ABCF->ab-angle b

Average Error: 81.67% → 63.07%

Time: 1.0min

Precision: binary64

Cost: 148300

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := {B}^{2} + C \cdot \left(A \cdot -4\right)\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_1}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot t_0}\right)}{t_0}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(A + \left(A + \frac{-0.5}{\frac{C}{B \cdot B + \left(A \cdot A - {\left(-A\right)}^{2}\right)}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_0}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)}^{0.25}\right)}^{2} \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]

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 (fma B B (* A (* C -4.0))))
        (t_1 (+ (pow B 2.0) (* C (* A -4.0))))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
          t_1)))
   (if (<= t_2 -2e-191)
     (/
      (* (sqrt (* F (+ A (- C (hypot B (- A C)))))) (- (sqrt (* 2.0 t_0))))
      t_0)
     (if (<= t_2 2e+224)
       (/
        (-
         (sqrt
          (*
           t_0
           (*
            (+ A (+ A (/ -0.5 (/ C (+ (* B B) (- (* A A) (pow (- A) 2.0)))))))
            (* 2.0 F)))))
        t_0)
       (if (<= t_2 INFINITY)
         (sqrt (/ 2.0 (* -2.0 (/ C F))))
         (* (pow (pow (* F (- C (hypot B C))) 0.25) 2.0) (/ (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 = fma(B, B, (A * (C * -4.0)));
	double t_1 = pow(B, 2.0) + (C * (A * -4.0));
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_1;
	double tmp;
	if (t_2 <= -2e-191) {
		tmp = (sqrt((F * (A + (C - hypot(B, (A - C)))))) * -sqrt((2.0 * t_0))) / t_0;
	} else if (t_2 <= 2e+224) {
		tmp = -sqrt((t_0 * ((A + (A + (-0.5 / (C / ((B * B) + ((A * A) - pow(-A, 2.0))))))) * (2.0 * F)))) / t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 / (-2.0 * (C / F))));
	} else {
		tmp = pow(pow((F * (C - hypot(B, C))), 0.25), 2.0) * (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 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_1 = Float64((B ^ 2.0) + Float64(C * Float64(A * -4.0)))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_1)
	tmp = 0.0
	if (t_2 <= -2e-191)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A + Float64(C - hypot(B, Float64(A - C)))))) * Float64(-sqrt(Float64(2.0 * t_0)))) / t_0);
	elseif (t_2 <= 2e+224)
		tmp = Float64(Float64(-sqrt(Float64(t_0 * Float64(Float64(A + Float64(A + Float64(-0.5 / Float64(C / Float64(Float64(B * B) + Float64(Float64(A * A) - (Float64(-A) ^ 2.0))))))) * Float64(2.0 * F))))) / t_0);
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(2.0 / Float64(-2.0 * Float64(C / F))));
	else
		tmp = Float64(((Float64(F * Float64(C - hypot(B, C))) ^ 0.25) ^ 2.0) * 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[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B, 2.0], $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-191], N[(N[(N[Sqrt[N[(F * N[(A + N[(C - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$2, 2e+224], N[((-N[Sqrt[N[(t$95$0 * N[(N[(A + N[(A + N[(-0.5 / N[(C / N[(N[(B * B), $MachinePrecision] + N[(N[(A * A), $MachinePrecision] - N[Power[(-A), 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(2.0 / N[(-2.0 * N[(C / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Power[N[Power[N[(F * N[(C - N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -2e-191

   1. Initial program 58.37

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

   2. Simplified 49.74

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

      [Start] 58.37	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 34.44

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

   4. Simplified 34.44

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

      [Start] 34.44	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 34.44	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 34.44	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if -2e-191 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 1.99999999999999994e224

   1. Initial program 77.49

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

   2. Simplified 73.52

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

      [Start] 77.49	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around inf 68.41

