Quadratic roots, medium range

Percentage Accurate: 32.0% → 99.2%

Time: 13.3s

Precision: binary64

Cost: 14400

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

↓

\[\frac{0 \cdot \left(b \cdot b\right) + \left(c \cdot a\right) \cdot -4}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (*
  (/
   (+ (* 0.0 (* b b)) (* (* c a) -4.0))
   (+ b (sqrt (fma c (* a -4.0) (* b b)))))
  (/ 0.5 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

↓

double code(double a, double b, double c) {
	return (((0.0 * (b * b)) + ((c * a) * -4.0)) / (b + sqrt(fma(c, (a * -4.0), (b * b))))) * (0.5 / a);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	return Float64(Float64(Float64(Float64(0.0 * Float64(b * b)) + Float64(Float64(c * a) * -4.0)) / Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * b))))) * Float64(0.5 / a))
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := N[(N[(N[(N[(0.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.6%

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

2. Simplified 30.6%

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

   [Start] 30.6%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   /-rgt-identity [<=] 30.6%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
   metadata-eval [<=] 30.6%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
   associate-/l* [<=] 30.6%	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
   associate-*r/ [<=] 30.6%	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]
   +-commutative [=>] 30.6%	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]
   unsub-neg [=>] 30.6%	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]
   fma-neg [=>] 30.6%	\[ \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]
   associate-*l* [=>] 30.6%	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
   *-commutative [=>] 30.6%	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
   distribute-rgt-neg-in [=>] 30.6%	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
   metadata-eval [=>] 30.6%	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
   associate-/r* [=>] 30.6%	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]
   metadata-eval [=>] 30.6%	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]
   metadata-eval [=>] 30.6%	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

3. Applied egg-rr 30.7%

   \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]
   Step-by-step derivation

   [Start] 30.6%	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a} \]
   fma-udef [=>] 30.6%	\[ \left(\sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -4}} - b\right) \cdot \frac{0.5}{a} \]
   *-commutative [=>] 30.6%	\[ \left(\sqrt{b \cdot b + \color{blue}{-4 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]
   metadata-eval [<=] 30.6%	\[ \left(\sqrt{b \cdot b + \color{blue}{\left(-4\right)} \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.5}{a} \]
   cancel-sign-sub-inv [<=] 30.6%	\[ \left(\sqrt{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]
   associate-*l* [<=] 30.6%	\[ \left(\sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}} - b\right) \cdot \frac{0.5}{a} \]
   *-un-lft-identity [=>] 30.6%	\[ \left(\sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{0.5}{a} \]
   prod-diff [=>] 30.6%	\[ \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right) + \mathsf{fma}\left(-\left(4 \cdot a\right) \cdot c, 1, \left(\left(4 \cdot a\right) \cdot c\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

4. Simplified 30.5%

   \[\leadsto \left(\sqrt{\color{blue}{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]
   Step-by-step derivation

   [Start] 30.7%	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]
   +-commutative [=>] 30.7%	\[ \left(\sqrt{\color{blue}{\mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]
   fma-udef [=>] 30.7%	\[ \left(\sqrt{\color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]
   *-rgt-identity [=>] 30.7%	\[ \left(\sqrt{\left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]
   *-rgt-identity [=>] 30.7%	\[ \left(\sqrt{\left(a \cdot \left(c \cdot -4\right) + \color{blue}{a \cdot \left(c \cdot -4\right)}\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]
   count-2 [=>] 30.7%	\[ \left(\sqrt{\color{blue}{2 \cdot \left(a \cdot \left(c \cdot -4\right)\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]
   *-commutative [=>] 30.7%	\[ \left(\sqrt{2 \cdot \color{blue}{\left(\left(c \cdot -4\right) \cdot a\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]
   *-commutative [=>] 30.7%	\[ \left(\sqrt{2 \cdot \left(\color{blue}{\left(-4 \cdot c\right)} \cdot a\right) + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]
   associate-*r* [<=] 30.7%	\[ \left(\sqrt{2 \cdot \color{blue}{\left(-4 \cdot \left(c \cdot a\right)\right)} + \mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]
   *-rgt-identity [=>] 30.7%	\[ \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -4\right)}\right)} - b\right) \cdot \frac{0.5}{a} \]
   fma-neg [<=] 30.5%	\[ \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right)}} - b\right) \cdot \frac{0.5}{a} \]
   *-commutative [=>] 30.5%	\[ \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - \color{blue}{\left(c \cdot -4\right) \cdot a}\right)} - b\right) \cdot \frac{0.5}{a} \]
   *-commutative [=>] 30.5%	\[ \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - \color{blue}{\left(-4 \cdot c\right)} \cdot a\right)} - b\right) \cdot \frac{0.5}{a} \]
   associate-*r* [<=] 30.5%	\[ \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - \color{blue}{-4 \cdot \left(c \cdot a\right)}\right)} - b\right) \cdot \frac{0.5}{a} \]

5. Applied egg-rr 31.3%

   \[\leadsto \color{blue}{\frac{\left(-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} + b}} \cdot \frac{0.5}{a} \]
   Step-by-step derivation

