ABCF->ab-angle b

Average Error: 52.3 → 37.4

Time: 37.9s

Precision: binary64

\[[A, C]=\mathsf{sort}([A, C])\]

Math

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

↓

\[\begin{array}{l} t_0 := -\frac{{\left(-F\right)}^{0.5}}{\sqrt{C}}\\ \mathbf{if}\;B \leq -2.9228459489893423 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_2 := 2 \cdot \left(F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\\ \mathbf{if}\;B \leq -8.101188799698452 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{t_1}}{\frac{t_1}{-\sqrt{t_2}}}\\ \mathbf{elif}\;B \leq 7.250586550384806 \cdot 10^{-280}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{C}}{\sqrt{-F}}}\\ \mathbf{elif}\;B \leq 1.0618671080824585 \cdot 10^{-266}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 4.975272536947154 \cdot 10^{-246}:\\ \;\;\;\;\begin{array}{l} t_3 := \sqrt{\sqrt{C}}\\ \frac{\frac{-1}{\frac{t_3}{\sqrt{F \cdot -0.5}}}}{\frac{t_3}{\sqrt{2}}} \end{array}\\ \mathbf{elif}\;B \leq 1.6492372060764243 \cdot 10^{-136}:\\ \;\;\;\;\frac{A \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot C\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 1.0293379345528187 \cdot 10^{-124}:\\ \;\;\;\;-\frac{\sqrt{t_1 \cdot t_2}}{t_1}\\ \mathbf{elif}\;B \leq 1.8232299960392075 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \end{array}\\ \end{array} \]

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 (- (/ (pow (- F) 0.5) (sqrt C)))))
   (if (<= B -2.9228459489893423e+102)
     t_0
     (let* ((t_1 (fma A (* C -4.0) (* B B)))
            (t_2 (* 2.0 (* F (- (+ C A) (hypot B (- A C)))))))
       (if (<= B -8.101188799698452e-29)
         (/ (sqrt t_1) (/ t_1 (- (sqrt t_2))))
         (if (<= B 7.250586550384806e-280)
           (/ -1.0 (/ (sqrt C) (sqrt (- F))))
           (if (<= B 1.0618671080824585e-266)
             (/
              (-
               (sqrt (* t_1 (* 2.0 (* F (fma 2.0 A (* -0.5 (/ (* B B) C))))))))
              t_1)
             (if (<= B 4.975272536947154e-246)
               (let* ((t_3 (sqrt (sqrt C))))
                 (/ (/ -1.0 (/ t_3 (sqrt (* F -0.5)))) (/ t_3 (sqrt 2.0))))
               (if (<= B 1.6492372060764243e-136)
                 (/ (* A (* (sqrt 2.0) (sqrt (* -8.0 (* F C))))) t_1)
                 (if (<= B 1.0293379345528187e-124)
                   (- (/ (sqrt (* t_1 t_2)) t_1))
                   (if (<= B 1.8232299960392075e+25)
                     t_0
                     (-
                      (*
                       (sqrt (* F (- A (hypot A B))))
                       (/ (sqrt 2.0) 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 = -(pow(-F, 0.5) / sqrt(C));
	double tmp;
	if (B <= -2.9228459489893423e+102) {
		tmp = t_0;
	} else {
		double t_1 = fma(A, (C * -4.0), (B * B));
		double t_2 = 2.0 * (F * ((C + A) - hypot(B, (A - C))));
		double tmp_1;
		if (B <= -8.101188799698452e-29) {
			tmp_1 = sqrt(t_1) / (t_1 / -sqrt(t_2));
		} else if (B <= 7.250586550384806e-280) {
			tmp_1 = -1.0 / (sqrt(C) / sqrt(-F));
		} else if (B <= 1.0618671080824585e-266) {
			tmp_1 = -sqrt(t_1 * (2.0 * (F * fma(2.0, A, (-0.5 * ((B * B) / C)))))) / t_1;
		} else if (B <= 4.975272536947154e-246) {
			double t_3 = sqrt(sqrt(C));
			tmp_1 = (-1.0 / (t_3 / sqrt(F * -0.5))) / (t_3 / sqrt(2.0));
		} else if (B <= 1.6492372060764243e-136) {
			tmp_1 = (A * (sqrt(2.0) * sqrt(-8.0 * (F * C)))) / t_1;
		} else if (B <= 1.0293379345528187e-124) {
			tmp_1 = -(sqrt(t_1 * t_2) / t_1);
		} else if (B <= 1.8232299960392075e+25) {
			tmp_1 = t_0;
		} else {
			tmp_1 = -(sqrt(F * (A - hypot(A, B))) * (sqrt(2.0) / 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 < -2.9228459489893423e102 or 1.02933793455281871e-124 < B < 1.82322999603920747e25

