ABCF->ab-angle a

Average Error: 52.5 → 44.8

Time: 41.4s

Precision: binary64

Cost: 30215

Math

\[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

↓

\[\begin{array}{l} \mathbf{if}\;C \leq -1.1667719819638848 \cdot 10^{-40}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;C \leq -3.762903277684874 \cdot 10^{-203}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \sqrt{\left(C + A\right) + \left(\sqrt{B \cdot B + C \cdot C} - \left(C \cdot A\right) \cdot \sqrt{\frac{1}{B \cdot B + C \cdot C}}\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq -7.717133116842071 \cdot 10^{-236}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;C \leq 1.8971049353173214 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + C \cdot C}}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 1.348626692408251 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \left(-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 7636043547696.668:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;C \leq 1.551151584627811 \cdot 10^{+89}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \left(-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 4.276475041946287 \cdot 10^{+196}:\\ \;\;\;\;\frac{-C \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot A\right)}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 1.5032216862996874 \cdot 10^{+286}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{C + \left(C + A\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

FPCore

(FPCore (A B C F)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
     (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
  (- (pow B 2.0) (* (* 4.0 A) C))))

↓

(FPCore (A B C F)
 :precision binary64
 (if (<= C -1.1667719819638848e-40)
   (- (* (sqrt 2.0) (sqrt (* -0.5 (/ F C)))))
   (if (<= C -3.762903277684874e-203)
     (/
      (-
       (*
        (sqrt (* 2.0 (* F (- (* B B) (* C (* A 4.0))))))
        (sqrt
         (+
          (+ C A)
          (-
           (sqrt (+ (* B B) (* C C)))
           (* (* C A) (sqrt (/ 1.0 (+ (* B B) (* C C))))))))))
      (- (* B B) (* C (* A 4.0))))
     (if (<= C -7.717133116842071e-236)
       (- (* (sqrt 2.0) (sqrt (* -0.5 (/ F C)))))
       (if (<= C 1.8971049353173214e-208)
         (*
          (sqrt (* 2.0 (* F (- (* B B) (* C (* A 4.0))))))
          (/
           (- (sqrt (+ (+ C A) (sqrt (+ (* B B) (* C C))))))
           (- (* B B) (* C (* A 4.0)))))
         (if (<= C 1.348626692408251e-98)
           (/
            (*
             (sqrt (* 2.0 (* F (- (* B B) (* C (* A 4.0))))))
             (- (sqrt (+ (+ C A) (sqrt (+ (* B B) (pow (- A C) 2.0)))))))
            (- (* B B) (* C (* A 4.0))))
           (if (<= C 7636043547696.668)
             (- (* (sqrt 2.0) (sqrt (* -0.5 (/ F A)))))
             (if (<= C 1.551151584627811e+89)
               (/
                (*
                 (sqrt (* 2.0 (* F (- (* B B) (* C (* A 4.0))))))
                 (- (sqrt (+ (+ C A) (sqrt (+ (* B B) (pow (- A C) 2.0)))))))
                (- (* B B) (* C (* A 4.0))))
               (if (<= C 4.276475041946287e+196)
                 (/
                  (- (* C (* (sqrt 2.0) (sqrt (* -8.0 (* F A))))))
                  (- (* B B) (* C (* A 4.0))))
                 (if (<= C 1.5032216862996874e+286)
                   (- (* (sqrt 2.0) (sqrt (* -0.5 (/ F A)))))
                   (*
                    (sqrt (* 2.0 (* F (- (* B B) (* C (* A 4.0))))))
                    (/
                     (- (sqrt (+ C (+ C A))))
                     (- (* B B) (* C (* A 4.0)))))))))))))))

C

double code(double A, double B, double C, double F) {
	return -sqrt((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + sqrt(pow((A - C), 2.0) + pow(B, 2.0)))) / (pow(B, 2.0) - ((4.0 * A) * C));
}

