4.5) {
		tmp = (sqrt(fma(b, b, ((a * c) * -4.0))) - b) * (0.5 / a);
	} else {
		tmp = (0.5 * (((pow(c, 3.0) * (-4.0 * ((a * a) / pow(b, 5.0)))) + ((c * c) * ((a / pow(b, 3.0)) + (2.0 * (((b * ((a * a) / pow(b, 4.0))) * -1.5) / a))))) + (pow(c, 4.0) * ((((pow(a, 4.0) / pow(b, 8.0)) * 2.25) / (a / b)) + (-12.25 * (pow(a, 3.0) / pow(b, 7.0))))))) - (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)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -4.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * c) * -4.0))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(Float64((c ^ 3.0) * Float64(-4.0 * Float64(Float64(a * a) / (b ^ 5.0)))) + Float64(Float64(c * c) * Float64(Float64(a / (b ^ 3.0)) + Float64(2.0 * Float64(Float64(Float64(b * Float64(Float64(a * a) / (b ^ 4.0))) * -1.5) / a))))) + Float64((c ^ 4.0) * Float64(Float64(Float64(Float64((a ^ 4.0) / (b ^ 8.0)) * 2.25) / Float64(a / b)) + Float64(-12.25 * Float64((a ^ 3.0) / (b ^ 7.0))))))) - 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_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -4.5], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(-4.0 * N[(N[(a * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[(b * N[(N[(a * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(N[(N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[b, 8.0], $MachinePrecision]), $MachinePrecision] * 2.25), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(-12.25 * N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $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)) < -4.5

   1. Initial program 88.0%

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

   2. Simplified 88.2%

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

      [Start] 88.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      /-rgt-identity [<=] 88.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 88.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [<=] 88.0	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
      associate-*r/ [<=] 88.1	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]
      +-commutative [=>] 88.1	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]
      unsub-neg [=>] 88.1	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]
      fma-neg [=>] 88.2	\[ \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      associate-*l* [=>] 88.2	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      *-commutative [=>] 88.2	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      distribute-rgt-neg-in [=>] 88.2	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      metadata-eval [=>] 88.2	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
      associate-/r* [=>] 88.2	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]
      metadata-eval [=>] 88.2	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]
      metadata-eval [=>] 88.2	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   if -4.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 48.8%

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

   2. Simplified 49.0%

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

      [Start] 48.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 48.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]
      +-commutative [=>] 48.8	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]
      unsub-neg [=>] 48.8	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]
      fma-neg [=>] 49.0	\[ \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]
      associate-*l* [=>] 49.0	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b}{a \cdot 2} \]
      *-commutative [=>] 49.0	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b}{a \cdot 2} \]
      distribute-rgt-neg-in [=>] 49.0	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b}{a \cdot 2} \]
      metadata-eval [=>] 49.0	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b}{a \cdot 2} \]

   3. Applied egg-rr 48.1%

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

      [Start] 49.0	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}{a \cdot 2} \]
      add-sqr-sqrt [=>] 48.1	\[ \frac{\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}} - b}{a \cdot 2} \]
      pow2 [=>] 48.1	\[ \frac{\color{blue}{{\left(\sqrt{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}\right)}^{2}} - b}{a \cdot 2} \]
      pow1/2 [=>] 48.1	\[ \frac{{\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}^{0.5}}}\right)}^{2} - b}{a \cdot 2} \]
      sqrt-pow1 [=>] 48.2	\[ \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} - b}{a \cdot 2} \]
      fma-udef [=>] 48.1	\[ \frac{{\left({\color{blue}{\left(b \cdot b + \left(a \cdot c\right) \cdot -4\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b}{a \cdot 2} \]
      +-commutative [=>] 48.1	\[ \frac{{\left({\color{blue}{\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b}{a \cdot 2} \]
      associate-*l* [=>] 48.1	\[ \frac{{\left({\left(\color{blue}{a \cdot \left(c \cdot -4\right)} + b \cdot b\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b}{a \cdot 2} \]
      fma-def [=>] 48.1	\[ \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b}{a \cdot 2} \]
      metadata-eval [=>] 48.1	\[ \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{\color{blue}{0.25}}\right)}^{2} - b}{a \cdot 2} \]

   4. Taylor expanded in c around 0 94.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + \left(0.5 \cdot \left({c}^{4} \cdot \left(\frac{{\left(0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right)}^{2} \cdot b}{a} + \left(-2 \cdot \frac{-0.16666666666666666 \cdot \frac{{a}^{3}}{{b}^{6}} + \left(2 \cdot \frac{{a}^{3}}{{b}^{6}} + -5.333333333333333 \cdot \frac{{a}^{3}}{{b}^{6}}\right)}{b} + 2 \cdot \frac{b \cdot \left(2 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(-1 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(5.333333333333333 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(-16 \cdot \frac{{a}^{4}}{{b}^{8}} + 0.041666666666666664 \cdot \frac{{a}^{4}}{{b}^{8}}\right)\right)\right)\right)}{a}\right)\right)\right) + \left(0.5 \cdot \left({c}^{3} \cdot \left(-2 \cdot \frac{0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}}{b} + 2 \cdot \frac{\left(-0.16666666666666666 \cdot \frac{{a}^{3}}{{b}^{6}} + \left(2 \cdot \frac{{a}^{3}}{{b}^{6}} + -5.333333333333333 \cdot \frac{{a}^{3}}{{b}^{6}}\right)\right) \cdot b}{a}\right)\right) + 0.5 \cdot \left({c}^{2} \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\left(0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) \cdot b}{a}\right)\right)\right)\right)} \]

