ABCF->ab-angle a

Average Error: 18.4% → 49.4%

Time: 1.2min

Precision: binary64

Cost: 162124.00

Math

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

↓

\[\begin{array}{l} t_0 := C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(\left(F \cdot \left(t_1 - {B}^{2}\right)\right) \cdot -2\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - t_1}\\ t_3 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_4 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_5 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_0}}{t_3} \cdot \left(\sqrt{t_3} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_4 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B}{\frac{C}{B}}\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\left|\frac{\sqrt{t_0}}{\frac{t_5}{\sqrt{t_5 \cdot \left(2 \cdot F\right)}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
     (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
  (- (pow B 2.0) (* (* 4.0 A) C))))

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ C (+ A (hypot B (- A C)))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* (* F (- t_1 (pow B 2.0))) -2.0)
             (+ (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
          (- (pow B 2.0) t_1)))
        (t_3 (fma -4.0 (* A C) (* B B)))
        (t_4 (fma B B (* C (* A -4.0))))
        (t_5 (fma B B (* A (* C -4.0)))))
   (if (<= t_2 -2e-180)
     (* (/ (sqrt (* 2.0 t_0)) t_3) (* (sqrt t_3) (- (sqrt F))))
     (if (<= t_2 0.0)
       (-
        (/
         (sqrt (* 2.0 (* t_4 (* F (fma 2.0 A (* -0.5 (/ B (/ C B))))))))
         t_4))
       (if (<= t_2 INFINITY)
         (fabs (/ (sqrt t_0) (/ t_5 (sqrt (* t_5 (* 2.0 F))))))
         (* (/ (- (sqrt 2.0)) B) (* (sqrt F) (sqrt (+ C (hypot C 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 = C + (A + hypot(B, (A - C)));
	double t_1 = (4.0 * A) * C;
	double t_2 = -sqrt((((F * (t_1 - pow(B, 2.0))) * -2.0) * ((A + C) + sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (pow(B, 2.0) - t_1);
	double t_3 = fma(-4.0, (A * C), (B * B));
	double t_4 = fma(B, B, (C * (A * -4.0)));
	double t_5 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (t_2 <= -2e-180) {
		tmp = (sqrt((2.0 * t_0)) / t_3) * (sqrt(t_3) * -sqrt(F));
	} else if (t_2 <= 0.0) {
		tmp = -(sqrt((2.0 * (t_4 * (F * fma(2.0, A, (-0.5 * (B / (C / B)))))))) / t_4);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fabs((sqrt(t_0) / (t_5 / sqrt((t_5 * (2.0 * F))))));
	} else {
		tmp = (-sqrt(2.0) / B) * (sqrt(F) * sqrt((C + hypot(C, 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 = Float64(C + Float64(A + hypot(B, Float64(A - C))))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(Float64(-sqrt(Float64(Float64(Float64(F * Float64(t_1 - (B ^ 2.0))) * -2.0) * Float64(Float64(A + C) + sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / Float64((B ^ 2.0) - t_1))
	t_3 = fma(-4.0, Float64(A * C), Float64(B * B))
	t_4 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_5 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (t_2 <= -2e-180)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) / t_3) * Float64(sqrt(t_3) * Float64(-sqrt(F))));
	elseif (t_2 <= 0.0)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(t_4 * Float64(F * fma(2.0, A, Float64(-0.5 * Float64(B / Float64(C / B)))))))) / t_4));
	elseif (t_2 <= Inf)
		tmp = abs(Float64(sqrt(t_0) / Float64(t_5 / sqrt(Float64(t_5 * Float64(2.0 * F))))));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B) * Float64(sqrt(F) * sqrt(Float64(C + hypot(C, 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[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(N[(F * N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $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]) / N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-180], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] * N[(N[Sqrt[t$95$3], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], (-N[(N[Sqrt[N[(2.0 * N[(t$95$4 * N[(F * N[(2.0 * A + N[(-0.5 * N[(B / N[(C / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), If[LessEqual[t$95$2, Infinity], N[Abs[N[(N[Sqrt[t$95$0], $MachinePrecision] / N[(t$95$5 / N[Sqrt[N[(t$95$5 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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-180

   1. Initial program 39.7

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

   2. Simplified 49.0

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

      [Start] 39.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 63.0

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

      [Start] 49.0	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      sqrt-prod [=>] 63.8	\[ \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-+r+ [=>] 63.0	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      +-commutative [=>] 63.0	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \color{blue}{\left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   4. Applied egg-rr 63.8

