Average Error: 52.3 → 30.9
Time: 36.6s
Precision: binary64
\[[A, C]=\mathsf{sort}([A, C])\]
\[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -2.243908304163857 \cdot 10^{+296}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{F \cdot -0.5}\right) \cdot \frac{-1}{\sqrt{C}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -5.768969992851784 \cdot 10^{-177}:\\ \;\;\;\;\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\ \;\;\;\;-\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}\right)\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(C \cdot F\right)} \cdot \left(A \cdot \sqrt{2}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}}\right)\\ \end{array}\]
\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}\;\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{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -2.243908304163857 \cdot 10^{+296}:\\
\;\;\;\;\left(\sqrt{2} \cdot \sqrt{F \cdot -0.5}\right) \cdot \frac{-1}{\sqrt{C}}\\

\mathbf{elif}\;\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{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -5.768969992851784 \cdot 10^{-177}:\\
\;\;\;\;\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\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{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}}\\

\mathbf{elif}\;\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{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\
\;\;\;\;-\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}\right)\\

\mathbf{elif}\;\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{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\
\;\;\;\;\frac{\sqrt{-8 \cdot \left(C \cdot F\right)} \cdot \left(A \cdot \sqrt{2}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\

\mathbf{else}:\\
\;\;\;\;-\sqrt{\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}}\right)\\

\end{array}
(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 (<=
      (/
       (-
        (sqrt
         (*
          (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
          (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
       (- (pow B 2.0) (* (* 4.0 A) C)))
      -2.243908304163857e+296)
   (* (* (sqrt 2.0) (sqrt (* F -0.5))) (/ -1.0 (sqrt C)))
   (if (<=
        (/
         (-
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
         (- (pow B 2.0) (* (* 4.0 A) C)))
        -5.768969992851784e-177)
     (/
      -1.0
      (/
       (- (pow B 2.0) (* (* 4.0 A) C))
       (sqrt
        (*
         (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
         (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))))
     (if (<=
          (/
           (-
            (sqrt
             (*
              (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
              (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
           (- (pow B 2.0) (* (* 4.0 A) C)))
          0.0)
       (-
        (*
         (* (cbrt (sqrt 2.0)) (cbrt (sqrt 2.0)))
         (* (cbrt (sqrt 2.0)) (/ (sqrt (* F -0.5)) (sqrt C)))))
       (if (<=
            (/
             (-
              (sqrt
               (*
                (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
                (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
             (- (pow B 2.0) (* (* 4.0 A) C)))
            INFINITY)
         (/
          (* (sqrt (* -8.0 (* C F))) (* A (sqrt 2.0)))
          (- (pow B 2.0) (* (* 4.0 A) C)))
         (-
          (*
           (sqrt (/ (sqrt (* F -0.5)) (sqrt C)))
           (* (sqrt 2.0) (sqrt (/ (sqrt (* F -0.5)) (sqrt 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 ((-sqrt((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt(pow(B, 2.0) + pow((A - C), 2.0)))) / (pow(B, 2.0) - ((4.0 * A) * C))) <= -2.243908304163857e+296) {
		tmp = (sqrt(2.0) * sqrt(F * -0.5)) * (-1.0 / sqrt(C));
	} else if ((-sqrt((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt(pow(B, 2.0) + pow((A - C), 2.0)))) / (pow(B, 2.0) - ((4.0 * A) * C))) <= -5.768969992851784e-177) {
		tmp = -1.0 / ((pow(B, 2.0) - ((4.0 * A) * C)) / sqrt((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt(pow(B, 2.0) + pow((A - C), 2.0)))));
	} else if ((-sqrt((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt(pow(B, 2.0) + pow((A - C), 2.0)))) / (pow(B, 2.0) - ((4.0 * A) * C))) <= 0.0) {
		tmp = -((cbrt(sqrt(2.0)) * cbrt(sqrt(2.0))) * (cbrt(sqrt(2.0)) * (sqrt(F * -0.5) / sqrt(C))));
	} else if ((-sqrt((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt(pow(B, 2.0) + pow((A - C), 2.0)))) / (pow(B, 2.0) - ((4.0 * A) * C))) <= ((double) INFINITY)) {
		tmp = (sqrt(-8.0 * (C * F)) * (A * sqrt(2.0))) / (pow(B, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = -(sqrt(sqrt(F * -0.5) / sqrt(C)) * (sqrt(2.0) * sqrt(sqrt(F * -0.5) / sqrt(C))));
	}
	return tmp;
}

Error

Bits error versus A

Bits error versus B

Bits error versus C

Bits error versus F

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -2.24390830416385684e296

    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. Taylor expanded around -inf 33.8

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

    3. Simplified33.8

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}}\]
    4. Using strategy rm

    5. Applied associate-*r/_binary64_172533.8

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

    6. Applied sqrt-div_binary64_180022.0

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

    7. Simplified22.0

      \[\leadsto -\sqrt{2} \cdot \frac{\color{blue}{\sqrt{F \cdot -0.5}}}{\sqrt{C}}\]
    8. Using strategy rm

    9. Applied div-inv_binary64_178022.0

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

    10. Applied associate-*r*_binary64_172322.0

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

    if -2.24390830416385684e296 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -5.76896999285178388e-177

    1. Initial program 1.2

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

    3. Applied neg-mul-1_binary64_17791.2

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

    4. Applied associate-/l*_binary64_17281.3

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

    if -5.76896999285178388e-177 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 0.0

    1. Initial program 58.4

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

    2. Taylor expanded around -inf 33.2

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

    3. Simplified33.2

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}}\]
    4. Using strategy rm

    5. Applied associate-*r/_binary64_172533.2

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

    6. Applied sqrt-div_binary64_180029.4

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

    7. Simplified29.4

      \[\leadsto -\sqrt{2} \cdot \frac{\color{blue}{\sqrt{F \cdot -0.5}}}{\sqrt{C}}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt_binary64_181829.4

      \[\leadsto -\color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}\]

    10. Applied associate-*l*_binary64_172429.4

      \[\leadsto -\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}\right)}\]

    if 0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

    1. Initial program 38.5

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

    2. Taylor expanded around -inf 13.2

      \[\leadsto \frac{-\color{blue}{-1 \cdot \left(\sqrt{-8 \cdot \left(C \cdot F\right)} \cdot \left(A \cdot \sqrt{2}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

    if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) 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. Taylor expanded around -inf 53.1

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

    3. Simplified53.1

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}}\]
    4. Using strategy rm

    5. Applied associate-*r/_binary64_172553.1

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

    6. Applied sqrt-div_binary64_180048.7

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

    7. Simplified48.7

      \[\leadsto -\sqrt{2} \cdot \frac{\color{blue}{\sqrt{F \cdot -0.5}}}{\sqrt{C}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt_binary64_180548.7

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

    10. Applied associate-*r*_binary64_172348.7

      \[\leadsto -\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}}\right) \cdot \sqrt{\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification30.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -2.243908304163857 \cdot 10^{+296}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{F \cdot -0.5}\right) \cdot \frac{-1}{\sqrt{C}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -5.768969992851784 \cdot 10^{-177}:\\ \;\;\;\;\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\ \;\;\;\;-\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}\right)\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(C \cdot F\right)} \cdot \left(A \cdot \sqrt{2}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :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))))