   5. Simplified 94.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left({c}^{3} \cdot \mathsf{fma}\left(-2, \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{b}, 2 \cdot \frac{\frac{{a}^{3}}{{b}^{6}} \cdot -3.5}{\frac{a}{b}}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\left(b \cdot \frac{a \cdot a}{{b}^{4}}\right) \cdot -1.5}{a}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + \mathsf{fma}\left(-2, \frac{\frac{{a}^{3}}{{b}^{6}} \cdot -3.5}{b}, 2 \cdot \frac{b}{\frac{a}{\frac{{a}^{4}}{{b}^{8}} \cdot 1 + \frac{{a}^{4}}{{b}^{8}} \cdot -10.625}}\right)\right)\right) - \frac{c}{b}} \]
      Step-by-step derivation

      [Start] 94.6

      \[ -1 \cdot \frac{c}{b} + \left(0.5 \cdot \left({c}^{4} \cdot \left(\frac{{\left(0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right)}^{2} \cdot b}{a} + \left(-2 \cdot \frac{-0.16666666666666666 \cdot \frac{{a}^{3}}{{b}^{6}} + \left(2 \cdot \frac{{a}^{3}}{{b}^{6}} + -5.333333333333333 \cdot \frac{{a}^{3}}{{b}^{6}}\right)}{b} + 2 \cdot \frac{b \cdot \left(2 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(-1 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(5.333333333333333 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(-16 \cdot \frac{{a}^{4}}{{b}^{8}} + 0.041666666666666664 \cdot \frac{{a}^{4}}{{b}^{8}}\right)\right)\right)\right)}{a}\right)\right)\right) + \left(0.5 \cdot \left({c}^{3} \cdot \left(-2 \cdot \frac{0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}}{b} + 2 \cdot \frac{\left(-0.16666666666666666 \cdot \frac{{a}^{3}}{{b}^{6}} + \left(2 \cdot \frac{{a}^{3}}{{b}^{6}} + -5.333333333333333 \cdot \frac{{a}^{3}}{{b}^{6}}\right)\right) \cdot b}{a}\right)\right) + 0.5 \cdot \left({c}^{2} \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\left(0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) \cdot b}{a}\right)\right)\right)\right) \]

   6. Taylor expanded in a around 0 94.6%

      \[\leadsto 0.5 \cdot \left(\left({c}^{3} \cdot \mathsf{fma}\left(-2, \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{b}, 2 \cdot \frac{\frac{{a}^{3}}{{b}^{6}} \cdot -3.5}{\frac{a}{b}}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\left(b \cdot \frac{a \cdot a}{{b}^{4}}\right) \cdot -1.5}{a}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + \color{blue}{-12.25 \cdot \frac{{a}^{3}}{{b}^{7}}}\right)\right) - \frac{c}{b} \]

   7. Taylor expanded in a around 0 94.6%

      \[\leadsto 0.5 \cdot \left(\left({c}^{3} \cdot \color{blue}{\left(-4 \cdot \frac{{a}^{2}}{{b}^{5}}\right)} + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\left(b \cdot \frac{a \cdot a}{{b}^{4}}\right) \cdot -1.5}{a}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b} \]

   8. Simplified 94.6%

      \[\leadsto 0.5 \cdot \left(\left({c}^{3} \cdot \color{blue}{\left(-4 \cdot \frac{a \cdot a}{{b}^{5}}\right)} + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\left(b \cdot \frac{a \cdot a}{{b}^{4}}\right) \cdot -1.5}{a}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b} \]
      Step-by-step derivation

      [Start] 94.6	\[ 0.5 \cdot \left(\left({c}^{3} \cdot \left(-4 \cdot \frac{{a}^{2}}{{b}^{5}}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\left(b \cdot \frac{a \cdot a}{{b}^{4}}\right) \cdot -1.5}{a}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b} \]
      unpow2 [=>] 94.6	\[ 0.5 \cdot \left(\left({c}^{3} \cdot \left(-4 \cdot \frac{\color{blue}{a \cdot a}}{{b}^{5}}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\left(b \cdot \frac{a \cdot a}{{b}^{4}}\right) \cdot -1.5}{a}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b} \]

3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -4.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left({c}^{3} \cdot \left(-4 \cdot \frac{a \cdot a}{{b}^{5}}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\left(b \cdot \frac{a \cdot a}{{b}^{4}}\right) \cdot -1.5}{a}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	92.0%
Cost	54660

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -4.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), -2 \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\\ \end{array} \]

Alternative 2
Accuracy	89.7%
Cost	28164

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -4.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 3
Accuracy	84.9%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;b \leq 0.355:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 4
Accuracy	84.9%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;b \leq 0.355:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 5
Accuracy	84.9%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 0.355:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 6
Accuracy	84.9%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 0.355:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 7
Accuracy	81.9%
Cost	7232

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

Alternative 8
Accuracy	81.6%
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
Accuracy	64.4%
Cost	256

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

Alternative 10
Accuracy	3.2%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023157 
(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)))