   1. Initial program 53.4

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

   2. Simplified 51.3

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

   3. Taylor expanded in A around -inf 50.5

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

   4. Simplified 50.5

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

   5. Applied associate-*r/_binary64 50.5

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

   6. Applied sqrt-div_binary64 46.2

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

   7. Applied associate-*l/_binary64 46.2

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

   8. Applied pow1/2_binary64 46.2

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

   9. Applied pow1/2_binary64 46.2

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

   10. Applied pow-prod-down_binary64 46.2

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

   11. Simplified 46.2

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

   if -2.9228459489893423e102 < B < -8.10118879969845242e-29

   1. Initial program 43.5

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

   2. Simplified 39.8

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

   3. Applied sqrt-prod_binary64 35.5

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

   4. Applied distribute-rgt-neg-in_binary64 35.5

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

   5. Applied associate-/l*_binary64 35.5

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

   if -8.10118879969845242e-29 < B < 7.25058655038480578e-280

   1. Initial program 50.9

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

   2. Simplified 46.2

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

   3. Taylor expanded in A around -inf 40.5

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

   4. Simplified 40.5

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

   5. Applied associate-*r/_binary64 40.5

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

   6. Applied sqrt-div_binary64 33.3

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

   7. Applied associate-*l/_binary64 33.3

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

   8. Applied clear-num_binary64 33.3

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

   9. Applied sqrt-unprod_binary64 33.3

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

   10. Simplified 33.3

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

   if 7.25058655038480578e-280 < B < 1.0618671080824585e-266

   1. Initial program 55.2

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

   2. Simplified 48.5

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

   3. Taylor expanded in C around inf 34.7

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

   4. Simplified 34.7

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

   if 1.0618671080824585e-266 < B < 4.9752725369471542e-246

   1. Initial program 56.4

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

   2. 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. Taylor expanded in A around -inf 40.9

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

   4. Simplified 40.9

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

   5. Applied associate-*r/_binary64 40.9

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

   6. Applied sqrt-div_binary64 36.8

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

   7. Applied associate-*l/_binary64 36.8

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

   8. Applied clear-num_binary64 36.8

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

   9. Applied add-sqr-sqrt_binary64 36.9

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

   10. Applied times-frac_binary64 36.9

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

   11. Applied associate-/r*_binary64 36.9

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

   if 4.9752725369471542e-246 < B < 1.6492372060764243e-136

   1. Initial program 51.8

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

   2. Simplified 46.2

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

   3. Taylor expanded in A around -inf 35.7

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

   4. Simplified 35.7

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

   if 1.6492372060764243e-136 < B < 1.02933793455281871e-124

   1. Initial program 53.1

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

   2. Simplified 47.8

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

   3. Applied *-un-lft-identity_binary64 47.8

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

   4. Applied neg-mul-1_binary64 47.8

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

   5. Applied times-frac_binary64 47.8

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

   6. Simplified 47.8

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

   if 1.82322999603920747e25 < B

   1. Initial program 56.6

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

   2. Simplified 55.1

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

   3. Taylor expanded in C around 0 52.6

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

   4. Simplified 32.0

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

4. Final simplification 37.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.9228459489893423 \cdot 10^{+102}:\\ \;\;\;\;-\frac{{\left(-F\right)}^{0.5}}{\sqrt{C}}\\ \mathbf{elif}\;B \leq -8.101188799698452 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{-\sqrt{2 \cdot \left(F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}\\ \mathbf{elif}\;B \leq 7.250586550384806 \cdot 10^{-280}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{C}}{\sqrt{-F}}}\\ \mathbf{elif}\;B \leq 1.0618671080824585 \cdot 10^{-266}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 4.975272536947154 \cdot 10^{-246}:\\ \;\;\;\;\frac{\frac{-1}{\frac{\sqrt{\sqrt{C}}}{\sqrt{F \cdot -0.5}}}}{\frac{\sqrt{\sqrt{C}}}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq 1.6492372060764243 \cdot 10^{-136}:\\ \;\;\;\;\frac{A \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot C\right)}\right)}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.0293379345528187 \cdot 10^{-124}:\\ \;\;\;\;-\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.8232299960392075 \cdot 10^{+25}:\\ \;\;\;\;-\frac{{\left(-F\right)}^{0.5}}{\sqrt{C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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))))