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

   4. Simplified 68.69

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

      [Start] 68.41	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r/ [=>] 68.41	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + \color{blue}{\frac{-0.5 \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)}{C}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-/l* [=>] 68.4	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + \color{blue}{\frac{-0.5}{\frac{C}{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate--l+ [=>] 68.68	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + \frac{-0.5}{\frac{C}{\color{blue}{{B}^{2} + \left({A}^{2} - {\left(-1 \cdot A\right)}^{2}\right)}}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 68.68	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + \frac{-0.5}{\frac{C}{\color{blue}{B \cdot B} + \left({A}^{2} - {\left(-1 \cdot A\right)}^{2}\right)}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 68.69	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + \frac{-0.5}{\frac{C}{B \cdot B + \left(\color{blue}{A \cdot A} - {\left(-1 \cdot A\right)}^{2}\right)}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      mul-1-neg [=>] 68.69	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + \frac{-0.5}{\frac{C}{B \cdot B + \left(A \cdot A - {\color{blue}{\left(-A\right)}}^{2}\right)}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 1.99999999999999994e224 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 94.42

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

   2. Simplified 66.57

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

      [Start] 94.42	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 80.94

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

   4. Simplified 73.53

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

      [Start] 80.94	\[ \sqrt{\frac{2 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right)\right)}{{\left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right)}^{2}}} \]
      associate-/l* [=>] 80.94	\[ \sqrt{\color{blue}{\frac{2}{\frac{{\left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right)}^{2}}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right)}}}} \]
      associate-/r* [=>] 73.53	\[ \sqrt{\frac{2}{\color{blue}{\frac{\frac{{\left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right)}^{2}}{A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F}}}} \]
      *-commutative [=>] 73.53	\[ \sqrt{\frac{2}{\frac{\frac{{\left(\mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)\right)}^{2}}{A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F}}} \]
      *-commutative [=>] 73.53	\[ \sqrt{\frac{2}{\frac{\frac{{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{2}}{A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)}}{\color{blue}{F \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}} \]
      *-commutative [=>] 73.53	\[ \sqrt{\frac{2}{\frac{\frac{{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{2}}{A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)}}{F \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}} \]

   5. Taylor expanded in A around -inf 61.14

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))

   1. Initial program 100

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

   2. Simplified 98.61

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

      [Start] 100	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 99.25

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

   4. Simplified 99.34

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

      [Start] 99.25	\[ \sqrt{\frac{2 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right)\right)}{{\left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right)}^{2}}} \]
      associate-/l* [=>] 99.26	\[ \sqrt{\color{blue}{\frac{2}{\frac{{\left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right)}^{2}}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right)}}}} \]
      associate-/r* [=>] 99.34	\[ \sqrt{\frac{2}{\color{blue}{\frac{\frac{{\left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right)}^{2}}{A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F}}}} \]
      *-commutative [=>] 99.34	\[ \sqrt{\frac{2}{\frac{\frac{{\left(\mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)\right)}^{2}}{A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F}}} \]
      *-commutative [=>] 99.34	\[ \sqrt{\frac{2}{\frac{\frac{{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{2}}{A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)}}{\color{blue}{F \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}} \]
      *-commutative [=>] 99.34	\[ \sqrt{\frac{2}{\frac{\frac{{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{2}}{A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)}}{F \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}} \]

   5. Taylor expanded in A around 0 99.41

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

   6. Simplified 99.41

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

      [Start] 99.41	\[ \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]
      *-commutative [=>] 99.41	\[ \sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]
      unpow2 [=>] 99.41	\[ \sqrt{F \cdot \left(C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B} \]
      unpow2 [=>] 99.41	\[ \sqrt{F \cdot \left(C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}\right)} \cdot \frac{\sqrt{2}}{B} \]

   7. Applied egg-rr 83.45

      \[\leadsto \color{blue}{{\left({\left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)}^{0.25}\right)}^{2}} \cdot \frac{\sqrt{2}}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.07

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq -2 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + \frac{-0.5}{\frac{C}{B \cdot B + \left(A \cdot A - {\left(-A\right)}^{2}\right)}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq \infty:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)}^{0.25}\right)}^{2} \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	63.38%
Cost	28820

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := \frac{-\sqrt{t_2 \cdot \left(\left(A + \left(C - t_1\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_2}\\ \mathbf{if}\;B \leq -3 \cdot 10^{+60}:\\ \;\;\;\;{\left({\left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)}^{0.25}\right)}^{2} \cdot t_0\\ \mathbf{elif}\;B \leq -5.9 \cdot 10^{-89}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -3.7 \cdot 10^{-171}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + A\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{-35}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(A + \left(A + \frac{-0.5}{\frac{C}{B \cdot B + \left(A \cdot A - {\left(-A\right)}^{2}\right)}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(A + C\right) - t_1\right)\right)}}{\frac{t_2}{B}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)}\right)\\ \end{array} \]

