ABCF->ab-angle b

Average Error: 52.5 → 37.5

Time: 29.5s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := 2 \cdot \left(F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;C \leq -4.127320517617318 \cdot 10^{-160}:\\ \;\;\;\;\frac{A \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}\right)}{t_1}\\ \mathbf{elif}\;C \leq -1.0003519539071463 \cdot 10^{-281}:\\ \;\;\;\;\frac{-1}{\frac{t_1}{\sqrt{t_1 \cdot t_0}}}\\ \mathbf{elif}\;C \leq 1.7199397216927585 \cdot 10^{-66}:\\ \;\;\;\;-\frac{\sqrt{t_0}}{\sqrt{t_1}}\\ \mathbf{elif}\;C \leq 1.780894702735933:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{-2 \cdot \frac{C}{F}}}{\sqrt{2}}}\\ \mathbf{elif}\;C \leq 8.349140996588686 \cdot 10^{+33}:\\ \;\;\;\;-\sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;C \leq 1.8496276951365653 \cdot 10^{+192}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{t_1}\\ \end{array} \]

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 (* 2.0 (* F (- (+ C A) (hypot B (- A C))))))
        (t_1 (fma A (* C -4.0) (* B B))))
   (if (<= C -4.127320517617318e-160)
     (/ (* A (* (sqrt 2.0) (sqrt (* -8.0 (* C F))))) t_1)
     (if (<= C -1.0003519539071463e-281)
       (/ -1.0 (/ t_1 (sqrt (* t_1 t_0))))
       (if (<= C 1.7199397216927585e-66)
         (- (/ (sqrt t_0) (sqrt t_1)))
         (if (<= C 1.780894702735933)
           (/ -1.0 (/ (sqrt (* -2.0 (/ C F))) (sqrt 2.0)))
           (if (<= C 8.349140996588686e+33)
             (- (* (sqrt (* F (- C (hypot B C)))) (/ (sqrt 2.0) B)))
             (if (<= C 1.8496276951365653e+192)
               (- (* (sqrt 2.0) (sqrt (* -0.5 (/ F C)))))
               (/
                (-
                 (sqrt
                  (* t_1 (* 2.0 (* F (fma 2.0 A (* -0.5 (/ (* B B) C))))))))
                t_1)))))))))

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 = 2.0 * (F * ((C + A) - hypot(B, (A - C))));
	double t_1 = fma(A, (C * -4.0), (B * B));
	double tmp;
	if (C <= -4.127320517617318e-160) {
		tmp = (A * (sqrt(2.0) * sqrt((-8.0 * (C * F))))) / t_1;
	} else if (C <= -1.0003519539071463e-281) {
		tmp = -1.0 / (t_1 / sqrt((t_1 * t_0)));
	} else if (C <= 1.7199397216927585e-66) {
		tmp = -(sqrt(t_0) / sqrt(t_1));
	} else if (C <= 1.780894702735933) {
		tmp = -1.0 / (sqrt((-2.0 * (C / F))) / sqrt(2.0));
	} else if (C <= 8.349140996588686e+33) {
		tmp = -(sqrt((F * (C - hypot(B, C)))) * (sqrt(2.0) / B));
	} else if (C <= 1.8496276951365653e+192) {
		tmp = -(sqrt(2.0) * sqrt((-0.5 * (F / C))));
	} else {
		tmp = -sqrt((t_1 * (2.0 * (F * fma(2.0, A, (-0.5 * ((B * B) / C))))))) / t_1;
	}
	return tmp;
}

Julia

function code(A, B, C, F)
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)))
end

↓

function code(A, B, C, F)
	t_0 = Float64(2.0 * Float64(F * Float64(Float64(C + A) - hypot(B, Float64(A - C)))))
	t_1 = fma(A, Float64(C * -4.0), Float64(B * B))
	tmp = 0.0
	if (C <= -4.127320517617318e-160)
		tmp = Float64(Float64(A * Float64(sqrt(2.0) * sqrt(Float64(-8.0 * Float64(C * F))))) / t_1);
	elseif (C <= -1.0003519539071463e-281)
		tmp = Float64(-1.0 / Float64(t_1 / sqrt(Float64(t_1 * t_0))));
	elseif (C <= 1.7199397216927585e-66)
		tmp = Float64(-Float64(sqrt(t_0) / sqrt(t_1)));
	elseif (C <= 1.780894702735933)
		tmp = Float64(-1.0 / Float64(sqrt(Float64(-2.0 * Float64(C / F))) / sqrt(2.0)));
	elseif (C <= 8.349140996588686e+33)
		tmp = Float64(-Float64(sqrt(Float64(F * Float64(C - hypot(B, C)))) * Float64(sqrt(2.0) / B)));
	elseif (C <= 1.8496276951365653e+192)
		tmp = Float64(-Float64(sqrt(2.0) * sqrt(Float64(-0.5 * Float64(F / C)))));
	else
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(2.0 * Float64(F * fma(2.0, A, Float64(-0.5 * Float64(Float64(B * B) / C)))))))) / t_1);
	end
	return tmp
end

