ABCF->ab-angle b

Average Error: 52.4 → 40.4

Time: 23.2s

Precision: binary64

\[ \begin{array}{c}[A, C] = \mathsf{sort}([A, C])\\ \end{array} \]

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(\left(-0.5 \cdot F\right) \cdot \frac{1}{C}\right)}\\ t_1 := \sqrt{-0.5 \cdot F}\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ t_4 := \sqrt{2 \cdot \frac{-0.5 \cdot F}{C}}\\ t_5 := \sqrt{F \cdot \left(C \cdot -8\right)}\\ t_6 := -t_1\\ t_7 := \sqrt{2} \cdot \left({C}^{-0.25} \cdot \frac{t_6}{{C}^{0.25}}\right)\\ t_8 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_9 := F \cdot t_8\\ t_10 := \frac{-\sqrt{2 \cdot \left(t_9 \cdot \left(A + \left(C - t_2\right)\right)\right)}}{t_8}\\ t_11 := \frac{-\sqrt{2 \cdot \left(t_9 \cdot \left(A + A\right)\right)}}{t_8}\\ t_12 := \frac{\sqrt{2} \cdot t_6}{\sqrt{C}}\\ \mathbf{if}\;B \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_6}{\sqrt{C}}\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-41}:\\ \;\;\;\;t_10\\ \mathbf{elif}\;B \leq -3.5 \cdot 10^{-116}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{2} \cdot \left(t_1 \cdot \left(-{C}^{-0.5}\right)\right)\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-306}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-235}:\\ \;\;\;\;t_11\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-225}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-222}:\\ \;\;\;\;t_11\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-202}:\\ \;\;\;\;t_12\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{t_5} \cdot \left(F \cdot \left(B \cdot B\right)\right) + \sqrt{2} \cdot \left(A \cdot t_5\right)}{t_8}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{-118}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 10^{-46}:\\ \;\;\;\;t_10\\ \mathbf{elif}\;B \leq 10^{-30}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;B \leq 100000:\\ \;\;\;\;\sqrt{2 \cdot \left(t_8 \cdot \left(F \cdot \left(\left(C + A\right) - t_2\right)\right)\right)} \cdot \frac{1}{-t_8}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{+70}:\\ \;\;\;\;t_12\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{+101}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 10^{+165}:\\ \;\;\;\;t_7\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \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 (* (* -0.5 F) (/ 1.0 C)))))
        (t_1 (sqrt (* -0.5 F)))
        (t_2 (hypot B (- A C)))
        (t_3 (* (sqrt (* F (- A (hypot A B)))) (/ (- (sqrt 2.0)) B)))
        (t_4 (sqrt (* 2.0 (/ (* -0.5 F) C))))
        (t_5 (sqrt (* F (* C -8.0))))
        (t_6 (- t_1))
        (t_7 (* (sqrt 2.0) (* (pow C -0.25) (/ t_6 (pow C 0.25)))))
        (t_8 (fma B B (* C (* A -4.0))))
        (t_9 (* F t_8))
        (t_10 (/ (- (sqrt (* 2.0 (* t_9 (+ A (- C t_2)))))) t_8))
        (t_11 (/ (- (sqrt (* 2.0 (* t_9 (+ A A))))) t_8))
        (t_12 (/ (* (sqrt 2.0) t_6) (sqrt C))))
   (if (<= B -1e+40)
     (* (sqrt 2.0) (/ t_6 (sqrt C)))
     (if (<= B -7e-41)
       t_10
       (if (<= B -3.5e-116)
         t_4
         (if (<= B -1e-280)
           (* (sqrt 2.0) (* t_1 (- (pow C -0.5))))
           (if (<= B -1.25e-306)
             t_4
             (if (<= B 5.2e-235)
               t_11
               (if (<= B 1.85e-225)
                 t_0
                 (if (<= B 6.5e-222)
                   t_11
                   (if (<= B 1.75e-202)
                     t_12
                     (if (<= B 2.1e-141)
                       t_0
                       (if (<= B 3.8e-125)
                         (/
                          (+
                           (* (/ (sqrt 2.0) t_5) (* F (* B B)))
                           (* (sqrt 2.0) (* A t_5)))
                          t_8)
                         (if (<= B 5.6e-118)
                           t_3
                           (if (<= B 1e-46)
                             t_10
                             (if (<= B 1e-30)
                               t_7
                               (if (<= B 100000.0)
                                 (*
                                  (sqrt (* 2.0 (* t_8 (* F (- (+ C A) t_2)))))
                                  (/ 1.0 (- t_8)))
                                 (if (<= B 4.8e+70)
                                   t_12
                                   (if (<= B 2.2e+101)
                                     t_3
                                     (if (<= B 1e+165)
                                       t_7
                                       t_3))))))))))))))))))))