Alternative 2
Error	62.56%
Cost	27984

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -1.3 \cdot 10^{+63}:\\ \;\;\;\;{\left({\left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)}^{0.25}\right)}^{2} \cdot t_0\\ \mathbf{elif}\;B \leq -1.4 \cdot 10^{-89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -9.2 \cdot 10^{-175}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{+69}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)}\right)\\ \end{array} \]

Alternative 3
Error	62.54%
Cost	27984

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{-\sqrt{t_0 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -4.5 \cdot 10^{+61}:\\ \;\;\;\;{\left({\left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)}^{0.25}\right)}^{2} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -4.6 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-170}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.12 \cdot 10^{+85}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Error	67.22%
Cost	26500

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -7.5 \cdot 10^{-80}:\\ \;\;\;\;{\left({\left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)}^{0.25}\right)}^{2} \cdot t_1\\ \mathbf{elif}\;B \leq -2.6 \cdot 10^{-174}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + A\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-255}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ \mathbf{elif}\;B \leq 2.95 \cdot 10^{-73}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)}\right)\\ \end{array} \]

Alternative 5
Error	67.15%
Cost	22228

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -2.2 \cdot 10^{-80}:\\ \;\;\;\;t_1 \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{elif}\;B \leq -2.8 \cdot 10^{-174}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + A\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 8.6 \cdot 10^{-257}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)}\right)\\ \end{array} \]

Alternative 6
Error	67.74%
Cost	21000

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -4.5 \cdot 10^{-80}:\\ \;\;\;\;t_1 \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{elif}\;B \leq -1.7 \cdot 10^{-173}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + A\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-256}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-113}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)}\right)\\ \end{array} \]

Alternative 7
Error	67.35%
Cost	20488

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -3.6 \cdot 10^{-79}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{elif}\;B \leq 1.46 \cdot 10^{-113}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, A \cdot \frac{A}{B}, B\right)\right)}\right)\\ \end{array} \]

Alternative 8
Error	67.1%
Cost	20488

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{-79}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)}\right)\\ \end{array} \]

Alternative 9
Error	72.65%
Cost	19972

\[\begin{array}{l} \mathbf{if}\;B \leq -3.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C - B\right)} \cdot \left(\sqrt{2} \cdot \left(-B\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 10
Error	76.09%
Cost	14872

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.95 \cdot 10^{+26}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(B + C\right)}\\ \mathbf{elif}\;B \leq -3.4 \cdot 10^{-63}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B}{\frac{C}{B}}\right)}\\ \mathbf{elif}\;B \leq -2.56 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-173}:\\ \;\;\;\;\frac{-\sqrt{\left(C \cdot F\right) \cdot \left(\left(A \cdot A\right) \cdot -16\right)}}{t_2}\\ \mathbf{elif}\;B \leq 8.6 \cdot 10^{-257}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C + C\right) \cdot \left(C \cdot F\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C - B\right)} \cdot \left(\sqrt{2} \cdot \left(-B\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 11
Error	74.66%
Cost	14736

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.95 \cdot 10^{+26}:\\ \;\;\;\;t_1 \cdot \sqrt{F \cdot \left(B + C\right)}\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-91}:\\ \;\;\;\;t_1 \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B}{\frac{C}{B}}\right)}\\ \mathbf{elif}\;B \leq -2.75 \cdot 10^{-182}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C - B\right)} \cdot \left(\sqrt{2} \cdot \left(-B\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 12
Error	81.29%
Cost	14689

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := -2 \cdot \frac{C}{F}\\ t_2 := \frac{\sqrt{2}}{B}\\ t_3 := t_2 \cdot \sqrt{F \cdot \left(B + C\right)}\\ \mathbf{if}\;C \leq -2.35 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 2 \cdot 10^{-274}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq 10^{-140}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B - C\right) - A\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 8 \cdot 10^{+63}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq 1.35 \cdot 10^{+120}:\\ \;\;\;\;\frac{-\sqrt{\left(C \cdot F\right) \cdot \left(\left(A \cdot A\right) \cdot -16\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;C \leq 7.2 \cdot 10^{+160}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(0.5, \frac{B \cdot B}{A \cdot F}, t_1\right)}}\\ \mathbf{elif}\;C \leq 2.3 \cdot 10^{+207} \lor \neg \left(C \leq 6.2 \cdot 10^{+234}\right):\\ \;\;\;\;t_2 \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B}{\frac{C}{B}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{t_1}}\\ \end{array} \]