↓

double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= -1.1667719819638848e-40) {
		tmp = -(sqrt(2.0) * sqrt(-0.5 * (F / C)));
	} else if (C <= -3.762903277684874e-203) {
		tmp = -(sqrt(2.0 * (F * ((B * B) - (C * (A * 4.0))))) * sqrt((C + A) + (sqrt((B * B) + (C * C)) - ((C * A) * sqrt(1.0 / ((B * B) + (C * C))))))) / ((B * B) - (C * (A * 4.0)));
	} else if (C <= -7.717133116842071e-236) {
		tmp = -(sqrt(2.0) * sqrt(-0.5 * (F / C)));
	} else if (C <= 1.8971049353173214e-208) {
		tmp = sqrt(2.0 * (F * ((B * B) - (C * (A * 4.0))))) * (-sqrt((C + A) + sqrt((B * B) + (C * C))) / ((B * B) - (C * (A * 4.0))));
	} else if (C <= 1.348626692408251e-98) {
		tmp = (sqrt(2.0 * (F * ((B * B) - (C * (A * 4.0))))) * -sqrt((C + A) + sqrt((B * B) + pow((A - C), 2.0)))) / ((B * B) - (C * (A * 4.0)));
	} else if (C <= 7636043547696.668) {
		tmp = -(sqrt(2.0) * sqrt(-0.5 * (F / A)));
	} else if (C <= 1.551151584627811e+89) {
		tmp = (sqrt(2.0 * (F * ((B * B) - (C * (A * 4.0))))) * -sqrt((C + A) + sqrt((B * B) + pow((A - C), 2.0)))) / ((B * B) - (C * (A * 4.0)));
	} else if (C <= 4.276475041946287e+196) {
		tmp = -(C * (sqrt(2.0) * sqrt(-8.0 * (F * A)))) / ((B * B) - (C * (A * 4.0)));
	} else if (C <= 1.5032216862996874e+286) {
		tmp = -(sqrt(2.0) * sqrt(-0.5 * (F / A)));
	} else {
		tmp = sqrt(2.0 * (F * ((B * B) - (C * (A * 4.0))))) * (-sqrt(C + (C + A)) / ((B * B) - (C * (A * 4.0))));
	}
	return tmp;
}

Error

Bits error versus A

Bits error versus B

Bits error versus C

Bits error versus F

Alternatives

Alternative 1
Error	44.5
Cost	30215

\[\begin{array}{l} \mathbf{if}\;C \leq -1.2972713832354343 \cdot 10^{-67}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;C \leq -6.34148830625923 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \left(-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq -6.553253316396736 \cdot 10^{-236}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;C \leq 2.0180754963362826 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + C \cdot C}}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 1.7579411442645465 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \left(-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 3894365731760.578:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;C \leq 1.1470329558487983 \cdot 10^{+89}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \left(-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 1.4361793469845245 \cdot 10^{+198}:\\ \;\;\;\;\frac{-C \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot A\right)}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 7.889896727992821 \cdot 10^{+286}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{C + \left(C + A\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 2
Error	44.6
Cost	30215

\[\begin{array}{l} \mathbf{if}\;C \leq -1.3117655194042228 \cdot 10^{-72}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;C \leq -1.020829083158243 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq -2.6559097095624615 \cdot 10^{-236}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;C \leq 2.0180754963362826 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + C \cdot C}}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 9.558429771912338 \cdot 10^{-99}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 4302573288998.992:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;C \leq 2.0370057584299153 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 2.3848522097125993 \cdot 10^{+198}:\\ \;\;\;\;\frac{-C \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot A\right)}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 4.244579846234397 \cdot 10^{+286}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{C + \left(C + A\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 3
Error	44.3
Cost	23172

