ABCF->ab-angle b

Average Error: 52.5 → 45.2

Time: 20.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := -\sqrt{\frac{-F}{C}}\\ t_1 := F \cdot \left(B \cdot B\right)\\ t_2 := -\sqrt{2}\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ t_4 := \mathsf{fma}\left(B, B, C \cdot C - C \cdot C\right)\\ t_5 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_6 := 2 \cdot \left(A + \left(C - t_3\right)\right)\\ t_7 := -\frac{{\left({t_5}^{0.25} \cdot {\left(F \cdot t_6\right)}^{0.25}\right)}^{2}}{t_5}\\ \mathbf{if}\;A \leq -1.9 \cdot 10^{+94}:\\ \;\;\;\;{\left({\left(t_5 \cdot \left(\left(F \cdot 2\right) \cdot \left(\left(A + C\right) - t_3\right)\right)\right)}^{0.25}\right)}^{2} \cdot \frac{-1}{t_5}\\ \mathbf{elif}\;A \leq -5.2 \cdot 10^{-53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -1.06 \cdot 10^{-116}:\\ \;\;\;\;\frac{-\sqrt{\left(t_5 \cdot F\right) \cdot t_6}}{t_5}\\ \mathbf{elif}\;A \leq -1.8 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\frac{F}{C} \cdot -0.5} \cdot t_2\\ \mathbf{elif}\;A \leq -8.8 \cdot 10^{-244}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;A \leq -6.3 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 1.4 \cdot 10^{-251}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)} \cdot \frac{t_2}{B}\\ \mathbf{elif}\;A \leq 9.2 \cdot 10^{-39}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;A \leq 2.25 \cdot 10^{+117}:\\ \;\;\;\;-\sqrt{-0.5 \cdot \frac{F}{\frac{A}{1 + A \cdot \sqrt{\frac{1}{\mathsf{fma}\left(A, A, B \cdot B\right)}}}}}\\ \mathbf{elif}\;A \leq 1.02 \cdot 10^{+185}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(2, \frac{\mathsf{fma}\left(2, \left(F \cdot \left(C \cdot C\right)\right) \cdot t_4, -0.5 \cdot \left(t_4 \cdot t_1\right)\right)}{A}, \mathsf{fma}\left(2, \mathsf{fma}\left(2, t_4 \cdot \left(C \cdot F\right), t_1 \cdot \left(C + C\right)\right), -8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)\right)}}{t_5}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \end{array} \]

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 (/ (- F) C))))
        (t_1 (* F (* B B)))
        (t_2 (- (sqrt 2.0)))
        (t_3 (hypot B (- A C)))
        (t_4 (fma B B (- (* C C) (* C C))))
        (t_5 (fma B B (* C (* A -4.0))))
        (t_6 (* 2.0 (+ A (- C t_3))))
        (t_7 (- (/ (pow (* (pow t_5 0.25) (pow (* F t_6) 0.25)) 2.0) t_5))))
   (if (<= A -1.9e+94)
     (*
      (pow (pow (* t_5 (* (* F 2.0) (- (+ A C) t_3))) 0.25) 2.0)
      (/ -1.0 t_5))
     (if (<= A -5.2e-53)
       t_0
       (if (<= A -1.06e-116)
         (/ (- (sqrt (* (* t_5 F) t_6))) t_5)
         (if (<= A -1.8e-150)
           (* (sqrt (* (/ F C) -0.5)) t_2)
           (if (<= A -8.8e-244)
             t_7
             (if (<= A -6.3e-308)
               t_0
               (if (<= A 1.4e-251)
                 (* (sqrt (* F (- C (hypot C B)))) (/ t_2 B))
                 (if (<= A 9.2e-39)
                   t_7
                   (if (<= A 2.25e+117)
                     (-
                      (sqrt
                       (*
                        -0.5
                        (/
                         F
                         (/
                          A
                          (+ 1.0 (* A (sqrt (/ 1.0 (fma A A (* B B)))))))))))
                     (if (<= A 1.02e+185)
                       (/
                        (-
                         (sqrt
                          (fma
                           2.0
                           (/
                            (fma
                             2.0
                             (* (* F (* C C)) t_4)
                             (* -0.5 (* t_4 t_1)))
                            A)
                           (fma
                            2.0
                            (fma 2.0 (* t_4 (* C F)) (* t_1 (+ C C)))
                            (* -8.0 (* (* C F) (* A (+ C C))))))))
                        t_5)
                       (- (sqrt (/ (- F) A)))))))))))))))

