ABCF->ab-angle b

Average Error: 52.4 → 42.3

Time: 42.0s

Precision: binary64

[A C]: =sort([A C])

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{B \cdot B - \left(4 \cdot A\right) \cdot C}{\sqrt{A \cdot \left(4 \cdot \left(F \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right)\right)}}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -2.291700553487221 \cdot 10^{-210}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}} \cdot \sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(4 \cdot \left(F \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \end{array}\]

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 (<=
      (/
       (-
        (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))
   (/
    -1.0
    (/
     (- (* B B) (* (* 4.0 A) C))
     (sqrt (* A (* 4.0 (* F (- (* B B) (* (* 4.0 A) 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)))
        -2.291700553487221e-210)
     (/
      (-
       (sqrt
        (*
         (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
         (-
          (+ A C)
          (*
           (sqrt (sqrt (+ (pow (- A C) 2.0) (* B B))))
           (sqrt (sqrt (+ (pow (- A C) 2.0) (* B B)))))))))
      (- (pow B 2.0) (* (* 4.0 A) C)))
     (/
      (sqrt (* A (* 4.0 (* F (- (* B B) (* (* 4.0 A) C))))))
      (- (* (* 4.0 A) C) (* B B))))))

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 = -1.0 / (((B * B) - ((4.0 * A) * C)) / sqrt(A * (4.0 * (F * ((B * B) - ((4.0 * A) * 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))) <= -2.291700553487221e-210) {
		tmp = -sqrt((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - (sqrt(sqrt(pow((A - C), 2.0) + (B * B))) * sqrt(sqrt(pow((A - C), 2.0) + (B * B)))))) / (pow(B, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = sqrt(A * (4.0 * (F * ((B * B) - ((4.0 * A) * C))))) / (((4.0 * A) * C) - (B * B));
	}
	return tmp;
}

Error

Bits error versus A

Bits error versus B

Bits error versus C

Bits error versus F

Derivation

1. Split input into 3 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. Using strategy rm

   3. Applied associate--l+_binary64_3425 64.0

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

   4. Applied distribute-rgt-in_binary64_3438 64.0

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

   5. Simplified 64.0

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

   6. Simplified 64.0

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

   7. Taylor expanded around -inf 53.0

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

   9. Applied add-sqr-sqrt_binary64_3510 53.0

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

   10. Simplified 52.9

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

   11. Simplified 45.7

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

   13. Applied neg-mul-1_binary64_3484 45.7

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

   14. Applied associate-/l*_binary64_3433 45.7

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

   15. Simplified 45.6

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

   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))) < -2.29170055348722083e-210

   1. Initial program 1.6

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

   3. Applied add-sqr-sqrt_binary64_3510 1.7

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

   4. Simplified 1.7

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

   5. Simplified 1.7

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

   if -2.29170055348722083e-210 < (/.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 60.1

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

   3. Applied associate--l+_binary64_3425 59.2

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

   4. Applied distribute-rgt-in_binary64_3438 59.2

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

   5. Simplified 59.2

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

   6. Simplified 59.2

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

   7. Taylor expanded around -inf 56.8

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

   9. Applied add-sqr-sqrt_binary64_3510 56.9

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

   10. Simplified 56.7

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

   11. Simplified 50.3

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

   13. Applied frac-2neg_binary64_3499 50.3

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

   14. Simplified 50.3

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

   15. Simplified 50.3

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

4. Final simplification 42.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{B \cdot B - \left(4 \cdot A\right) \cdot C}{\sqrt{A \cdot \left(4 \cdot \left(F \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right)\right)}}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -2.291700553487221 \cdot 10^{-210}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}} \cdot \sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(4 \cdot \left(F \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020358 
(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))))