\[\begin{array}{l} \mathbf{if}\;A \leq -1.7496110785806515 \cdot 10^{-20}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;A \leq 6.243242224790431 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + C \cdot C}}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;A \leq 1.6975214123994237 \cdot 10^{-41}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;A \leq 2.02730538299883 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{\left(C + A\right) + \sqrt{\left(C \cdot C + A \cdot A\right) - 2 \cdot \left(C \cdot A\right)}}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;A \leq 4.346792882910052 \cdot 10^{+25}:\\ \;\;\;\;\frac{-\sqrt{2} \cdot \left(A \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;A \leq 7.806109043084902 \cdot 10^{+49}:\\ \;\;\;\;-\sqrt{F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \sqrt{A + \left(C + A\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 4
Error	44.3
Cost	22150

\[\begin{array}{l} \mathbf{if}\;A \leq -1.683869395181302 \cdot 10^{-20}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;A \leq 6.566063835698995 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + C \cdot C}}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;A \leq 1.3319858820885497 \cdot 10^{-52}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;A \leq 3.272603846781531 \cdot 10^{-21}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(C + A\right) + \sqrt{\left(C \cdot C + A \cdot A\right) - 2 \cdot \left(C \cdot A\right)}\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;A \leq 4.597491714676719 \cdot 10^{+32}:\\ \;\;\;\;\frac{-\sqrt{2} \cdot \left(A \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;A \leq 7.806109043084902 \cdot 10^{+49}:\\ \;\;\;\;-\sqrt{F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \sqrt{A + \left(C + A\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 5
Error	46.7
Cost	15683

\[\begin{array}{l} \mathbf{if}\;C \leq -9.79153399683607 \cdot 10^{-308}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;C \leq 9.19377001290605 \cdot 10^{-117}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \sqrt{A + \left(C + A\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 1.4439851968970833 \cdot 10^{+195}:\\ \;\;\;\;\frac{-C \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot A\right)}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{C + \left(C + A\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 6
Error	47.1
Cost	16004

\[\begin{array}{l} \mathbf{if}\;C \leq -9.636626628530083 \cdot 10^{-270}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;C \leq 8.8384979074682 \cdot 10^{-70}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;C \leq 4.37355896041663 \cdot 10^{+195}:\\ \;\;\;\;\frac{-C \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot A\right)}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 6.582667715211156 \cdot 10^{+286}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{C + \left(C + A\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 7
Error	47.2
Cost	16004

\[\begin{array}{l} \mathbf{if}\;C \leq -1.3698981427682082 \cdot 10^{-272}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;C \leq 3.675885697099643 \cdot 10^{-68}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;C \leq 1.1776615866996405 \cdot 10^{+196}:\\ \;\;\;\;\frac{-C \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot A\right)}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 2.2856784189714687 \cdot 10^{+281}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + A\right)}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 8
Error	47.3
Cost	14861

\[\begin{array}{l} \mathbf{if}\;C \leq -1.7145530100948862 \cdot 10^{-268}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;C \leq 2.2645652338554536 \cdot 10^{-73} \lor \neg \left(C \leq 4.037849062667424 \cdot 10^{+198}\right) \land C \leq 2.6301746459990342 \cdot 10^{+280}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-C \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot A\right)}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 9
Error	46.9
Cost	14722

\[\begin{array}{l} \mathbf{if}\;A \leq -3.968571333694865 \cdot 10^{-247}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;A \leq 1.2382252259873166 \cdot 10^{-56}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2} \cdot \left(A \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 10
Error	47.5
Cost	13961

\[\begin{array}{l} \mathbf{if}\;A \leq -5.364802292552887 \cdot 10^{-248}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;A \leq 2.8994078830207907 \cdot 10^{+85} \lor \neg \left(A \leq 4.529002837703455 \cdot 10^{+209}\right):\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 11
Error	49.1
Cost	13633

\[\begin{array}{l} \mathbf{if}\;A \leq -1.718342448941985 \cdot 10^{-307}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;A \leq 3.1582310936383745 \cdot 10^{-21}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(C + \left(C + A\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(C + A\right) + \left(A - C\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 12
Error	55.2
Cost	10049

