ABCF->ab-angle b

Average Error: 52.6 → 37.6

Time: 28.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-16 \cdot \left(C \cdot F\right)}\\ t_1 := \sqrt{t_0}\\ t_2 := 2 \cdot \left(F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\\ t_3 := \sqrt{t_2}\\ t_4 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_5 := \sqrt{t_4}\\ \mathbf{if}\;{B}^{2} \leq 1.850236647975572 \cdot 10^{-176}:\\ \;\;\;\;\frac{-\sqrt{t_4 \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;{B}^{2} \leq 3.351092992184262 \cdot 10^{-94}:\\ \;\;\;\;-0.25 \cdot \frac{1}{\frac{C}{t_0}}\\ \mathbf{elif}\;{B}^{2} \leq 3.0541871978177776 \cdot 10^{-47}:\\ \;\;\;\;\frac{-t_5}{\frac{t_4}{t_3}}\\ \mathbf{elif}\;{B}^{2} \leq 6.061880387862554 \cdot 10^{-11}:\\ \;\;\;\;-0.25 \cdot \frac{t_0}{C}\\ \mathbf{elif}\;{B}^{2} \leq 1.8727221069173937 \cdot 10^{+113}:\\ \;\;\;\;\frac{-1}{\frac{t_4}{\sqrt{t_4 \cdot t_2}}}\\ \mathbf{elif}\;{B}^{2} \leq 2.4938781602879654 \cdot 10^{+143}:\\ \;\;\;\;-0.25 \cdot \frac{t_1}{\frac{C}{t_1}}\\ \mathbf{elif}\;{B}^{2} \leq 1.011066639248309 \cdot 10^{+307}:\\ \;\;\;\;-\frac{t_3}{t_5}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]

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 (* -16.0 (* C F))))
        (t_1 (sqrt t_0))
        (t_2 (* 2.0 (* F (- (+ A C) (hypot B (- A C))))))
        (t_3 (sqrt t_2))
        (t_4 (fma A (* C -4.0) (* B B)))
        (t_5 (sqrt t_4)))
   (if (<= (pow B 2.0) 1.850236647975572e-176)
     (/ (- (sqrt (* t_4 (* 2.0 (* F (* 2.0 A)))))) t_4)
     (if (<= (pow B 2.0) 3.351092992184262e-94)
       (* -0.25 (/ 1.0 (/ C t_0)))
       (if (<= (pow B 2.0) 3.0541871978177776e-47)
         (/ (- t_5) (/ t_4 t_3))
         (if (<= (pow B 2.0) 6.061880387862554e-11)
           (* -0.25 (/ t_0 C))
           (if (<= (pow B 2.0) 1.8727221069173937e+113)
             (/ -1.0 (/ t_4 (sqrt (* t_4 t_2))))
             (if (<= (pow B 2.0) 2.4938781602879654e+143)
               (* -0.25 (/ t_1 (/ C t_1)))
               (if (<= (pow B 2.0) 1.011066639248309e+307)
                 (- (/ t_3 t_5))
                 (-
                  (* (sqrt (* F (- A (hypot A B)))) (/ (sqrt 2.0) B))))))))))))

C

double code(double A, double B, double C, double F) {
	return -sqrt(((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / (pow(B, 2.0) - ((4.0 * A) * C));
}

