ABCF->ab-angle a

Average Error: 52.5 → 40.2

Time: 49.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{2 \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ \mathbf{if}\;B \leq -1.5180475354614873 \cdot 10^{+53}:\\ \;\;\;\;-\frac{t_0}{-B}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -1.6993691026398643 \cdot 10^{-24}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot C - 0.5 \cdot \frac{{B}^{2}}{A}\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \sqrt{t_1}\\ t_3 := \frac{-t_0}{t_2}\\ t_4 := -t_0\\ \mathbf{if}\;B \leq -5.281598276878717 \cdot 10^{-147}:\\ \;\;\;\;\frac{t_2}{t_2} \cdot t_3\\ \mathbf{elif}\;B \leq -5.219385286668219 \cdot 10^{-272}:\\ \;\;\;\;\begin{array}{l} t_5 := \sqrt{-8 \cdot \left(F \cdot A\right)}\\ \frac{-\mathsf{fma}\left(t_5, C \cdot \sqrt{2}, \frac{F \cdot \left(\left(B \cdot B\right) \cdot \sqrt{2}\right)}{t_5}\right)}{t_1} \end{array}\\ \mathbf{elif}\;B \leq -3.853481712582081 \cdot 10^{-298}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A - 0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 2.0497063361135274 \cdot 10^{-260}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(0.5, \frac{B \cdot B}{C}, 2 \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.264548681151659 \cdot 10^{-128}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;B \leq 9.192513924446474 \cdot 10^{-33}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t_4}{B}\\ \end{array}\\ \end{array}\\ \end{array} \]

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 -1.5180475354614873e+53)
     (- (/ t_0 (- B)))
     (let* ((t_1 (fma A (* C -4.0) (* B B))))
       (if (<= B -1.6993691026398643e-24)
         (/
          (-
           (sqrt
            (* t_1 (* 2.0 (* F (- (* 2.0 C) (* 0.5 (/ (pow B 2.0) A))))))))
          t_1)
         (let* ((t_2 (sqrt t_1)) (t_3 (/ (- t_0) t_2)) (t_4 (- t_0)))
           (if (<= B -5.281598276878717e-147)
             (* (/ t_2 t_2) t_3)
             (if (<= B -5.219385286668219e-272)
               (let* ((t_5 (sqrt (* -8.0 (* F A)))))
                 (/
                  (-
                   (fma
                    t_5
                    (* C (sqrt 2.0))
                    (/ (* F (* (* B B) (sqrt 2.0))) t_5)))
                  t_1))
               (if (<= B -3.853481712582081e-298)
                 (/
                  (-
                   (sqrt
                    (*
                     t_1
                     (* 2.0 (* F (- (* 2.0 A) (* 0.5 (/ (pow B 2.0) C))))))))
                  t_1)
                 (if (<= B 2.0497063361135274e-260)
                   (/
                    (-
                     (sqrt
                      (* t_1 (* 2.0 (* F (fma 0.5 (/ (* B B) C) (* 2.0 C)))))))
                    t_1)
                   (if (<= B 1.264548681151659e-128)
                     (- (* (sqrt 2.0) (sqrt (* -0.5 (/ F A)))))
                     (if (<= B 9.192513924446474e-33) t_3 (/ t_4 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 <= -1.5180475354614873e+53) {
		tmp = -(t_0 / -B);
	} else {
		double t_1 = fma(A, (C * -4.0), (B * B));
		double tmp_1;
		if (B <= -1.6993691026398643e-24) {
			tmp_1 = -sqrt(t_1 * (2.0 * (F * ((2.0 * C) - (0.5 * (pow(B, 2.0) / A)))))) / t_1;
		} else {
			double t_2 = sqrt(t_1);
			double t_3 = -t_0 / t_2;
			double t_4 = -t_0;
			double tmp_2;
			if (B <= -5.281598276878717e-147) {
				tmp_2 = (t_2 / t_2) * t_3;
			} else if (B <= -5.219385286668219e-272) {
				double t_5 = sqrt(-8.0 * (F * A));
				tmp_2 = -fma(t_5, (C * sqrt(2.0)), ((F * ((B * B) * sqrt(2.0))) / t_5)) / t_1;
			} else if (B <= -3.853481712582081e-298) {
				tmp_2 = -sqrt(t_1 * (2.0 * (F * ((2.0 * A) - (0.5 * (pow(B, 2.0) / C)))))) / t_1;
			} else if (B <= 2.0497063361135274e-260) {
				tmp_2 = -sqrt(t_1 * (2.0 * (F * fma(0.5, ((B * B) / C), (2.0 * C))))) / t_1;
			} else if (B <= 1.264548681151659e-128) {
				tmp_2 = -(sqrt(2.0) * sqrt(-0.5 * (F / A)));
			} else if (B <= 9.192513924446474e-33) {
				tmp_2 = t_3;
			} else {
				tmp_2 = t_4 / B;
			}
			tmp_1 = tmp_2;
		}
		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 9 regimes

