Quadratic roots, medium range

Average Error: 67.97% → 3.9%

Time: 16.6s

Precision: binary64

Cost: 48260

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.08:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{b + \sqrt{t_0}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - t_0}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, \frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right)\\ \end{array} \]

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 (fma c (* a -4.0) (* b b))))
   (if (<= (/ (- (sqrt (+ (* b b) (* c (* a -4.0)))) b) (* a 2.0)) -0.08)
     (/
      (*
       (/ (* c (* 4.0 a)) (+ b (sqrt t_0)))
       (/ (- (* b b) (fma b b (* a (* 4.0 c)))) (- (* b b) t_0)))
      (* a 2.0))
     (-
      (-
       (fma
        -0.25
        (/ (pow a 3.0) (/ (pow b 7.0) (* (pow c 4.0) 20.0)))
        (/ (* (* a (* a -2.0)) (pow c 3.0)) (pow b 5.0)))
       (/ c b))
      (* a (* (/ c b) (/ c (* b 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 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a * 2.0)) <= -0.08) {
		tmp = (((c * (4.0 * a)) / (b + sqrt(t_0))) * (((b * b) - fma(b, b, (a * (4.0 * c)))) / ((b * b) - t_0))) / (a * 2.0);
	} else {
		tmp = (fma(-0.25, (pow(a, 3.0) / (pow(b, 7.0) / (pow(c, 4.0) * 20.0))), (((a * (a * -2.0)) * pow(c, 3.0)) / pow(b, 5.0))) - (c / b)) - (a * ((c / b) * (c / (b * 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 = 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)) <= -0.08)
		tmp = Float64(Float64(Float64(Float64(c * Float64(4.0 * a)) / Float64(b + sqrt(t_0))) * Float64(Float64(Float64(b * b) - fma(b, b, Float64(a * Float64(4.0 * c)))) / Float64(Float64(b * b) - t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(-0.25, Float64((a ^ 3.0) / Float64((b ^ 7.0) / Float64((c ^ 4.0) * 20.0))), Float64(Float64(Float64(a * Float64(a * -2.0)) * (c ^ 3.0)) / (b ^ 5.0))) - Float64(c / b)) - Float64(a * Float64(Float64(c / b) * Float64(c / Float64(b * 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] + 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], -0.08], N[(N[(N[(N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b * b), $MachinePrecision] - N[(b * b + N[(a * N[(4.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * N[(N[Power[a, 3.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $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)) < -0.0800000000000000017

   1. Initial program 30.64

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

   2. Simplified 30.64

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

      [Start] 30.64	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 30.64	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Applied egg-rr 31.02

      \[\leadsto \frac{\color{blue}{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{\frac{1}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}}{a \cdot 2} \]

   4. Simplified 31.05

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

      [Start] 31.02	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{\frac{1}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
      associate-/r/ [=>] 31.02	\[ \frac{\color{blue}{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{1} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}{a \cdot 2} \]
      *-commutative [=>] 31.02	\[ \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{1}}}{a \cdot 2} \]
      fma-def [<=] 31.66	\[ \frac{\left(b - \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}\right) \cdot \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{1}}{a \cdot 2} \]
      +-commutative [=>] 31.66	\[ \frac{\left(b - \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}\right) \cdot \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{1}}{a \cdot 2} \]
      fma-def [=>] 31.65	\[ \frac{\left(b - \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\right) \cdot \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{1}}{a \cdot 2} \]
      /-rgt-identity [=>] 31.65	\[ \frac{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
      /-rgt-identity [<=] 31.65	\[ \frac{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{1}}}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot 2} \]
      /-rgt-identity [=>] 31.65	\[ \frac{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \frac{\color{blue}{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot 2} \]
      associate-*r* [=>] 31.65	\[ \frac{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{\left(4 \cdot a\right) \cdot c}\right)}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot 2} \]
      *-commutative [=>] 31.65	\[ \frac{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot 4\right)} \cdot c\right)}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot 2} \]
      associate-*l* [=>] 31.65	\[ \frac{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(4 \cdot c\right)}\right)}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot 2} \]

   5. Applied egg-rr 29.45

      \[\leadsto \frac{\color{blue}{\left(\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right) \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\right)} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]