↓

double code(double A, double B, double C, double F) {
	double t_0 = sqrt((-16.0 * (C * F)));
	double t_1 = sqrt(t_0);
	double t_2 = 2.0 * (F * ((A + C) - hypot(B, (A - C))));
	double t_3 = sqrt(t_2);
	double t_4 = fma(A, (C * -4.0), (B * B));
	double t_5 = sqrt(t_4);
	double tmp;
	if (pow(B, 2.0) <= 1.850236647975572e-176) {
		tmp = -sqrt((t_4 * (2.0 * (F * (2.0 * A))))) / t_4;
	} else if (pow(B, 2.0) <= 3.351092992184262e-94) {
		tmp = -0.25 * (1.0 / (C / t_0));
	} else if (pow(B, 2.0) <= 3.0541871978177776e-47) {
		tmp = -t_5 / (t_4 / t_3);
	} else if (pow(B, 2.0) <= 6.061880387862554e-11) {
		tmp = -0.25 * (t_0 / C);
	} else if (pow(B, 2.0) <= 1.8727221069173937e+113) {
		tmp = -1.0 / (t_4 / sqrt((t_4 * t_2)));
	} else if (pow(B, 2.0) <= 2.4938781602879654e+143) {
		tmp = -0.25 * (t_1 / (C / t_1));
	} else if (pow(B, 2.0) <= 1.011066639248309e+307) {
		tmp = -(t_3 / t_5);
	} else {
		tmp = -(sqrt((F * (A - hypot(A, B)))) * (sqrt(2.0) / B));
	}
	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(-16.0 * Float64(C * F)))
	t_1 = sqrt(t_0)
	t_2 = Float64(2.0 * Float64(F * Float64(Float64(A + C) - hypot(B, Float64(A - C)))))
	t_3 = sqrt(t_2)
	t_4 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_5 = sqrt(t_4)
	tmp = 0.0
	if ((B ^ 2.0) <= 1.850236647975572e-176)
		tmp = Float64(Float64(-sqrt(Float64(t_4 * Float64(2.0 * Float64(F * Float64(2.0 * A)))))) / t_4);
	elseif ((B ^ 2.0) <= 3.351092992184262e-94)
		tmp = Float64(-0.25 * Float64(1.0 / Float64(C / t_0)));
	elseif ((B ^ 2.0) <= 3.0541871978177776e-47)
		tmp = Float64(Float64(-t_5) / Float64(t_4 / t_3));
	elseif ((B ^ 2.0) <= 6.061880387862554e-11)
		tmp = Float64(-0.25 * Float64(t_0 / C));
	elseif ((B ^ 2.0) <= 1.8727221069173937e+113)
		tmp = Float64(-1.0 / Float64(t_4 / sqrt(Float64(t_4 * t_2))));
	elseif ((B ^ 2.0) <= 2.4938781602879654e+143)
		tmp = Float64(-0.25 * Float64(t_1 / Float64(C / t_1)));
	elseif ((B ^ 2.0) <= 1.011066639248309e+307)
		tmp = Float64(-Float64(t_3 / t_5));
	else
		tmp = Float64(-Float64(sqrt(Float64(F * Float64(A - hypot(A, B)))) * Float64(sqrt(2.0) / B)));
	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[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[t$95$4], $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1.850236647975572e-176], N[((-N[Sqrt[N[(t$95$4 * N[(2.0 * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 3.351092992184262e-94], N[(-0.25 * N[(1.0 / N[(C / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 3.0541871978177776e-47], N[((-t$95$5) / N[(t$95$4 / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 6.061880387862554e-11], N[(-0.25 * N[(t$95$0 / C), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1.8727221069173937e+113], N[(-1.0 / N[(t$95$4 / N[Sqrt[N[(t$95$4 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2.4938781602879654e+143], N[(-0.25 * N[(t$95$1 / N[(C / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1.011066639248309e+307], (-N[(t$95$3 / t$95$5), $MachinePrecision]), (-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])]]]]]]]]]]]]]

Error

Bits error versus A

Bits error versus B

Bits error versus C

Bits error versus F

Derivation

1. Split input into 8 regimes

2. if (pow.f64 B 2) < 1.8502366479755722e-176

   1. Initial program 52.5

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

   2. Simplified 48.0

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

   3. Taylor expanded in A around -inf 35.0

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

   4. Applied *-un-lft-identity_binary64 35.0

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

   if 1.8502366479755722e-176 < (pow.f64 B 2) < 3.351092992184262e-94

   1. Initial program 47.9

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

   2. Simplified 42.6

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

   3. Taylor expanded in A around -inf 38.8

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

   4. Taylor expanded in A around -inf 34.5

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]