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 * ((-0.5 * F) * (1.0 / C))));
	double t_1 = sqrt((-0.5 * F));
	double t_2 = hypot(B, (A - C));
	double t_3 = sqrt((F * (A - hypot(A, B)))) * (-sqrt(2.0) / B);
	double t_4 = sqrt((2.0 * ((-0.5 * F) / C)));
	double t_5 = sqrt((F * (C * -8.0)));
	double t_6 = -t_1;
	double t_7 = sqrt(2.0) * (pow(C, -0.25) * (t_6 / pow(C, 0.25)));
	double t_8 = fma(B, B, (C * (A * -4.0)));
	double t_9 = F * t_8;
	double t_10 = -sqrt((2.0 * (t_9 * (A + (C - t_2))))) / t_8;
	double t_11 = -sqrt((2.0 * (t_9 * (A + A)))) / t_8;
	double t_12 = (sqrt(2.0) * t_6) / sqrt(C);
	double tmp;
	if (B <= -1e+40) {
		tmp = sqrt(2.0) * (t_6 / sqrt(C));
	} else if (B <= -7e-41) {
		tmp = t_10;
	} else if (B <= -3.5e-116) {
		tmp = t_4;
	} else if (B <= -1e-280) {
		tmp = sqrt(2.0) * (t_1 * -pow(C, -0.5));
	} else if (B <= -1.25e-306) {
		tmp = t_4;
	} else if (B <= 5.2e-235) {
		tmp = t_11;
	} else if (B <= 1.85e-225) {
		tmp = t_0;
	} else if (B <= 6.5e-222) {
		tmp = t_11;
	} else if (B <= 1.75e-202) {
		tmp = t_12;
	} else if (B <= 2.1e-141) {
		tmp = t_0;
	} else if (B <= 3.8e-125) {
		tmp = (((sqrt(2.0) / t_5) * (F * (B * B))) + (sqrt(2.0) * (A * t_5))) / t_8;
	} else if (B <= 5.6e-118) {
		tmp = t_3;
	} else if (B <= 1e-46) {
		tmp = t_10;
	} else if (B <= 1e-30) {
		tmp = t_7;
	} else if (B <= 100000.0) {
		tmp = sqrt((2.0 * (t_8 * (F * ((C + A) - t_2))))) * (1.0 / -t_8);
	} else if (B <= 4.8e+70) {
		tmp = t_12;
	} else if (B <= 2.2e+101) {
		tmp = t_3;
	} else if (B <= 1e+165) {
		tmp = t_7;
	} else {
		tmp = t_3;
	}
	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 = sqrt(Float64(2.0 * Float64(Float64(-0.5 * F) * Float64(1.0 / C))))
	t_1 = sqrt(Float64(-0.5 * F))
	t_2 = hypot(B, Float64(A - C))
	t_3 = Float64(sqrt(Float64(F * Float64(A - hypot(A, B)))) * Float64(Float64(-sqrt(2.0)) / B))
	t_4 = sqrt(Float64(2.0 * Float64(Float64(-0.5 * F) / C)))
	t_5 = sqrt(Float64(F * Float64(C * -8.0)))
	t_6 = Float64(-t_1)
	t_7 = Float64(sqrt(2.0) * Float64((C ^ -0.25) * Float64(t_6 / (C ^ 0.25))))
	t_8 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_9 = Float64(F * t_8)
	t_10 = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_9 * Float64(A + Float64(C - t_2)))))) / t_8)
	t_11 = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_9 * Float64(A + A))))) / t_8)
	t_12 = Float64(Float64(sqrt(2.0) * t_6) / sqrt(C))
	tmp = 0.0
	if (B <= -1e+40)
		tmp = Float64(sqrt(2.0) * Float64(t_6 / sqrt(C)));
	elseif (B <= -7e-41)
		tmp = t_10;
	elseif (B <= -3.5e-116)
		tmp = t_4;
	elseif (B <= -1e-280)
		tmp = Float64(sqrt(2.0) * Float64(t_1 * Float64(-(C ^ -0.5))));
	elseif (B <= -1.25e-306)
		tmp = t_4;
	elseif (B <= 5.2e-235)
		tmp = t_11;
	elseif (B <= 1.85e-225)
		tmp = t_0;
	elseif (B <= 6.5e-222)
		tmp = t_11;
	elseif (B <= 1.75e-202)
		tmp = t_12;
	elseif (B <= 2.1e-141)