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((-F / C));
	double t_1 = F * (B * B);
	double t_2 = -sqrt(2.0);
	double t_3 = hypot(B, (A - C));
	double t_4 = fma(B, B, ((C * C) - (C * C)));
	double t_5 = fma(B, B, (C * (A * -4.0)));
	double t_6 = 2.0 * (A + (C - t_3));
	double t_7 = -(pow((pow(t_5, 0.25) * pow((F * t_6), 0.25)), 2.0) / t_5);
	double tmp;
	if (A <= -1.9e+94) {
		tmp = pow(pow((t_5 * ((F * 2.0) * ((A + C) - t_3))), 0.25), 2.0) * (-1.0 / t_5);
	} else if (A <= -5.2e-53) {
		tmp = t_0;
	} else if (A <= -1.06e-116) {
		tmp = -sqrt(((t_5 * F) * t_6)) / t_5;
	} else if (A <= -1.8e-150) {
		tmp = sqrt(((F / C) * -0.5)) * t_2;
	} else if (A <= -8.8e-244) {
		tmp = t_7;
	} else if (A <= -6.3e-308) {
		tmp = t_0;
	} else if (A <= 1.4e-251) {
		tmp = sqrt((F * (C - hypot(C, B)))) * (t_2 / B);
	} else if (A <= 9.2e-39) {
		tmp = t_7;
	} else if (A <= 2.25e+117) {
		tmp = -sqrt((-0.5 * (F / (A / (1.0 + (A * sqrt((1.0 / fma(A, A, (B * B))))))))));
	} else if (A <= 1.02e+185) {
		tmp = -sqrt(fma(2.0, (fma(2.0, ((F * (C * C)) * t_4), (-0.5 * (t_4 * t_1))) / A), fma(2.0, fma(2.0, (t_4 * (C * F)), (t_1 * (C + C))), (-8.0 * ((C * F) * (A * (C + C))))))) / t_5;
	} else {
		tmp = -sqrt((-F / A));
	}
	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(-sqrt(Float64(Float64(-F) / C)))
	t_1 = Float64(F * Float64(B * B))
	t_2 = Float64(-sqrt(2.0))
	t_3 = hypot(B, Float64(A - C))
	t_4 = fma(B, B, Float64(Float64(C * C) - Float64(C * C)))
	t_5 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_6 = Float64(2.0 * Float64(A + Float64(C - t_3)))
	t_7 = Float64(-Float64((Float64((t_5 ^ 0.25) * (Float64(F * t_6) ^ 0.25)) ^ 2.0) / t_5))
	tmp = 0.0
	if (A <= -1.9e+94)
		tmp = Float64(((Float64(t_5 * Float64(Float64(F * 2.0) * Float64(Float64(A + C) - t_3))) ^ 0.25) ^ 2.0) * Float64(-1.0 / t_5));
	elseif (A <= -5.2e-53)
		tmp = t_0;
	elseif (A <= -1.06e-116)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_5 * F) * t_6))) / t_5);
	elseif (A <= -1.8e-150)
		tmp = Float64(sqrt(Float64(Float64(F / C) * -0.5)) * t_2);
	elseif (A <= -8.8e-244)
		tmp = t_7;
	elseif (A <= -6.3e-308)
		tmp = t_0;
	elseif (A <= 1.4e-251)
		tmp = Float64(sqrt(Float64(F * Float64(C - hypot(C, B)))) * Float64(t_2 / B));
	elseif (A <= 9.2e-39)
		tmp = t_7;
	elseif (A <= 2.25e+117)
		tmp = Float64(-sqrt(Float64(-0.5 * Float64(F / Float64(A / Float64(1.0 + Float64(A * sqrt(Float64(1.0 / fma(A, A, Float64(B * B)))))))))));
	elseif (A <= 1.02e+185)
		tmp = Float64(Float64(-sqrt(fma(2.0, Float64(fma(2.0, Float64(Float64(F * Float64(C * C)) * t_4), Float64(-0.5 * Float64(t_4 * t_1))) / A), fma(2.0, fma(2.0, Float64(t_4 * Float64(C * F)), Float64(t_1 * Float64(C + C))), Float64(-8.0 * Float64(Float64(C * F) * Float64(A * Float64(C + C)))))))) / t_5);
	else
		tmp = Float64(-sqrt(Float64(Float64(-F) / A)));
	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[((-F) / C), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$1 = N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$3 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[(B * B + N[(N[(C * C), $MachinePrecision] - N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(2.0 * N[(A + N[(C - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = (-N[(N[Power[N[(N[Power[t$95$5, 0.25], $MachinePrecision] * N[Power[N[(F * t$95$6), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$5), $MachinePrecision])}, If[LessEqual[A, -1.9e+94], N[(N[Power[N[Power[N[(t$95$5 * N[(N[(F * 2.0), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, -5.2e-53], t$95$0, If[LessEqual[A, -1.06e-116], N[((-N[Sqrt[N[(N[(t$95$5 * F), $MachinePrecision] * t$95$6), $MachinePrecision]], $MachinePrecision]) / t$95$5), $MachinePrecision], If[LessEqual[A, -1.8e-150], N[(N[Sqrt[N[(N[(F / C), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[A, -8.8e-244], t$95$7, If[LessEqual[A, -6.3e-308], t$95$0, If[LessEqual[A, 1.4e-251], N[(N[Sqrt[N[(F * N[(C - N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 9.2e-39], t$95$7, If[LessEqual[A, 2.25e+117], (-N[Sqrt[N[(-0.5 * N[(F / N[(A / N[(1.0 + N[(A * N[Sqrt[N[(1.0 / N[(A * A + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[A, 1.02e+185], N[((-N[Sqrt[N[(2.0 * N[(N[(2.0 * N[(N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(-0.5 * N[(t$95$4 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * N[(2.0 * N[(t$95$4 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-8.0 * N[(N[(C * F), $MachinePrecision] * N[(A * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$5), $MachinePrecision], (-N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision])]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 9 regimes