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

      [Start] 63.0	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      distribute-rgt-neg-in [=>] 63.0	\[ \frac{\color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F} \cdot \left(-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-un-lft-identity [=>] 63.0	\[ \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F} \cdot \left(-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}\right)}{\color{blue}{1 \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]
      times-frac [=>] 63.0	\[ \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{1} \cdot \frac{-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]
      associate-+r+ [=>] 63.8	\[ \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{1} \cdot \frac{-\sqrt{2 \cdot \color{blue}{\left(\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      +-commutative [=>] 63.8	\[ \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{1} \cdot \frac{-\sqrt{2 \cdot \color{blue}{\left(C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      +-commutative [=>] 63.8	\[ \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{1} \cdot \frac{-\sqrt{2 \cdot \left(C + \color{blue}{\left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   5. Simplified 63.8

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

      [Start] 63.8	\[ \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{1} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      /-rgt-identity [=>] 63.8	\[ \color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 63.8	\[ \sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 63.8	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 63.8	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \]

   6. Applied egg-rr 79.7

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

      [Start] 63.8	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
      sqrt-prod [=>] 79.7	\[ \color{blue}{\left(\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
      *-commutative [=>] 79.7	\[ \color{blue}{\left(\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F}\right)} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -2e-180 < (/.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))) < -0.0

   1. Initial program 10.4

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

   2. Simplified 11.7

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

      [Start] 10.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around -inf 28.0

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

   4. Simplified 28.0

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

      [Start] 28.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      fma-def [=>] 28.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(2, A, -0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 28.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{\color{blue}{B \cdot B}}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-/l* [=>] 28.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \color{blue}{\frac{B}{\frac{C}{B}}}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if -0.0 < (/.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))) < +inf.0

   1. Initial program 39.4

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

   2. Simplified 39.3

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

      [Start] 39.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 36.5

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

      [Start] 39.3	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      add-sqr-sqrt [=>] 39.2	\[ \color{blue}{\sqrt{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}}} \]
      sqrt-unprod [=>] 29.4	\[ \color{blue}{\sqrt{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}}} \]
      frac-times [=>] 23.4	\[ \sqrt{\color{blue}{\frac{\left(-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}\right) \cdot \left(-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}\right)}{\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)}}} \]

   4. Simplified 39.7

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

      [Start] 36.5	\[ \sqrt{\frac{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)\right)}{{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}^{2}}} \]
      associate-/l* [=>] 39.7	\[ \sqrt{\color{blue}{\frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}^{2}}{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}} \]
      *-commutative [=>] 39.7	\[ \sqrt{\frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}^{2}}{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}} \]

   5. Applied egg-rr 79.3

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

      [Start] 39.7	\[ \sqrt{\frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}^{2}}{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}} \]
      add-sqr-sqrt [=>] 39.7	\[ \sqrt{\color{blue}{\sqrt{\frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}^{2}}{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}} \cdot \sqrt{\frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}^{2}}{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}}} \]
      rem-sqrt-square [=>] 39.7	\[ \color{blue}{\left|\sqrt{\frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}^{2}}{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}\right|} \]
      sqrt-div [=>] 46.9	\[ \left|\color{blue}{\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{\sqrt{\frac{{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}^{2}}{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}}\right| \]
      +-commutative [=>] 46.9	\[ \left|\frac{\sqrt{A + \color{blue}{\left(\mathsf{hypot}\left(B, A - C\right) + C\right)}}}{\sqrt{\frac{{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}^{2}}{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}\right| \]
      associate-+r+ [=>] 46.9	\[ \left|\frac{\sqrt{\color{blue}{\left(A + \mathsf{hypot}\left(B, A - C\right)\right) + C}}}{\sqrt{\frac{{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}^{2}}{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}\right| \]
      +-commutative [<=] 46.9	\[ \left|\frac{\sqrt{\color{blue}{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}}{\sqrt{\frac{{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}^{2}}{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}\right| \]
      sqrt-div [=>] 52.7	\[ \left|\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}^{2}}}{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}}\right| \]

   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 0.0

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

   2. Simplified 1.3

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

      [Start] 0.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around 0 0.6

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

   4. Simplified 0.6

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

      [Start] 0.6	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right) \]
      associate-*r* [=>] 0.6	\[ \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]
      associate-*r/ [=>] 0.6	\[ \color{blue}{\frac{-1 \cdot \sqrt{2}}{B}} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F} \]
      mul-1-neg [=>] 0.6	\[ \frac{\color{blue}{-\sqrt{2}}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F} \]
      *-commutative [=>] 0.6	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]
      +-commutative [=>] 0.6	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{{C}^{2} + {B}^{2}}}\right)} \]
      unpow2 [=>] 0.6	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{C \cdot C} + {B}^{2}}\right)} \]
      unpow2 [=>] 0.6	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{C \cdot C + \color{blue}{B \cdot B}}\right)} \]