Alternative 13
Error	76.33%
Cost	14608

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -7.8 \cdot 10^{+25}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(B + C\right)}\\ \mathbf{elif}\;B \leq -4.1 \cdot 10^{-92}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B}{\frac{C}{B}}\right)}\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-183}:\\ \;\;\;\;\frac{-\sqrt{\left(C \cdot F\right) \cdot \left(\left(A \cdot A\right) \cdot -16\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C - B\right)} \cdot \left(\sqrt{2} \cdot \left(-B\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 14
Error	76.23%
Cost	14608

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -3.7 \cdot 10^{+25}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(B + C\right)}\\ \mathbf{elif}\;B \leq -5.1 \cdot 10^{-92}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B}{\frac{C}{B}}\right)}\\ \mathbf{elif}\;B \leq -9.4 \cdot 10^{-269}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C - B\right)} \cdot \left(\sqrt{2} \cdot \left(-B\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 15
Error	77.83%
Cost	13896

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := t_0 \cdot \sqrt{F \cdot \left(B + C\right)}\\ t_2 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -4.2 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -6 \cdot 10^{-62}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B}{\frac{C}{B}}\right)}\\ \mathbf{elif}\;B \leq -1.8 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.55 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(A + \left(C + \left(-0.5 \cdot \frac{C \cdot C}{B} - B\right)\right)\right)\right)}}{t_2}\\ \end{array} \]

Alternative 16
Error	77.28%
Cost	13508

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.45 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + \left(-0.5 \cdot \frac{C \cdot C}{B} - B\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 17
Error	82.57%
Cost	9228

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ \mathbf{if}\;A \leq -3.5 \cdot 10^{+183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -7 \cdot 10^{-77}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(\left(C - A\right) - C\right) - A\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 3.7 \cdot 10^{-237}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(\left(B + 0.5 \cdot \frac{A \cdot A}{B}\right) - C\right) - A\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 9.8 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{A}{F}}}\\ \end{array} \]

Alternative 18
Error	84.91%
Cost	8717

\[\begin{array}{l} \mathbf{if}\;C \leq -2.6 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{A}{F}}}\\ \mathbf{elif}\;C \leq -2.1 \cdot 10^{-43} \lor \neg \left(C \leq 2.8 \cdot 10^{-40}\right):\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B - C\right) - A\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 19
Error	82.8%
Cost	8716

\[\begin{array}{l} t_0 := \sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -3 \cdot 10^{+184}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -1.22 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 3.9 \cdot 10^{-237}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B - C\right) - A\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 1.7 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{A}{F}}}\\ \end{array} \]

Alternative 20
Error	82.34%
Cost	8716

\[\begin{array}{l} t_0 := \sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -1.8 \cdot 10^{+184}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -0.2:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 6.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 2.25 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{A}{F}}}\\ \end{array} \]

Alternative 21
Error	82.59%
Cost	8716

\[\begin{array}{l} t_0 := \sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -3.4 \cdot 10^{+183}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -3.4 \cdot 10^{-79}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(C + \left(A - C\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 3.7 \cdot 10^{-237}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B - C\right) - A\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 2.9 \cdot 10^{+49}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{A}{F}}}\\ \end{array} \]

Alternative 22
Error	84.5%
Cost	7245

\[\begin{array}{l} \mathbf{if}\;A \leq -1.9 \cdot 10^{+82} \lor \neg \left(A \leq -2 \cdot 10^{-297}\right) \land A \leq 10^{+46}:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{C}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{-2 \cdot \frac{A}{F}}}\\ \end{array} \]

Alternative 23
Error	88.84%
Cost	6848

\[\sqrt{\frac{2}{-2 \cdot \frac{A}{F}}} \]

Alternative 24
Error	98.16%
Cost	6720

\[\sqrt{F \cdot \frac{2}{B}} \]

Alternative 25
Error	98.13%
Cost	6720

\[\sqrt{\frac{2}{\frac{B}{F}}} \]

Reproduce

herbie shell --seed 2023090 
(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))))