		tmp = t_0;
	elseif (B <= 3.8e-125)
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) / t_5) * Float64(F * Float64(B * B))) + Float64(sqrt(2.0) * Float64(A * t_5))) / t_8);
	elseif (B <= 5.6e-118)
		tmp = t_3;
	elseif (B <= 1e-46)
		tmp = t_10;
	elseif (B <= 1e-30)
		tmp = t_7;
	elseif (B <= 100000.0)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_8 * Float64(F * Float64(Float64(C + A) - t_2))))) * Float64(1.0 / Float64(-t_8)));
	elseif (B <= 4.8e+70)
		tmp = t_12;
	elseif (B <= 2.2e+101)
		tmp = t_3;
	elseif (B <= 1e+165)
		tmp = t_7;
	else
		tmp = t_3;
	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[N[(2.0 * N[(N[(-0.5 * F), $MachinePrecision] * N[(1.0 / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(-0.5 * F), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(2.0 * N[(N[(-0.5 * F), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(F * N[(C * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = (-t$95$1)}, Block[{t$95$7 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[C, -0.25], $MachinePrecision] * N[(t$95$6 / N[Power[C, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(F * t$95$8), $MachinePrecision]}, Block[{t$95$10 = N[((-N[Sqrt[N[(2.0 * N[(t$95$9 * N[(A + N[(C - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$8), $MachinePrecision]}, Block[{t$95$11 = N[((-N[Sqrt[N[(2.0 * N[(t$95$9 * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$8), $MachinePrecision]}, Block[{t$95$12 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$6), $MachinePrecision] / N[Sqrt[C], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1e+40], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$6 / N[Sqrt[C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -7e-41], t$95$10, If[LessEqual[B, -3.5e-116], t$95$4, If[LessEqual[B, -1e-280], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * (-N[Power[C, -0.5], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.25e-306], t$95$4, If[LessEqual[B, 5.2e-235], t$95$11, If[LessEqual[B, 1.85e-225], t$95$0, If[LessEqual[B, 6.5e-222], t$95$11, If[LessEqual[B, 1.75e-202], t$95$12, If[LessEqual[B, 2.1e-141], t$95$0, If[LessEqual[B, 3.8e-125], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / t$95$5), $MachinePrecision] * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(A * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$8), $MachinePrecision], If[LessEqual[B, 5.6e-118], t$95$3, If[LessEqual[B, 1e-46], t$95$10, If[LessEqual[B, 1e-30], t$95$7, If[LessEqual[B, 100000.0], N[(N[Sqrt[N[(2.0 * N[(t$95$8 * N[(F * N[(N[(C + A), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / (-t$95$8)), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 4.8e+70], t$95$12, If[LessEqual[B, 2.2e+101], t$95$3, If[LessEqual[B, 1e+165], t$95$7, t$95$3]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 11 regimes