Wolfram

code[A_, B_, C_, F_] := N[((-N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(2.0 * N[(F * N[(N[(C + A), $MachinePrecision] - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -4.127320517617318e-160], N[(N[(A * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(-8.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[C, -1.0003519539071463e-281], N[(-1.0 / N[(t$95$1 / N[Sqrt[N[(t$95$1 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 1.7199397216927585e-66], (-N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), If[LessEqual[C, 1.780894702735933], N[(-1.0 / N[(N[Sqrt[N[(-2.0 * N[(C / F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 8.349140996588686e+33], (-N[(N[Sqrt[N[(F * N[(C - N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]), If[LessEqual[C, 1.8496276951365653e+192], (-N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), N[((-N[Sqrt[N[(t$95$1 * N[(2.0 * N[(F * N[(2.0 * A + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]]]]]]]

Error

Bits error versus A

Bits error versus B

Bits error versus C

Bits error versus F

Derivation

1. Split input into 7 regimes

2. if C < -4.1273205176173177e-160

   1. Initial program 48.7

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

   2. Simplified 43.7

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

   3. Taylor expanded in A around -inf 40.1

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

   4. Simplified 40.1

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

   if -4.1273205176173177e-160 < C < -1.00035195390714629e-281

   1. Initial program 46.9

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

   2. Simplified 39.6

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

   3. Applied neg-mul-1_binary64 39.6

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

   4. Applied associate-/l*_binary64 39.6

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

   if -1.00035195390714629e-281 < C < 1.7199397216927585e-66

   1. Initial program 43.4

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

   2. Simplified 38.5

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

   3. Applied add-sqr-sqrt_binary64 39.2

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

   4. Applied sqrt-prod_binary64 30.8

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

   5. Applied distribute-lft-neg-in_binary64 30.8

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

   6. Applied times-frac_binary64 30.8

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

   7. Simplified 30.3

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

   if 1.7199397216927585e-66 < C < 1.780894702735933

   1. Initial program 46.5

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

   2. Simplified 43.6

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

   3. Taylor expanded in C around inf 46.6

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

   4. Simplified 46.6

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

   5. Applied neg-mul-1_binary64 46.6

      \[\leadsto \frac{\color{blue}{-1 \cdot \sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   6. Applied associate-/l*_binary64 46.6

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}}} \]

   7. Taylor expanded in A around 0 38.2

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

   if 1.780894702735933 < C < 8.3491409965886856e33

   1. Initial program 51.3

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

   2. Simplified 49.9

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

   3. Taylor expanded in A around 0 57.8

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

   4. Simplified 52.2

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

   if 8.3491409965886856e33 < C < 1.8496276951365653e192

   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. Simplified 59.5

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

   3. Taylor expanded in A around -inf 36.0

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

   4. Simplified 36.0

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

   if 1.8496276951365653e192 < 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. Simplified 62.6

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

   3. Taylor expanded in C around inf 43.3

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

   4. Simplified 43.3

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 37.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -4.127320517617318 \cdot 10^{-160}:\\ \;\;\;\;\frac{A \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}\right)}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;C \leq -1.0003519539071463 \cdot 10^{-281}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}\\ \mathbf{elif}\;C \leq 1.7199397216927585 \cdot 10^{-66}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}\\ \mathbf{elif}\;C \leq 1.780894702735933:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{-2 \cdot \frac{C}{F}}}{\sqrt{2}}}\\ \mathbf{elif}\;C \leq 8.349140996588686 \cdot 10^{+33}:\\ \;\;\;\;-\sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;C \leq 1.8496276951365653 \cdot 10^{+192}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))))