ABCF->ab-angle b

Average Error: 52.2 → 42.0

Time: 20.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{-8 \cdot \left(A \cdot F\right)}\\ t_2 := C \cdot \left(A \cdot -4\right)\\ t_3 := \mathsf{fma}\left(B, B, t_2\right)\\ \mathbf{if}\;C \leq -4.9 \cdot 10^{+117}:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{2} \cdot t_1}{A}\\ \mathbf{elif}\;C \leq -1.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_3\right)\right) \cdot \left(A + \left(C - t_0\right)\right)}}{t_3}\\ \mathbf{elif}\;C \leq -4.3 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;C \leq -1.55 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \mathbf{elif}\;C \leq -3.05 \cdot 10^{-292}:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{2}}{\frac{A}{t_1}}\\ \mathbf{elif}\;C \leq 1.7 \cdot 10^{-120}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(B, \sqrt{t_2}\right) \cdot \left(-\sqrt{F \cdot \left(2 \cdot \left(\left(C + A\right) - t_0\right)\right)}\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{C}}\right)\\ \end{array} \]

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 (hypot B (- A C)))
        (t_1 (sqrt (* -8.0 (* A F))))
        (t_2 (* C (* A -4.0)))
        (t_3 (fma B B t_2)))
   (if (<= C -4.9e+117)
     (* -0.25 (/ (* (sqrt 2.0) t_1) A))
     (if (<= C -1.5e-106)
       (/ (- (sqrt (* (* 2.0 (* F t_3)) (+ A (- C t_0))))) t_3)
       (if (<= C -4.3e-151)
         (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
         (if (<= C -1.55e-239)
           (* (/ (sqrt 2.0) B) (- (sqrt (* F (- A (hypot A B))))))
           (if (<= C -3.05e-292)
             (* -0.25 (/ (sqrt 2.0) (/ A t_1)))
             (if (<= C 1.7e-120)
               (/
                (*
                 (hypot B (sqrt t_2))
                 (- (sqrt (* F (* 2.0 (- (+ C A) t_0))))))
                t_3)
               (* (sqrt 2.0) (- (sqrt (* -0.5 (/ F C)))))))))))))

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 = hypot(B, (A - C));
	double t_1 = sqrt((-8.0 * (A * F)));
	double t_2 = C * (A * -4.0);
	double t_3 = fma(B, B, t_2);
	double tmp;
	if (C <= -4.9e+117) {
		tmp = -0.25 * ((sqrt(2.0) * t_1) / A);
	} else if (C <= -1.5e-106) {
		tmp = -sqrt(((2.0 * (F * t_3)) * (A + (C - t_0)))) / t_3;
	} else if (C <= -4.3e-151) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else if (C <= -1.55e-239) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - hypot(A, B))));
	} else if (C <= -3.05e-292) {
		tmp = -0.25 * (sqrt(2.0) / (A / t_1));
	} else if (C <= 1.7e-120) {
		tmp = (hypot(B, sqrt(t_2)) * -sqrt((F * (2.0 * ((C + A) - t_0))))) / t_3;
	} else {
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / C)));
	}
	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 = hypot(B, Float64(A - C))
	t_1 = sqrt(Float64(-8.0 * Float64(A * F)))
	t_2 = Float64(C * Float64(A * -4.0))
	t_3 = fma(B, B, t_2)
	tmp = 0.0
	if (C <= -4.9e+117)
		tmp = Float64(-0.25 * Float64(Float64(sqrt(2.0) * t_1) / A));
	elseif (C <= -1.5e-106)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_3)) * Float64(A + Float64(C - t_0))))) / t_3);
	elseif (C <= -4.3e-151)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	elseif (C <= -1.55e-239)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(A - hypot(A, B))))));
	elseif (C <= -3.05e-292)
		tmp = Float64(-0.25 * Float64(sqrt(2.0) / Float64(A / t_1)));
	elseif (C <= 1.7e-120)
		tmp = Float64(Float64(hypot(B, sqrt(t_2)) * Float64(-sqrt(Float64(F * Float64(2.0 * Float64(Float64(C + A) - t_0)))))) / t_3);
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-0.5 * Float64(F / C)))));
	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[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(-8.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(B * B + t$95$2), $MachinePrecision]}, If[LessEqual[C, -4.9e+117], N[(-0.25 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, -1.5e-106], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[C, -4.3e-151], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[C, -1.55e-239], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[C, -3.05e-292], N[(-0.25 * N[(N[Sqrt[2.0], $MachinePrecision] / N[(A / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 1.7e-120], N[(N[(N[Sqrt[B ^ 2 + N[Sqrt[t$95$2], $MachinePrecision] ^ 2], $MachinePrecision] * (-N[Sqrt[N[(F * N[(2.0 * N[(N[(C + A), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if C < -4.9000000000000001e117

