Average Error: 52.8 → 31.1
Time: 32.5s
Precision: binary64
\[[A, C]=\mathsf{sort}([A, C])\]
\[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -\infty:\\ \;\;\;\;\sqrt{-F} \cdot \frac{-1}{\sqrt{C}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1.9623311228892837 \cdot 10^{-197}:\\ \;\;\;\;\frac{-1}{\frac{B \cdot B - 4 \cdot \left(A \cdot C\right)}{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}\right)}}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\ \;\;\;\;-\frac{{\left(-F\right)}^{0.5}}{\sqrt{C}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{\left(A \cdot \sqrt{2}\right) \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2} \cdot \sqrt{F \cdot -0.5}}{\sqrt{C}}\\ \end{array}\]
\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 -\infty:\\
\;\;\;\;\sqrt{-F} \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 -1.9623311228892837 \cdot 10^{-197}:\\
\;\;\;\;\frac{-1}{\frac{B \cdot B - 4 \cdot \left(A \cdot C\right)}{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}\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:\\
\;\;\;\;-\frac{{\left(-F\right)}^{0.5}}{\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 \infty:\\
\;\;\;\;\frac{\left(A \cdot \sqrt{2}\right) \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\

\mathbf{else}:\\
\;\;\;\;-\frac{\sqrt{2} \cdot \sqrt{F \cdot -0.5}}{\sqrt{C}}\\

\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)))
      (- INFINITY))
   (* (sqrt (- F)) (/ -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)))
        -1.9623311228892837e-197)
     (/
      -1.0
      (/
       (- (* B B) (* 4.0 (* A C)))
       (sqrt
        (*
         (* 2.0 (* F (- (* B B) (* 4.0 (* A C)))))
         (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (* B B))))))))
     (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)
       (- (/ (pow (- 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)
         (/
          (* (* A (sqrt 2.0)) (sqrt (* -8.0 (* C F))))
          (- (pow B 2.0) (* (* 4.0 A) C)))
         (- (/ (* (sqrt 2.0) (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))) <= -((double) INFINITY)) {
		tmp = sqrt(-F) * (-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))) <= -1.9623311228892837e-197) {
		tmp = -1.0 / (((B * B) - (4.0 * (A * C))) / sqrt((2.0 * (F * ((B * B) - (4.0 * (A * C))))) * ((A + C) - sqrt(pow((A - C), 2.0) + (B * B)))));
	} 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 = -(pow(-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 = ((A * sqrt(2.0)) * sqrt(-8.0 * (C * F))) / (pow(B, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = -((sqrt(2.0) * 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))) < -inf.0

    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 33.9

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

    3. Simplified33.9

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

    5. Applied associate-*r/_binary64_274833.9

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

    6. Applied sqrt-div_binary64_282323.3

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

    7. Applied associate-*r/_binary64_274823.3

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

    8. Simplified23.3

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

    10. Applied sqrt-unprod_binary64_282623.2

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

    11. Simplified23.2

      \[\leadsto -\frac{\sqrt{\color{blue}{-F}}}{\sqrt{C}}\]
    12. Using strategy rm

    13. Applied div-inv_binary64_280323.2

      \[\leadsto -\color{blue}{\sqrt{-F} \cdot \frac{1}{\sqrt{C}}}\]

    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))) < -1.96233112288928375e-197

    1. Initial program 1.3

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

    3. Applied neg-mul-1_binary64_28021.3

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

    4. Applied associate-/l*_binary64_27511.4

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

    5. Simplified1.4

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

    if -1.96233112288928375e-197 < (/.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 60.3

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

    2. Taylor expanded around -inf 32.1

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

    3. Simplified32.1

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

    5. Applied associate-*r/_binary64_274832.1

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

    6. Applied sqrt-div_binary64_282328.4

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

    7. Applied associate-*r/_binary64_274828.4

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

    8. Simplified28.4

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

    10. Applied pow1/2_binary64_288628.4

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

    11. Applied pow1/2_binary64_288628.4

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

    12. Applied pow-prod-down_binary64_287728.3

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

    13. Simplified28.3

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

    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 39.3

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

    2. Taylor expanded around -inf 13.5

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

    3. Simplified13.5

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

    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 52.7

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

    3. Simplified52.7

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

    5. Applied associate-*r/_binary64_274852.7

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

    6. Applied sqrt-div_binary64_282348.4

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

    7. Applied associate-*r/_binary64_274848.4

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

    8. Simplified48.4

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

  4. Final simplification31.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -\infty:\\ \;\;\;\;\sqrt{-F} \cdot \frac{-1}{\sqrt{C}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1.9623311228892837 \cdot 10^{-197}:\\ \;\;\;\;\frac{-1}{\frac{B \cdot B - 4 \cdot \left(A \cdot C\right)}{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}\right)}}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\ \;\;\;\;-\frac{{\left(-F\right)}^{0.5}}{\sqrt{C}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{\left(A \cdot \sqrt{2}\right) \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2} \cdot \sqrt{F \cdot -0.5}}{\sqrt{C}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021050 
(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))))