2. if B < -1.5180475354614873e53

   1. Initial program 57.7

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

   2. Simplified 56.3

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

   3. Applied add-sqr-sqrt_binary64 56.5

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

   4. Applied sqrt-prod_binary64 50.3

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

   5. Applied distribute-lft-neg-in_binary64 50.3

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

   6. Applied times-frac_binary64 50.3

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

   7. Simplified 48.9

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

   8. Taylor expanded in B around -inf 29.8

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

   if -1.5180475354614873e53 < B < -1.6993691026398643e-24

   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. Simplified 35.9

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

   3. Taylor expanded in A around -inf 54.5

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

   if -1.6993691026398643e-24 < B < -5.28159827687871672e-147

   1. Initial program 47.2

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

   2. Simplified 42.3

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

   3. Applied add-sqr-sqrt_binary64 47.7

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

   4. Applied sqrt-prod_binary64 43.8

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

   5. Applied distribute-rgt-neg-in_binary64 43.8

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

   6. Applied times-frac_binary64 43.7

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

   if -5.28159827687871672e-147 < B < -5.2193852866682191e-272

   1. Initial program 53.8

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

   2. Simplified 48.7

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

   3. Taylor expanded in C around inf 51.6

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

   4. Simplified 51.6

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

   if -5.2193852866682191e-272 < B < -3.85348171258208062e-298

   1. Initial program 52.1

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

   2. Simplified 47.6

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

   3. Taylor expanded in C around -inf 46.5

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

   if -3.85348171258208062e-298 < B < 2.04970633611352736e-260

   1. Initial program 53.6

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

   2. Simplified 48.5

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

   3. Taylor expanded in C around inf 50.1

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

   4. Simplified 50.1

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

   if 2.04970633611352736e-260 < B < 1.26454868115165894e-128

   1. Initial program 52.2

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

   2. Simplified 47.6

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

   3. Taylor expanded in C around inf 49.3

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

   4. Simplified 49.3

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

   if 1.26454868115165894e-128 < B < 9.19251392444647417e-33

   1. Initial program 48.5

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

   2. Simplified 43.8

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

   3. Applied add-sqr-sqrt_binary64 49.1

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

   4. Applied sqrt-prod_binary64 43.8

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

   5. Applied distribute-lft-neg-in_binary64 43.8

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

   6. Applied times-frac_binary64 43.7

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

   7. Simplified 43.6

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

   if 9.19251392444647417e-33 < B

   1. Initial program 54.2

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

   2. Simplified 52.2

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

   3. Applied add-sqr-sqrt_binary64 53.0

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

   4. Applied sqrt-prod_binary64 47.3

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

   5. Applied distribute-lft-neg-in_binary64 47.3

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

   6. Applied times-frac_binary64 47.3

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

   7. Simplified 46.2

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

   8. Taylor expanded in A around 0 32.5

      \[\leadsto -1 \cdot \frac{\sqrt{2 \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\color{blue}{B}} \]
3. Recombined 9 regimes into one program.

4. Final simplification 40.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.5180475354614873 \cdot 10^{+53}:\\ \;\;\;\;-\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 -1.6993691026398643 \cdot 10^{-24}:\\ \;\;\;\;\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 -5.281598276878717 \cdot 10^{-147}:\\ \;\;\;\;\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)}}\\ \mathbf{elif}\;B \leq -5.219385286668219 \cdot 10^{-272}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\sqrt{-8 \cdot \left(F \cdot A\right)}, C \cdot \sqrt{2}, \frac{F \cdot \left(\left(B \cdot B\right) \cdot \sqrt{2}\right)}{\sqrt{-8 \cdot \left(F \cdot A\right)}}\right)}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq -3.853481712582081 \cdot 10^{-298}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A - 0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 2.0497063361135274 \cdot 10^{-260}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(0.5, \frac{B \cdot B}{C}, 2 \cdot C\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.264548681151659 \cdot 10^{-128}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;B \leq 9.192513924446474 \cdot 10^{-33}:\\ \;\;\;\;\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)}}\\ \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 2021340 
(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))))