   1. Initial program 59.5

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

   2. Simplified 50.4

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

   3. Applied egg-rr 50.6

      \[\leadsto \frac{-\sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F}\right)}^{3}} \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Applied egg-rr 50.5

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

   5. Taylor expanded in F around 0 59.5

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

   6. Simplified 50.4

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

   7. Taylor expanded in C around -inf 34.4

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{2} \cdot \sqrt{-8 \cdot \left(A \cdot F\right)}}{A}} \]

   if -4.9000000000000001e117 < C < -1.50000000000000009e-106

   1. Initial program 42.3

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

   2. Simplified 40.7

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

   3. Applied egg-rr 40.9

      \[\leadsto \frac{-\sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F}\right)}^{3}} \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Applied egg-rr 40.8

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

   5. Applied egg-rr 40.7

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

   if -1.50000000000000009e-106 < C < -4.30000000000000018e-151

   1. Initial program 45.8

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

   2. Simplified 42.9

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

   3. Taylor expanded in B around 0 52.1

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

   4. Simplified 52.1

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

   if -4.30000000000000018e-151 < C < -1.54999999999999993e-239

   1. Initial program 47.3

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

   2. Simplified 43.7

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

   3. Taylor expanded in C around 0 54.1

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

   4. Simplified 46.6

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

   if -1.54999999999999993e-239 < C < -3.04999999999999982e-292

   1. Initial program 49.0

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

   2. Simplified 46.5

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

   3. Applied egg-rr 46.6

      \[\leadsto \frac{-\sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F}\right)}^{3}} \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Applied egg-rr 46.5

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

   5. Taylor expanded in F around 0 48.9

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

   6. Simplified 46.8

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

   7. Taylor expanded in C around -inf 50.0

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{2} \cdot \sqrt{-8 \cdot \left(A \cdot F\right)}}{A}} \]

   8. Simplified 50.0

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{2}}{\frac{A}{\sqrt{-8 \cdot \left(F \cdot A\right)}}}} \]

   if -3.04999999999999982e-292 < C < 1.70000000000000005e-120

   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 45.3

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

   3. Applied egg-rr 43.3

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

   if 1.70000000000000005e-120 < C

   1. Initial program 58.1

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

   2. Simplified 55.3

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

   3. Taylor expanded in A around -inf 42.3

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

   4. Simplified 42.3

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

4. Final simplification 42.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -4.9 \cdot 10^{+117}:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{2} \cdot \sqrt{-8 \cdot \left(A \cdot F\right)}}{A}\\ \mathbf{elif}\;C \leq -1.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;C \leq -4.3 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;C \leq -1.55 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \mathbf{elif}\;C \leq -3.05 \cdot 10^{-292}:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{2}}{\frac{A}{\sqrt{-8 \cdot \left(A \cdot F\right)}}}\\ \mathbf{elif}\;C \leq 1.7 \cdot 10^{-120}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(B, \sqrt{C \cdot \left(A \cdot -4\right)}\right) \cdot \left(-\sqrt{F \cdot \left(2 \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{C}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(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))))