   5. Applied egg-rr 26.7

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

      [Start] 0.6	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{C \cdot C + B \cdot B}\right)} \]
      sqrt-prod [=>] 0.5	\[ \frac{-\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{C + \sqrt{C \cdot C + B \cdot B}}\right)} \]
      *-commutative [=>] 0.5	\[ \frac{-\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot \sqrt{F}\right)} \]
      hypot-def [=>] 26.7	\[ \frac{-\sqrt{2}}{B} \cdot \left(\sqrt{C + \color{blue}{\mathsf{hypot}\left(C, B\right)}} \cdot \sqrt{F}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 49.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(\left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right) \cdot -2\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{elif}\;\frac{-\sqrt{\left(\left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right) \cdot -2\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B}{\frac{C}{B}}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(\left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right) \cdot -2\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\left|\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	48.3%
Cost	27860.00

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot \sqrt{F}\\ t_2 := A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_3 := \sqrt{2 \cdot t_2}\\ t_4 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_5 := B \cdot B + t_0\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \mathbf{elif}\;B \leq -5 \cdot 10^{-98}:\\ \;\;\;\;t_1 \cdot \frac{t_3}{t_4}\\ \mathbf{elif}\;B \leq -3.9 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)} \cdot \left(-\sqrt{t_2}\right)}{t_5}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-10}:\\ \;\;\;\;\frac{-{\left(t_2 \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)\right)\right)}^{0.5}}{t_5}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{+131}:\\ \;\;\;\;t_1 \cdot \left(t_3 \cdot \frac{1}{-t_4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\\ \end{array} \]

Alternative 2
Error	52.5%
Cost	27848.00

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -3.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)} \cdot \left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\\ \end{array} \]

Alternative 3
Error	48.4%
Cost	27732.00

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot \sqrt{F}\\ t_2 := A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_3 := \sqrt{2 \cdot t_2}\\ t_4 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_5 := B \cdot B + t_0\\ \mathbf{if}\;B \leq -1.05 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \mathbf{elif}\;B \leq -1.06 \cdot 10^{-97}:\\ \;\;\;\;t_1 \cdot \frac{t_3}{t_4}\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)} \cdot \left(-\sqrt{t_2}\right)}{t_5}\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{-{\left(t_2 \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)\right)\right)}^{0.5}}{t_5}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{+139}:\\ \;\;\;\;\frac{-t_3}{\frac{t_4}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\\ \end{array} \]

Alternative 4
Error	48.1%
Cost	27596.00

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;B \leq -2.7 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-55}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(C + \left(A + t_0\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \sqrt{F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{+130}:\\ \;\;\;\;\left(B \cdot \sqrt{F}\right) \cdot \left(\sqrt{2 \cdot \left(A + \left(C + t_0\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\\ \end{array} \]

Alternative 5
Error	47.9%
Cost	27272.00

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B + t_0\\ t_2 := A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\\ \mathbf{if}\;B \leq -1 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \mathbf{elif}\;B \leq -1.28 \cdot 10^{-97}:\\ \;\;\;\;\left(B \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2 \cdot t_2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq -1.95 \cdot 10^{-179}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)} \cdot \left(-\sqrt{t_2}\right)}{t_1}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{+53}:\\ \;\;\;\;\frac{-{\left(t_2 \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)\right)\right)}^{0.5}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\\ \end{array} \]

Alternative 6
Error	48.0%
Cost	26568.00

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{-{\left(\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)\right)\right)}^{0.5}}{B \cdot B + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\\ \end{array} \]

Alternative 7
Error	43.4%
Cost	26372.00

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B + t_0\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;B \leq -5.9 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{+19}:\\ \;\;\;\;\frac{-{\left(\left(A + \left(C + t_2\right)\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)\right)\right)}^{0.5}}{t_1}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+100}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \left(A + t_2\right)\right)} \cdot \left(B \cdot \left(-\sqrt{F}\right)\right)}{t_1}\\ \mathbf{elif}\;B \leq 1.18 \cdot 10^{+240} \lor \neg \left(B \leq 1.95 \cdot 10^{+292}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \frac{1}{\frac{1}{C + \mathsf{hypot}\left(C, B\right)}}}\\ \end{array} \]

Alternative 8
Error	36.6%
Cost	22120.00

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)}\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := A + \left(C + t_2\right)\\ t_4 := B \cdot B + t_0\\ t_5 := -\frac{t_1 \cdot \sqrt{2 \cdot C}}{t_4}\\ t_6 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -6 \cdot 10^{+158}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \mathbf{elif}\;B \leq -4.65 \cdot 10^{-98}:\\ \;\;\;\;\frac{\left(B \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot t_3}}{t_4}\\ \mathbf{elif}\;B \leq -3.8 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)} \cdot \left(-\sqrt{t_3}\right)}{t_4}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-240}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-144}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, C \cdot 0\right)\right)\right)\right)}}{t_6}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-94}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 1.36 \cdot 10^{-45}:\\ \;\;\;\;\frac{t_1 \cdot \left(-\sqrt{2 \cdot A}\right)}{t_4}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_6 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_6}\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{+100}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \left(A + t_2\right)\right)} \cdot \left(B \cdot \left(-\sqrt{F}\right)\right)}{t_4}\\ \mathbf{elif}\;B \leq 1.18 \cdot 10^{+240} \lor \neg \left(B \leq 3 \cdot 10^{+292}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \frac{1}{\frac{1}{C + \mathsf{hypot}\left(C, B\right)}}}\\ \end{array} \]

