ABCF->ab-angle a

Average Error: 52.1 → 40.2

Time: 41.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{2 \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ \mathbf{if}\;B \leq -2.1809907538999357 \cdot 10^{-29}:\\ \;\;\;\;-\frac{t_0}{-B}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -7.450543718790283 \cdot 10^{-121}:\\ \;\;\;\;\begin{array}{l} t_2 := \sqrt{\left(A \cdot A + \left(B \cdot B + C \cdot C\right)\right) - 2 \cdot \left(A \cdot C\right)}\\ \frac{-\sqrt{F \cdot \left(\mathsf{fma}\left(2, C \cdot \left(B \cdot B\right), 2 \cdot \left(\left(B \cdot B\right) \cdot t_2 + A \cdot \left(B \cdot B\right)\right)\right) - \mathsf{fma}\left(8, C \cdot \left(A \cdot A\right), 8 \cdot \left(A \cdot \left(C \cdot C\right) + \left(A \cdot C\right) \cdot t_2\right)\right)\right)}}{t_1} \end{array}\\ \mathbf{elif}\;B \leq -1.3918271591250041 \cdot 10^{-139}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;B \leq -7.195204907113419 \cdot 10^{-174}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot C - 0.5 \cdot \frac{{B}^{2}}{A}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 2.478409119982122 \cdot 10^{-220}:\\ \;\;\;\;\frac{-A \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot C\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 1.310184971611859 \cdot 10^{-166}:\\ \;\;\;\;\begin{array}{l} t_3 := \sqrt{\mathsf{hypot}\left(\sqrt{A \cdot \left(C \cdot -4\right)}, B\right)}\\ -\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{t_3 \cdot t_3} \end{array}\\ \mathbf{elif}\;B \leq 2.852086878622813 \cdot 10^{-23}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t_0}{B}\\ \end{array}\\ \end{array} \]

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
 (let* ((t_0 (sqrt (* 2.0 (* F (+ (+ A C) (hypot B (- A C))))))))
   (if (<= B -2.1809907538999357e-29)
     (- (/ t_0 (- B)))
     (let* ((t_1 (fma A (* C -4.0) (* B B))))
       (if (<= B -7.450543718790283e-121)
         (let* ((t_2
                 (sqrt (- (+ (* A A) (+ (* B B) (* C C))) (* 2.0 (* A C))))))
           (/
            (-
             (sqrt
              (*
               F
               (-
                (fma
                 2.0
                 (* C (* B B))
                 (* 2.0 (+ (* (* B B) t_2) (* A (* B B)))))
                (fma
                 8.0
                 (* C (* A A))
                 (* 8.0 (+ (* A (* C C)) (* (* A C) t_2))))))))
            t_1))
         (if (<= B -1.3918271591250041e-139)
           (- (* (sqrt 2.0) (sqrt (* -0.5 (/ F A)))))
           (if (<= B -7.195204907113419e-174)
             (/
              (-
               (sqrt
                (* t_1 (* 2.0 (* F (- (* 2.0 C) (* 0.5 (/ (pow B 2.0) A))))))))
              t_1)
             (if (<= B 2.478409119982122e-220)
               (/ (- (* A (* (sqrt 2.0) (sqrt (* -8.0 (* F C)))))) t_1)
               (if (<= B 1.310184971611859e-166)
                 (let* ((t_3 (sqrt (hypot (sqrt (* A (* C -4.0))) B))))
                   (-
                    (/
                     (* (sqrt 2.0) (sqrt (* F (+ C (hypot B C)))))
                     (* t_3 t_3))))
                 (if (<= B 2.852086878622813e-23)
                   (/ (- (sqrt (* t_1 (* 2.0 (* F (* 2.0 A)))))) t_1)
                   (- (/ t_0 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 t_0 = sqrt(2.0 * (F * ((A + C) + hypot(B, (A - C)))));
	double tmp;
	if (B <= -2.1809907538999357e-29) {
		tmp = -(t_0 / -B);
	} else {
		double t_1 = fma(A, (C * -4.0), (B * B));
		double tmp_1;
		if (B <= -7.450543718790283e-121) {
			double t_2_2 = sqrt(((A * A) + ((B * B) + (C * C))) - (2.0 * (A * C)));
			tmp_1 = -sqrt(F * (fma(2.0, (C * (B * B)), (2.0 * (((B * B) * t_2_2) + (A * (B * B))))) - fma(8.0, (C * (A * A)), (8.0 * ((A * (C * C)) + ((A * C) * t_2_2)))))) / t_1;
		} else if (B <= -1.3918271591250041e-139) {
			tmp_1 = -(sqrt(2.0) * sqrt(-0.5 * (F / A)));
		} else if (B <= -7.195204907113419e-174) {
			tmp_1 = -sqrt(t_1 * (2.0 * (F * ((2.0 * C) - (0.5 * (pow(B, 2.0) / A)))))) / t_1;
		} else if (B <= 2.478409119982122e-220) {
			tmp_1 = -(A * (sqrt(2.0) * sqrt(-8.0 * (F * C)))) / t_1;
		} else if (B <= 1.310184971611859e-166) {
			double t_3 = sqrt(hypot(sqrt(A * (C * -4.0)), B));
			tmp_1 = -((sqrt(2.0) * sqrt(F * (C + hypot(B, C)))) / (t_3 * t_3));
		} else if (B <= 2.852086878622813e-23) {
			tmp_1 = -sqrt(t_1 * (2.0 * (F * (2.0 * A)))) / t_1;
		} else {
			tmp_1 = -(t_0 / B);
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus A