2. if A < -1.8999999999999998e94

   1. Initial program 58.3

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

   2. Simplified 50.3

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

   3. Applied egg-rr 50.4

      \[\leadsto \frac{-\color{blue}{{\left({\left(\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Applied egg-rr 51.6

      \[\leadsto \frac{-{\color{blue}{\left({\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(F \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{0.25}\right)}}^{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Applied egg-rr 50.1

      \[\leadsto \color{blue}{\left(-{\left({\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]

   if -1.8999999999999998e94 < A < -5.19999999999999993e-53 or -8.79999999999999939e-244 < A < -6.2999999999999995e-308

   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 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 41.5

      \[\leadsto \frac{-\color{blue}{{\left({\left(\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Applied egg-rr 44.0

      \[\leadsto \frac{-{\color{blue}{\left({\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(F \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{0.25}\right)}}^{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Taylor expanded in A around -inf 49.5

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

   6. Simplified 49.5

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

   if -5.19999999999999993e-53 < A < -1.05999999999999997e-116

   1. Initial program 44.2

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

   2. Simplified 41.5

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

   if -1.05999999999999997e-116 < A < -1.8000000000000001e-150

   1. Initial program 45.9

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

   2. Simplified 42.1

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

   3. Taylor expanded in A around -inf 50.5

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

   4. Simplified 50.5

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

   if -1.8000000000000001e-150 < A < -8.79999999999999939e-244 or 1.39999999999999994e-251 < A < 9.20000000000000033e-39

   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 44.6

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

   3. Applied egg-rr 45.6

      \[\leadsto \frac{-\color{blue}{{\left({\left(\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Applied egg-rr 41.4

      \[\leadsto \frac{-{\color{blue}{\left({\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(F \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{0.25}\right)}}^{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if -6.2999999999999995e-308 < A < 1.39999999999999994e-251

   1. Initial program 49.3

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

   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. Applied egg-rr 44.1

      \[\leadsto \frac{-\color{blue}{{\left({\left(\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Applied egg-rr 38.4

      \[\leadsto \frac{-{\color{blue}{\left({\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(F \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{0.25}\right)}}^{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Taylor expanded in A around 0 55.0

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

   6. Simplified 47.7

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

   if 9.20000000000000033e-39 < A < 2.25e117

   1. Initial program 56.3

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

   2. Simplified 54.8

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

   3. Applied egg-rr 55.4

      \[\leadsto \frac{-\color{blue}{{\left({\left(\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Applied egg-rr 51.5

      \[\leadsto \frac{-{\color{blue}{\left({\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(F \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{0.25}\right)}}^{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Taylor expanded in C around 0 52.6

      \[\leadsto \frac{-{\left({\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(F \cdot \left(2 \cdot \left(A + \color{blue}{\left(C \cdot \left(1 - -1 \cdot \left(A \cdot \sqrt{\frac{1}{{B}^{2} + {A}^{2}}}\right)\right) + -1 \cdot \sqrt{{B}^{2} + {A}^{2}}\right)}\right)\right)\right)}^{0.25}\right)}^{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   6. Simplified 52.6

      \[\leadsto \frac{-{\left({\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(F \cdot \left(2 \cdot \left(A + \color{blue}{\mathsf{fma}\left(C, 1 - \left(-A\right) \cdot \sqrt{\frac{1}{B \cdot B + A \cdot A}}, -\mathsf{hypot}\left(A, B\right)\right)}\right)\right)\right)}^{0.25}\right)}^{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   7. Taylor expanded in C around inf 43.3