   6. Simplified 8.89

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

      [Start] 29.45	\[ \frac{\left(\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right) \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\right) \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      associate-*r/ [=>] 29.45	\[ \frac{\color{blue}{\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right) \cdot 1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      *-rgt-identity [=>] 29.45	\[ \frac{\frac{\color{blue}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      fma-udef [=>] 29.47	\[ \frac{\frac{b \cdot b - \color{blue}{\left(c \cdot \left(a \cdot -4\right) + b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      associate-*l* [<=] 29.47	\[ \frac{\frac{b \cdot b - \left(\color{blue}{\left(c \cdot a\right) \cdot -4} + b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      associate--r+ [=>] 29.47	\[ \frac{\frac{\color{blue}{\left(b \cdot b - \left(c \cdot a\right) \cdot -4\right) - b \cdot b}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      unsub-neg [<=] 29.47	\[ \frac{\frac{\color{blue}{\left(b \cdot b + \left(-\left(c \cdot a\right) \cdot -4\right)\right)} - b \cdot b}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      unsub-neg [<=] 29.47	\[ \frac{\frac{\color{blue}{\left(b \cdot b + \left(-\left(c \cdot a\right) \cdot -4\right)\right) + \left(-b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      +-commutative [=>] 29.47	\[ \frac{\frac{\color{blue}{\left(-b \cdot b\right) + \left(b \cdot b + \left(-\left(c \cdot a\right) \cdot -4\right)\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      associate-+r+ [=>] 8.89	\[ \frac{\frac{\color{blue}{\left(\left(-b \cdot b\right) + b \cdot b\right) + \left(-\left(c \cdot a\right) \cdot -4\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      +-commutative [<=] 8.89	\[ \frac{\frac{\color{blue}{\left(b \cdot b + \left(-b \cdot b\right)\right)} + \left(-\left(c \cdot a\right) \cdot -4\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      unsub-neg [=>] 8.89	\[ \frac{\frac{\color{blue}{\left(b \cdot b - b \cdot b\right)} + \left(-\left(c \cdot a\right) \cdot -4\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      +-inverses [=>] 8.89	\[ \frac{\frac{\color{blue}{0} + \left(-\left(c \cdot a\right) \cdot -4\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      sub-neg [<=] 8.89	\[ \frac{\frac{\color{blue}{0 - \left(c \cdot a\right) \cdot -4}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      neg-sub0 [<=] 8.89	\[ \frac{\frac{\color{blue}{-\left(c \cdot a\right) \cdot -4}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      distribute-rgt-neg-in [=>] 8.89	\[ \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot \left(--4\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      metadata-eval [=>] 8.89	\[ \frac{\frac{\left(c \cdot a\right) \cdot \color{blue}{4}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      associate-*l* [=>] 8.89	\[ \frac{\frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]
      *-commutative [=>] 8.89	\[ \frac{\frac{c \cdot \left(a \cdot 4\right)}{b + \sqrt{\mathsf{fma}\left(c, \color{blue}{-4 \cdot a}, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} \]

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 75.54

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

   2. Simplified 75.54

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

      [Start] 75.54	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 75.54	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in a around 0 2.89

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

   4. Simplified 2.89

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

      [Start] 2.89	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
      +-commutative [=>] 2.89	\[ \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
      mul-1-neg [=>] 2.89	\[ \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
      unsub-neg [=>] 2.89	\[ \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

   5. Taylor expanded in b around 0 2.89

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

   6. Simplified 2.89

      \[\leadsto \left(\mathsf{fma}\left(-0.25, \color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}}, \frac{\left(\left(a \cdot -2\right) \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{{b}^{3}} \cdot a \]
      Proof

      [Start] 2.89	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3} \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}, \frac{\left(\left(a \cdot -2\right) \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{{b}^{3}} \cdot a \]
      associate-/l* [=>] 2.89	\[ \left(\mathsf{fma}\left(-0.25, \color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{4 \cdot {c}^{4} + 16 \cdot {c}^{4}}}}, \frac{\left(\left(a \cdot -2\right) \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{{b}^{3}} \cdot a \]
      distribute-rgt-out [=>] 2.89	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{\color{blue}{{c}^{4} \cdot \left(4 + 16\right)}}}, \frac{\left(\left(a \cdot -2\right) \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{{b}^{3}} \cdot a \]
      metadata-eval [=>] 2.89	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot \color{blue}{20}}}, \frac{\left(\left(a \cdot -2\right) \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{{b}^{3}} \cdot a \]

   7. Applied egg-rr 2.89

      \[\leadsto \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, \frac{\left(\left(a \cdot -2\right) \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \color{blue}{\left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right)} \cdot a \]
3. Recombined 2 regimes into one program.

4. Final simplification 3.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.08:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, \frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	4.74%
Cost	35780

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.08:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{b + \sqrt{t_0}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b \cdot b - t_0}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot {c}^{3}}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 2
Error	9.24%
Cost	22020

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -100000:\\ \;\;\;\;\frac{\sqrt{\frac{{b}^{4} + \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -16}{b \cdot b + c \cdot \left(4 \cdot a\right)}} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right)\\ \end{array} \]

Alternative 3
Error	9.21%
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -100000:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right)\\ \end{array} \]

Alternative 4
Error	6.35%
Cost	20736

\[\left(\frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot {c}^{3}}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}} \]

Alternative 5
Error	9.21%
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -100000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right)\\ \end{array} \]

Alternative 6
Error	9.6%
Cost	1024

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

Alternative 7
Error	19.16%
Cost	256

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

Alternative 8
Error	98.36%
Cost	192

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

Reproduce

herbie shell --seed 2023121 
(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)))