Bits error versus B

Bits error versus C

Bits error versus F

Derivation

1. Split input into 8 regimes

2. if B < -2.1809907538999357e-29

   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 50.8

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

   3. Applied add-sqr-sqrt_binary64 51.6

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

   4. Applied sqrt-prod_binary64 46.1

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

   5. Applied distribute-lft-neg-in_binary64 46.1

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

   6. Applied times-frac_binary64 46.1

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

   7. Simplified 45.1

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

   8. Taylor expanded in B around -inf 31.8

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

   9. Simplified 31.8

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

   if -2.1809907538999357e-29 < B < -7.4505437187902831e-121

   1. Initial program 45.2

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

   2. Simplified 40.0

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

   3. Taylor expanded in F around 0 44.9

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

   4. Simplified 44.9

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

   if -7.4505437187902831e-121 < B < -1.39182715912500414e-139

   1. Initial program 49.7

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

   2. Simplified 45.7

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

   3. Taylor expanded in C around inf 51.7

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

   4. Simplified 51.7

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

   if -1.39182715912500414e-139 < B < -7.19520490711341923e-174

   1. Initial program 51.5

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

   2. Simplified 47.1

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

   3. Taylor expanded in A around -inf 45.7

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \color{blue}{\left(2 \cdot C - 0.5 \cdot \frac{{B}^{2}}{A}\right)}\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if -7.19520490711341923e-174 < B < 2.47840911998212199e-220

   1. Initial program 53.2

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

   2. Simplified 48.2

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

   3. Taylor expanded in A around inf 49.3

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

   if 2.47840911998212199e-220 < B < 1.31018497161185898e-166

   1. Initial program 56.0

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

   2. Simplified 50.6

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

   3. Applied add-sqr-sqrt_binary64 57.2

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

   4. Applied sqrt-prod_binary64 54.2

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

   5. Applied distribute-lft-neg-in_binary64 54.2

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

   6. Applied times-frac_binary64 54.2

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

   7. Simplified 54.0

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

   8. Applied add-sqr-sqrt_binary64 54.1

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

   9. Simplified 54.1

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

   10. Simplified 52.6

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

   11. Taylor expanded in A around 0 55.8

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

   12. Simplified 52.9

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

   if 1.31018497161185898e-166 < B < 2.85208687862281319e-23

   1. Initial program 48.1

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

   2. Simplified 43.2

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

   3. Taylor expanded in A around inf 48.9

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

   if 2.85208687862281319e-23 < B

   1. Initial program 53.9

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

   2. Simplified 51.9

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

   3. Applied add-sqr-sqrt_binary64 52.9

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

   4. Applied sqrt-prod_binary64 47.6

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

   5. Applied distribute-lft-neg-in_binary64 47.6

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

   6. Applied times-frac_binary64 47.5

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

   7. Simplified 46.5

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

   8. Taylor expanded in A around 0 33.6

      \[\leadsto -1 \cdot \frac{\sqrt{2 \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\color{blue}{B}} \]
3. Recombined 8 regimes into one program.

4. Final simplification 40.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.1809907538999357 \cdot 10^{-29}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{-B}\\ \mathbf{elif}\;B \leq -7.450543718790283 \cdot 10^{-121}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(\mathsf{fma}\left(2, C \cdot \left(B \cdot B\right), 2 \cdot \left(\left(B \cdot B\right) \cdot \sqrt{\left(A \cdot A + \left(B \cdot B + C \cdot C\right)\right) - 2 \cdot \left(A \cdot C\right)} + A \cdot \left(B \cdot B\right)\right)\right) - \mathsf{fma}\left(8, C \cdot \left(A \cdot A\right), 8 \cdot \left(A \cdot \left(C \cdot C\right) + \left(A \cdot C\right) \cdot \sqrt{\left(A \cdot A + \left(B \cdot B + C \cdot C\right)\right) - 2 \cdot \left(A \cdot C\right)}\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq -1.3918271591250041 \cdot 10^{-139}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;B \leq -7.195204907113419 \cdot 10^{-174}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot C - 0.5 \cdot \frac{{B}^{2}}{A}\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 2.478409119982122 \cdot 10^{-220}:\\ \;\;\;\;\frac{-A \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot C\right)}\right)}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.310184971611859 \cdot 10^{-166}:\\ \;\;\;\;-\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\sqrt{\mathsf{hypot}\left(\sqrt{A \cdot \left(C \cdot -4\right)}, B\right)} \cdot \sqrt{\mathsf{hypot}\left(\sqrt{A \cdot \left(C \cdot -4\right)}, B\right)}}\\ \mathbf{elif}\;B \leq 2.852086878622813 \cdot 10^{-23}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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))))