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

   8. Simplified 43.3

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

   if 2.25e117 < A < 1.0200000000000001e185

   1. Initial program 62.8

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

   2. Simplified 61.9

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

   3. Taylor expanded in A around inf 48.2

      \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \frac{-0.5 \cdot \left(\left(\left({B}^{2} + {C}^{2}\right) - {\left(-1 \cdot C\right)}^{2}\right) \cdot \left(F \cdot {B}^{2}\right)\right) + 2 \cdot \left(\left(\left({B}^{2} + {C}^{2}\right) - {\left(-1 \cdot C\right)}^{2}\right) \cdot \left({C}^{2} \cdot F\right)\right)}{A} + \left(-8 \cdot \left(A \cdot \left(\left(C - -1 \cdot C\right) \cdot \left(C \cdot F\right)\right)\right) + 2 \cdot \left(2 \cdot \left(\left(\left({B}^{2} + {C}^{2}\right) - {\left(-1 \cdot C\right)}^{2}\right) \cdot \left(C \cdot F\right)\right) + \left(C - -1 \cdot C\right) \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Simplified 45.1

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{\mathsf{fma}\left(2, \left(F \cdot \left(C \cdot C\right)\right) \cdot \mathsf{fma}\left(B, B, C \cdot C - C \cdot C\right), -0.5 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \mathsf{fma}\left(B, B, C \cdot C - C \cdot C\right)\right)\right)}{A}, \mathsf{fma}\left(2, \mathsf{fma}\left(2, \left(F \cdot C\right) \cdot \mathsf{fma}\left(B, B, C \cdot C - C \cdot C\right), \left(F \cdot \left(B \cdot B\right)\right) \cdot \left(C + C\right)\right), -8 \cdot \left(\left(A \cdot \left(C + C\right)\right) \cdot \left(F \cdot C\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 1.0200000000000001e185 < A

   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 62.8

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

   3. Applied egg-rr 62.8

      \[\leadsto \frac{-\color{blue}{{\left({\left(\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Taylor expanded in B around 0 40.6

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

   5. Simplified 40.6

      \[\leadsto \color{blue}{-\sqrt{-\frac{F}{A}}} \]
3. Recombined 9 regimes into one program.

4. Final simplification 45.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.9 \cdot 10^{+94}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.25}\right)}^{2} \cdot \frac{-1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq -5.2 \cdot 10^{-53}:\\ \;\;\;\;-\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq -1.06 \cdot 10^{-116}:\\ \;\;\;\;\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)}\\ \mathbf{elif}\;A \leq -1.8 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\frac{F}{C} \cdot -0.5} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;A \leq -8.8 \cdot 10^{-244}:\\ \;\;\;\;-\frac{{\left({\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(F \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq -6.3 \cdot 10^{-308}:\\ \;\;\;\;-\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq 1.4 \cdot 10^{-251}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;A \leq 9.2 \cdot 10^{-39}:\\ \;\;\;\;-\frac{{\left({\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(F \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq 2.25 \cdot 10^{+117}:\\ \;\;\;\;-\sqrt{-0.5 \cdot \frac{F}{\frac{A}{1 + A \cdot \sqrt{\frac{1}{\mathsf{fma}\left(A, A, B \cdot B\right)}}}}}\\ \mathbf{elif}\;A \leq 1.02 \cdot 10^{+185}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(2, \frac{\mathsf{fma}\left(2, \left(F \cdot \left(C \cdot C\right)\right) \cdot \mathsf{fma}\left(B, B, C \cdot C - C \cdot C\right), -0.5 \cdot \left(\mathsf{fma}\left(B, B, C \cdot C - C \cdot C\right) \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)\right)}{A}, \mathsf{fma}\left(2, \mathsf{fma}\left(2, \mathsf{fma}\left(B, B, C \cdot C - C \cdot C\right) \cdot \left(C \cdot F\right), \left(F \cdot \left(B \cdot B\right)\right) \cdot \left(C + C\right)\right), -8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{-F}{A}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022206 
(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))))