   5. Applied clear-num_binary64 34.5

      \[\leadsto -0.25 \cdot \color{blue}{\frac{1}{\frac{C}{\sqrt{-16 \cdot \left(C \cdot F\right)}}}} \]

   if 3.351092992184262e-94 < (pow.f64 B 2) < 3.0541871978177776e-47

   1. Initial program 43.5

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

   2. Simplified 39.9

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

   3. Applied sqrt-prod_binary64 40.4

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

   4. Applied distribute-lft-neg-in_binary64 40.4

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

   5. Applied associate-/l*_binary64 40.5

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

   if 3.0541871978177776e-47 < (pow.f64 B 2) < 6.0618803878625541e-11

   1. Initial program 42.6

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

   2. Simplified 37.7

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

   3. Taylor expanded in A around -inf 41.2

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

   4. Taylor expanded in A around -inf 34.5

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]

   5. Applied div-inv_binary64 34.6

      \[\leadsto -0.25 \cdot \color{blue}{\left(\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \frac{1}{C}\right)} \]

   6. Applied *-un-lft-identity_binary64 34.6

      \[\leadsto -0.25 \cdot \left(\color{blue}{\left(1 \cdot \sqrt{-16 \cdot \left(C \cdot F\right)}\right)} \cdot \frac{1}{C}\right) \]

   7. Applied associate-*l*_binary64 34.6

      \[\leadsto -0.25 \cdot \color{blue}{\left(1 \cdot \left(\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \frac{1}{C}\right)\right)} \]

   8. Simplified 34.5

      \[\leadsto -0.25 \cdot \left(1 \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}}\right) \]

   if 6.0618803878625541e-11 < (pow.f64 B 2) < 1.87272210691739365e113

   1. Initial program 41.0

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

   2. Simplified 37.2

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

   3. Applied neg-mul-1_binary64 37.2

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

   4. Applied associate-/l*_binary64 37.2

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

   if 1.87272210691739365e113 < (pow.f64 B 2) < 2.49387816028796536e143

   1. Initial program 46.7

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

   2. Simplified 41.6

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

   3. Taylor expanded in A around -inf 51.5

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

   4. Taylor expanded in A around -inf 43.8

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]

   5. Applied add-sqr-sqrt_binary64 43.9

      \[\leadsto -0.25 \cdot \frac{\color{blue}{\sqrt{\sqrt{-16 \cdot \left(C \cdot F\right)}} \cdot \sqrt{\sqrt{-16 \cdot \left(C \cdot F\right)}}}}{C} \]

   6. Applied associate-/l*_binary64 43.9

      \[\leadsto -0.25 \cdot \color{blue}{\frac{\sqrt{\sqrt{-16 \cdot \left(C \cdot F\right)}}}{\frac{C}{\sqrt{\sqrt{-16 \cdot \left(C \cdot F\right)}}}}} \]

   if 2.49387816028796536e143 < (pow.f64 B 2) < 1.01106663924830905e307

   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 44.8

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

   3. Applied add-sqr-sqrt_binary64 45.1

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

   4. Applied sqrt-prod_binary64 30.0

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

   5. Applied distribute-lft-neg-in_binary64 30.0

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

   6. Applied times-frac_binary64 29.9

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

   7. Simplified 29.7

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

   if 1.01106663924830905e307 < (pow.f64 B 2)

   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 64.0

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

   3. Taylor expanded in C around 0 64.0

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

   4. Simplified 46.1

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

4. Final simplification 37.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 1.850236647975572 \cdot 10^{-176}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 3.351092992184262 \cdot 10^{-94}:\\ \;\;\;\;-0.25 \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \left(C \cdot F\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 3.0541871978177776 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{2 \cdot \left(F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 6.061880387862554 \cdot 10^{-11}:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}\\ \mathbf{elif}\;{B}^{2} \leq 1.8727221069173937 \cdot 10^{+113}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 2.4938781602879654 \cdot 10^{+143}:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{\sqrt{-16 \cdot \left(C \cdot F\right)}}}{\frac{C}{\sqrt{\sqrt{-16 \cdot \left(C \cdot F\right)}}}}\\ \mathbf{elif}\;{B}^{2} \leq 1.011066639248309 \cdot 10^{+307}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]

Reproduce

herbie shell --seed 2022137 
(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))))