\[\begin{array}{l} \mathbf{if}\;B \leq -3.8481650581774855 \cdot 10^{-118}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(C + A\right) + \left(\frac{C \cdot A}{B} - \left(\left(B + 0.5 \cdot \frac{A \cdot A}{B}\right) + 0.5 \cdot \frac{C \cdot C}{B}\right)\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 3.4253030403005424 \cdot 10^{-55}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(B + \left(C + A\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 13
Error	54.9
Cost	8962

\[\begin{array}{l} \mathbf{if}\;B \leq -9.812862966233216 \cdot 10^{-82}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(C + A\right) - B\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 6.3262005158442605 \cdot 10^{-55}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(B + \left(C + A\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 14
Error	56.5
Cost	8962

\[\begin{array}{l} \mathbf{if}\;B \leq -1.0094198241230907 \cdot 10^{-27}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(C + A\right) - B\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 8.35043699247265 \cdot 10^{-232}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(C + \left(C + A\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 15
Error	55.4
Cost	8641

\[\begin{array}{l} \mathbf{if}\;A \leq 4.4497381856068945 \cdot 10^{-24}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(C + \left(C + A\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 16
Error	57.2
Cost	8641

\[\begin{array}{l} \mathbf{if}\;C \leq 1.4678054841862924 \cdot 10^{-47}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(C + \left(C + A\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 17
Error	60.6
Cost	64

\[-1\]

Alternative 18
Error	61.6
Cost	64

\[0\]

Alternative 19
Error	62.1
Cost	64

\[1\]

Derivation

1. Split input into 7 regimes

2. if C < -1.1667719819638848e-40 or -3.7629032776848743e-203 < C < -7.7171331168420709e-236

   1. Initial program 59.0

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   2. Simplified 59.0

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}}\]

   3. Taylor expanded around inf 42.6

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2}\right)}\]

   4. Simplified 42.6

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}}\]

   5. Simplified 42.6

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}}\]

   if -1.1667719819638848e-40 < C < -3.7629032776848743e-203

   1. Initial program 48.8

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   2. Simplified 48.8

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}}\]
   3. Using strategy rm

   4. Applied sqrt-prod_binary64_2140 46.1

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\]

   5. Simplified 46.1

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)}} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\]

   6. Simplified 46.1

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \color{blue}{\sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}} + \left(A + C\right)}}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\]

   7. Taylor expanded around 0 47.2

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\color{blue}{\left(\sqrt{{C}^{2} + {B}^{2}} - \sqrt{\frac{1}{{C}^{2} + {B}^{2}}} \cdot \left(A \cdot C\right)\right)} + \left(A + C\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\]

   8. Simplified 47.2

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\color{blue}{\left(\sqrt{B \cdot B + C \cdot C} - \left(A \cdot C\right) \cdot \sqrt{\frac{1}{B \cdot B + C \cdot C}}\right)} + \left(A + C\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\]

   9. Simplified 47.2

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\left(C + A\right) + \left(\sqrt{B \cdot B + C \cdot C} - \left(C \cdot A\right) \cdot \sqrt{\frac{1}{B \cdot B + C \cdot C}}\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}}\]

   if -7.7171331168420709e-236 < C < 1.89710493531732136e-208

   1. Initial program 48.6

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   2. Simplified 48.6

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity_binary64_2124 48.6

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{1 \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)}}\]

   5. Applied sqrt-prod_binary64_2140 45.1

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}}{1 \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)}\]

   6. Applied distribute-rgt-neg-in_binary64_2082 45.1

      \[\leadsto \frac{\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \left(-\sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}\right)}}{1 \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)}\]

   7. Applied times-frac_binary64_2130 45.1

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}{1} \cdot \frac{-\sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - \left(4 \cdot A\right) \cdot C}}\]

   8. Simplified 45.1

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)}} \cdot \frac{-\sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\]

   9. Simplified 45.1

      \[\leadsto \sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \color{blue}{\frac{-\sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}} + \left(A + C\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}}\]

   10. Taylor expanded around 0 48.4

       \[\leadsto \sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \frac{-\sqrt{\color{blue}{\sqrt{{C}^{2} + {B}^{2}}} + \left(A + C\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\]