2. if B < -1.00000000000000003e40

   1. Initial program 56.9

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

   2. Simplified 55.5

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

   3. Taylor expanded in A around -inf 54.6

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

   4. Simplified 54.6

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

   5. Applied egg-rr 51.0

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

   if -1.00000000000000003e40 < B < -6.9999999999999999e-41 or 5.6e-118 < B < 1.00000000000000002e-46

   1. Initial program 43.8

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

   2. Simplified 37.4

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

   if -6.9999999999999999e-41 < B < -3.49999999999999984e-116 or -9.9999999999999996e-281 < B < -1.25e-306

   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 41.7

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

   3. Taylor expanded in A around -inf 40.0

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

   4. Simplified 40.0

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

   5. Applied egg-rr 43.3

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

   if -3.49999999999999984e-116 < B < -9.9999999999999996e-281

   1. Initial program 53.3

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

   2. Simplified 46.4

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

   3. Taylor expanded in A around -inf 39.1

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

   4. Simplified 39.1

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

   5. Applied egg-rr 32.4

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

   6. Applied egg-rr 32.4

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

   if -1.25e-306 < B < 5.2000000000000001e-235 or 1.84999999999999994e-225 < B < 6.5000000000000005e-222

   1. Initial program 52.0

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

   2. Simplified 44.8

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

   3. Taylor expanded in C around inf 34.5

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

   if 5.2000000000000001e-235 < B < 1.84999999999999994e-225 or 1.75e-202 < B < 2.0999999999999999e-141

   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 44.3

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

   3. Taylor expanded in A around -inf 38.3

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

   4. Simplified 38.3

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

   5. Applied egg-rr 30.0

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

   6. Applied egg-rr 30.0

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

   7. Applied egg-rr 42.6

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

   if 6.5000000000000005e-222 < B < 1.75e-202 or 1e5 < B < 4.79999999999999974e70

   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 40.2

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

   3. Taylor expanded in A around -inf 45.0

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

   4. Simplified 45.0

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

   5. Applied egg-rr 39.8

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

   6. Applied egg-rr 39.8

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

   if 2.0999999999999999e-141 < B < 3.8000000000000001e-125

   1. Initial program 47.5

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

   2. Simplified 38.8

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

   3. Taylor expanded in A around -inf 35.0

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

   4. Simplified 34.9

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

   if 3.8000000000000001e-125 < B < 5.6e-118 or 4.79999999999999974e70 < B < 2.2000000000000001e101 or 9.99999999999999899e164 < B

   1. Initial program 59.8

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

   2. Simplified 58.9

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

   3. Taylor expanded in C around 0 58.8

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

   4. Simplified 31.3

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

   if 1.00000000000000002e-46 < B < 1e-30 or 2.2000000000000001e101 < B < 9.99999999999999899e164

   1. Initial program 50.4

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

   2. Simplified 47.5

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

   3. Taylor expanded in A around -inf 52.9

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

   4. Simplified 52.9

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

   5. Applied egg-rr 48.2

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

   6. Applied egg-rr 48.2

      \[\leadsto \sqrt{2} \cdot \left(-\color{blue}{{C}^{-0.25} \cdot \frac{\sqrt{-0.5 \cdot F}}{{C}^{0.25}}}\right) \]

   if 1e-30 < B < 1e5

   1. Initial program 44.9

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

   2. Simplified 36.7

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

   3. Applied egg-rr 38.5

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
3. Recombined 11 regimes into one program.

4. Final simplification 40.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-\sqrt{-0.5 \cdot F}}{\sqrt{C}}\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \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)}\\ \mathbf{elif}\;B \leq -3.5 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2 \cdot \frac{-0.5 \cdot F}{C}}\\ \mathbf{elif}\;B \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{-0.5 \cdot F} \cdot \left(-{C}^{-0.5}\right)\right)\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{2 \cdot \frac{-0.5 \cdot F}{C}}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-235}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(-0.5 \cdot F\right) \cdot \frac{1}{C}\right)}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-222}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot F}\right)}{\sqrt{C}}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(-0.5 \cdot F\right) \cdot \frac{1}{C}\right)}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\sqrt{F \cdot \left(C \cdot -8\right)}} \cdot \left(F \cdot \left(B \cdot B\right)\right) + \sqrt{2} \cdot \left(A \cdot \sqrt{F \cdot \left(C \cdot -8\right)}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{-118}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 10^{-46}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \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)}\\ \mathbf{elif}\;B \leq 10^{-30}:\\ \;\;\;\;\sqrt{2} \cdot \left({C}^{-0.25} \cdot \frac{-\sqrt{-0.5 \cdot F}}{{C}^{0.25}}\right)\\ \mathbf{elif}\;B \leq 100000:\\ \;\;\;\;\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(C + A\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)}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot F}\right)}{\sqrt{C}}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 10^{+165}:\\ \;\;\;\;\sqrt{2} \cdot \left({C}^{-0.25} \cdot \frac{-\sqrt{-0.5 \cdot F}}{{C}^{0.25}}\right)\\ \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 2022210 
(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))))