Alternative 9
Error	36.8%
Cost	21196.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.35 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-54}:\\ \;\;\;\;-\frac{\sqrt{\left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right) \cdot -2}}{t_0}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+100}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(B \cdot \left(-\sqrt{F}\right)\right)}{t_0}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{+240} \lor \neg \left(B \leq 1.5 \cdot 10^{+292}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \frac{1}{\frac{1}{C + \mathsf{hypot}\left(C, B\right)}}}\\ \end{array} \]

Alternative 10
Error	36.0%
Cost	20689.00

\[\begin{array}{l} \mathbf{if}\;B \leq -1.35 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+47}:\\ \;\;\;\;-\frac{\sqrt{\left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right) \cdot -2}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{+240} \lor \neg \left(B \leq 1.45 \cdot 10^{+292}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \frac{1}{\frac{1}{C + \mathsf{hypot}\left(C, B\right)}}}\\ \end{array} \]

Alternative 11
Error	35.9%
Cost	15176.00

\[\begin{array}{l} \mathbf{if}\;B \leq -1.35 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{+54}:\\ \;\;\;\;-\frac{\sqrt{\left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right) \cdot -2}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{+238} \lor \neg \left(B \leq 1.55 \cdot 10^{+292}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{B \cdot F}\\ \end{array} \]

Alternative 12
Error	22.9%
Cost	14105.00

\[\begin{array}{l} t_0 := F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -5.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(\left(A + C\right) - B\right) \cdot t_0\right) \cdot -2}}{t_1}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(C + \left(A + C\right)\right) \cdot t_0\right) \cdot -2}}{t_1}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(\left(C - A\right) - \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{+237} \lor \neg \left(B \leq 1.65 \cdot 10^{+292}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{B \cdot F}\\ \end{array} \]

Alternative 13
Error	30.4%
Cost	13841.00

\[\begin{array}{l} \mathbf{if}\;B \leq -6.7 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-84}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 1.62 \cdot 10^{+240} \lor \neg \left(B \leq 5.5 \cdot 10^{+292}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{B \cdot F}\\ \end{array} \]

Alternative 14
Error	23.2%
Cost	13712.00

\[\begin{array}{l} t_0 := F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -2 \cdot 10^{-91}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(\left(A + C\right) - B\right) \cdot t_0\right) \cdot -2}}{t_1}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-93}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(C + \left(A + C\right)\right) \cdot t_0\right) \cdot -2}}{t_1}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-39}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(\left(C - A\right) - \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 15
Error	17.4%
Cost	8848.00

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{A}}\\ t_1 := \sqrt{\frac{-F}{C}}\\ t_2 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -8.2 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 10^{-255}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 4.5 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 6 \cdot 10^{+230}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	17.9%
Cost	8844.00

\[\begin{array}{l} t_0 := F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -1.45 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;A \leq 6500000000000:\\ \;\;\;\;\frac{-\sqrt{\left(\left(C + \left(A + C\right)\right) \cdot t_0\right) \cdot -2}}{t_1}\\ \mathbf{elif}\;A \leq 1.8 \cdot 10^{+229}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(\left(C - A\right) - \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 17
Error	17.8%
Cost	8716.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -1.45 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;A \leq 7400000000000:\\ \;\;\;\;\frac{-\sqrt{\left(\left(C + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right) \cdot -2}}{t_0}\\ \mathbf{elif}\;A \leq 2 \cdot 10^{+229}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 18
Error	17.5%
Cost	8336.00

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{A}}\\ t_1 := \sqrt{\frac{-F}{C}}\\ \mathbf{if}\;A \leq -2.3 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 1.7 \cdot 10^{-255}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 1.4 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 4.9 \cdot 10^{+145}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Error	15.5%
Cost	7053.00

\[\begin{array}{l} \mathbf{if}\;A \leq -4.05 \cdot 10^{+21} \lor \neg \left(A \leq 6.2 \cdot 10^{-256}\right) \land A \leq 6 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 20
Error	11.1%
Cost	6656.00

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

Alternative 21
Error	1.2%
Cost	6592.00

\[\sqrt{\frac{F}{A}} \]

Reproduce

herbie shell --seed 2023104 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :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))))