   [Start] 30.5%	\[ \left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} - b\right) \cdot \frac{0.5}{a} \]
   flip-- [=>] 30.3%	\[ \color{blue}{\frac{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} \cdot \sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}} \cdot \frac{0.5}{a} \]
   add-sqr-sqrt [<=] 31.3%	\[ \frac{\color{blue}{\left(2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right)} - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b} \cdot \frac{0.5}{a} \]
   associate-*r* [=>] 31.3%	\[ \frac{\left(\color{blue}{\left(2 \cdot -4\right) \cdot \left(c \cdot a\right)} + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b} \cdot \frac{0.5}{a} \]
   metadata-eval [=>] 31.3%	\[ \frac{\left(\color{blue}{-8} \cdot \left(c \cdot a\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b} \cdot \frac{0.5}{a} \]
   cancel-sign-sub-inv [=>] 31.3%	\[ \frac{\left(-8 \cdot \left(c \cdot a\right) + \color{blue}{\left(b \cdot b + \left(--4\right) \cdot \left(c \cdot a\right)\right)}\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b} \cdot \frac{0.5}{a} \]
   metadata-eval [=>] 31.3%	\[ \frac{\left(-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + \color{blue}{4} \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b} \cdot \frac{0.5}{a} \]

6. Simplified 31.5%

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

   [Start] 31.3%	\[ \frac{\left(-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} + b} \cdot \frac{0.5}{a} \]

7. Applied egg-rr 31.5%

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

   [Start] 31.5%	\[ \frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   sub-neg [=>] 31.5%	\[ \frac{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) + \left(-b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   *-commutative [=>] 31.5%	\[ \frac{\mathsf{fma}\left(c, \color{blue}{-4 \cdot a}, b \cdot b\right) + \left(-b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]

8. Simplified 99.3%

   \[\leadsto \frac{\color{blue}{0 \cdot \left(b \cdot b\right) + \left(c \cdot a\right) \cdot -4}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   Step-by-step derivation

   [Start] 31.5%	\[ \frac{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right) + \left(-b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   +-commutative [=>] 31.5%	\[ \frac{\color{blue}{\left(-b \cdot b\right) + \mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   fma-def [<=] 31.5%	\[ \frac{\left(-b \cdot b\right) + \color{blue}{\left(c \cdot \left(-4 \cdot a\right) + b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   +-commutative [=>] 31.5%	\[ \frac{\left(-b \cdot b\right) + \color{blue}{\left(b \cdot b + c \cdot \left(-4 \cdot a\right)\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   associate-+r+ [=>] 99.3%	\[ \frac{\color{blue}{\left(\left(-b \cdot b\right) + b \cdot b\right) + c \cdot \left(-4 \cdot a\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   neg-mul-1 [=>] 99.3%	\[ \frac{\left(\color{blue}{-1 \cdot \left(b \cdot b\right)} + b \cdot b\right) + c \cdot \left(-4 \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   distribute-lft1-in [=>] 99.3%	\[ \frac{\color{blue}{\left(-1 + 1\right) \cdot \left(b \cdot b\right)} + c \cdot \left(-4 \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   metadata-eval [=>] 99.3%	\[ \frac{\color{blue}{0} \cdot \left(b \cdot b\right) + c \cdot \left(-4 \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   associate-*r* [=>] 99.3%	\[ \frac{0 \cdot \left(b \cdot b\right) + \color{blue}{\left(c \cdot -4\right) \cdot a}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   rem-square-sqrt [<=] 0.0%	\[ \frac{0 \cdot \left(b \cdot b\right) + \left(c \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}\right) \cdot a}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   unpow2 [<=] 0.0%	\[ \frac{0 \cdot \left(b \cdot b\right) + \left(c \cdot \color{blue}{{\left(\sqrt{-4}\right)}^{2}}\right) \cdot a}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   *-commutative [=>] 0.0%	\[ \frac{0 \cdot \left(b \cdot b\right) + \color{blue}{\left({\left(\sqrt{-4}\right)}^{2} \cdot c\right)} \cdot a}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   unpow2 [=>] 0.0%	\[ \frac{0 \cdot \left(b \cdot b\right) + \left(\color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)} \cdot c\right) \cdot a}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   rem-square-sqrt [=>] 99.3%	\[ \frac{0 \cdot \left(b \cdot b\right) + \left(\color{blue}{-4} \cdot c\right) \cdot a}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   associate-*r* [<=] 99.3%	\[ \frac{0 \cdot \left(b \cdot b\right) + \color{blue}{-4 \cdot \left(c \cdot a\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]
   *-commutative [=>] 99.3%	\[ \frac{0 \cdot \left(b \cdot b\right) + \color{blue}{\left(c \cdot a\right) \cdot -4}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]

9. Final simplification 99.3%

   \[\leadsto \frac{0 \cdot \left(b \cdot b\right) + \left(c \cdot a\right) \cdot -4}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	14400

\[\frac{0 \cdot \left(b \cdot b\right) + \left(c \cdot a\right) \cdot -4}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a} \]

Alternative 2
Accuracy	90.9%
Cost	16452

\[\begin{array}{l} t_0 := \left(c \cdot a\right) \cdot -8 + \left(b \cdot b + \left(c \cdot a\right) \cdot 4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -1000:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{t_0 - b \cdot b}{b + \sqrt{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c \cdot c}{\frac{{b}^{3}}{a}} - \frac{c}{b}\\ \end{array} \]

Alternative 3
Accuracy	90.8%
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -1000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c \cdot c}{\frac{{b}^{3}}{a}} - \frac{c}{b}\\ \end{array} \]

Alternative 4
Accuracy	90.7%
Cost	7232

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

Alternative 5
Accuracy	90.7%
Cost	1344

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

Alternative 6
Accuracy	81.0%
Cost	256

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

Reproduce

herbie shell --seed 2023165 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))