   11. Simplified 48.4

       \[\leadsto \sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \frac{-\sqrt{\color{blue}{\sqrt{B \cdot B + C \cdot C}} + \left(A + C\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\]

   12. Simplified 48.4

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \frac{-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + C \cdot C}}}{B \cdot B - C \cdot \left(A \cdot 4\right)}}\]

   if 1.89710493531732136e-208 < C < 1.34862669240825097e-98 or 7636043547696.668 < C < 1.55115158462781088e89

   1. Initial program 45.8

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   2. Simplified 45.8

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}}\]
   3. Using strategy rm

   4. Applied sqrt-prod_binary64_2140 42.6

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\]

   5. Simplified 42.6

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)}} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\]

   6. Simplified 42.6

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \color{blue}{\sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}} + \left(A + C\right)}}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\]

   7. Simplified 42.6

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \left(-\sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}} + \left(C + A\right)}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}}\]

   if 1.34862669240825097e-98 < C < 7636043547696.668 or 4.27647504194628672e196 < C < 1.50322168629968736e286

   1. Initial program 52.8

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   2. Simplified 52.8

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}}\]

   3. Taylor expanded around inf 48.0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2}\right)}\]

   4. Simplified 48.0

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}}\]

   5. Simplified 48.0

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}}\]

   if 1.55115158462781088e89 < C < 4.27647504194628672e196

   1. Initial program 49.9

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   2. Simplified 49.9

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}}\]

   3. Taylor expanded around inf 41.4

      \[\leadsto \frac{-\color{blue}{C \cdot \left(\sqrt{-8 \cdot \left(A \cdot F\right)} \cdot \sqrt{2}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\]

   4. Simplified 41.4

      \[\leadsto \frac{-\color{blue}{C \cdot \left(\sqrt{2} \cdot \sqrt{\left(A \cdot F\right) \cdot -8}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\]

   5. Simplified 41.4

      \[\leadsto \color{blue}{\frac{-C \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(A \cdot F\right)}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}}\]

   if 1.50322168629968736e286 < C

   1. Initial program 64.0

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity_binary64_2124 64.0

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{1 \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)}}\]

   5. Applied sqrt-prod_binary64_2140 64.0

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}}{1 \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)}\]

   6. Applied distribute-rgt-neg-in_binary64_2082 64.0

      \[\leadsto \frac{\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \left(-\sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}\right)}}{1 \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)}\]

   7. Applied times-frac_binary64_2130 64.0

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}{1} \cdot \frac{-\sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - \left(4 \cdot A\right) \cdot C}}\]

   8. Simplified 64.0

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)}} \cdot \frac{-\sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\]

   9. Simplified 64.0

      \[\leadsto \sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \color{blue}{\frac{-\sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}} + \left(A + C\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}}\]

   10. Taylor expanded around inf 46.3

       \[\leadsto \sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \frac{-\sqrt{\color{blue}{C} + \left(A + C\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\]

   11. Simplified 46.3

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \frac{-\sqrt{C + \left(C + A\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}}\]
3. Recombined 7 regimes into one program.

4. Final simplification 44.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.1667719819638848 \cdot 10^{-40}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;C \leq -3.762903277684874 \cdot 10^{-203}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \sqrt{\left(C + A\right) + \left(\sqrt{B \cdot B + C \cdot C} - \left(C \cdot A\right) \cdot \sqrt{\frac{1}{B \cdot B + C \cdot C}}\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq -7.717133116842071 \cdot 10^{-236}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;C \leq 1.8971049353173214 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + C \cdot C}}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 1.348626692408251 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \left(-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 7636043547696.668:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;C \leq 1.551151584627811 \cdot 10^{+89}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \left(-\sqrt{\left(C + A\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 4.276475041946287 \cdot 10^{+196}:\\ \;\;\;\;\frac{-C \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot A\right)}\right)}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 1.5032216862996874 \cdot 10^{+286}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)} \cdot \frac{-\sqrt{C + \left(C + A\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))