ABCF->ab-angle b

Average Accuracy: 17.8% → 43.5%

Time: 52.6s

Precision: binary64

Cost: 47568

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

Math

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

↓

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_3 := 2 \cdot \left(F \cdot t_2\right)\\ t_4 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(A + A\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(A + \left(C - t_4\right)\right)\right)} \cdot \left(-\mathsf{hypot}\left(B, \sqrt{t_0}\right)\right)}{t_1}\\ \mathbf{elif}\;B \leq 45000000000000:\\ \;\;\;\;\frac{-\sqrt[3]{{\left(t_3 \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{hypot}\left(B, A\right)\right)}^{2} - A \cdot A}{C}, A\right)\right)\right)}^{1.5}}}{t_2}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \left(A - t_4\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

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 (* A (* C -4.0)))
        (t_1 (fma B B t_0))
        (t_2 (fma B B (* C (* A -4.0))))
        (t_3 (* 2.0 (* F t_2)))
        (t_4 (hypot B (- A C))))
   (if (<= B -1.1e-70)
     (/ (sqrt (* (- A (hypot B A)) (* F 2.0))) B)
     (if (<= B 2.3e-151)
       (/ (- (sqrt (* t_3 (+ A A)))) t_2)
       (if (<= B 2.6e-47)
         (/
          (* (sqrt (* F (* 2.0 (+ A (- C t_4))))) (- (hypot B (sqrt t_0))))
          t_1)
         (if (<= B 45000000000000.0)
           (/
            (-
             (cbrt
              (pow
               (*
                t_3
                (+ A (fma -0.5 (/ (- (pow (hypot B A) 2.0) (* A A)) C) A)))
               1.5)))
            t_2)
           (if (<= B 7e+148)
             (/
              (*
               (sqrt (* F (+ C (- A t_4))))
               (- (sqrt (fma C (* A -8.0) (* B (* B 2.0))))))
              t_1)
             (* (sqrt (* F (- C (hypot B C)))) (- (/ (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 = A * (C * -4.0);
	double t_1 = fma(B, B, t_0);
	double t_2 = fma(B, B, (C * (A * -4.0)));
	double t_3 = 2.0 * (F * t_2);
	double t_4 = hypot(B, (A - C));
	double tmp;
	if (B <= -1.1e-70) {
		tmp = sqrt(((A - hypot(B, A)) * (F * 2.0))) / B;
	} else if (B <= 2.3e-151) {
		tmp = -sqrt((t_3 * (A + A))) / t_2;
	} else if (B <= 2.6e-47) {
		tmp = (sqrt((F * (2.0 * (A + (C - t_4))))) * -hypot(B, sqrt(t_0))) / t_1;
	} else if (B <= 45000000000000.0) {
		tmp = -cbrt(pow((t_3 * (A + fma(-0.5, ((pow(hypot(B, A), 2.0) - (A * A)) / C), A))), 1.5)) / t_2;
	} else if (B <= 7e+148) {
		tmp = (sqrt((F * (C + (A - t_4)))) * -sqrt(fma(C, (A * -8.0), (B * (B * 2.0))))) / t_1;
	} else {
		tmp = sqrt((F * (C - hypot(B, C)))) * -(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 = Float64(A * Float64(C * -4.0))
	t_1 = fma(B, B, t_0)
	t_2 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_3 = Float64(2.0 * Float64(F * t_2))
	t_4 = hypot(B, Float64(A - C))
	tmp = 0.0
	if (B <= -1.1e-70)
		tmp = Float64(sqrt(Float64(Float64(A - hypot(B, A)) * Float64(F * 2.0))) / B);
	elseif (B <= 2.3e-151)
		tmp = Float64(Float64(-sqrt(Float64(t_3 * Float64(A + A)))) / t_2);
	elseif (B <= 2.6e-47)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(2.0 * Float64(A + Float64(C - t_4))))) * Float64(-hypot(B, sqrt(t_0)))) / t_1);
	elseif (B <= 45000000000000.0)
		tmp = Float64(Float64(-cbrt((Float64(t_3 * Float64(A + fma(-0.5, Float64(Float64((hypot(B, A) ^ 2.0) - Float64(A * A)) / C), A))) ^ 1.5))) / t_2);
	elseif (B <= 7e+148)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(C + Float64(A - t_4)))) * Float64(-sqrt(fma(C, Float64(A * -8.0), Float64(B * Float64(B * 2.0)))))) / t_1);
	else
		tmp = Float64(sqrt(Float64(F * Float64(C - hypot(B, C)))) * Float64(-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[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * B + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(F * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[B, -1.1e-70], N[(N[Sqrt[N[(N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 2.3e-151], N[((-N[Sqrt[N[(t$95$3 * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 2.6e-47], N[(N[(N[Sqrt[N[(F * N[(2.0 * N[(A + N[(C - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[B ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B, 45000000000000.0], N[((-N[Power[N[Power[N[(t$95$3 * N[(A + N[(-0.5 * N[(N[(N[Power[N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision], 2.0], $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 7e+148], N[(N[(N[Sqrt[N[(F * N[(C + N[(A - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C * N[(A * -8.0), $MachinePrecision] + N[(B * N[(B * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(F * N[(C - N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision])), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if B < -1.0999999999999999e-70

   1. Initial program 17.4%

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

   2. Simplified 16.3%

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

      [Start] 17.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around 0 0.9%

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

   4. Simplified 2.0%

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

      [Start] 0.9	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \]
      mul-1-neg [=>] 0.9	\[ \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}} \]
      distribute-rgt-neg-in [=>] 0.9	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
      *-commutative [=>] 0.9	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}\right) \]
      unpow2 [=>] 0.9	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)}\right) \]
      unpow2 [=>] 0.9	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right)}\right) \]
      hypot-def [=>] 2.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \color{blue}{\mathsf{hypot}\left(B, A\right)}\right)}\right) \]

   5. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)}^{3}}} \]
      Proof

      [Start] 2.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right) \]
      add-cbrt-cube [=>] 2.4	\[ \color{blue}{\sqrt[3]{\left(\left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\right) \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\right)\right) \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\right)}} \]
      pow3 [=>] 2.4	\[ \sqrt[3]{\color{blue}{{\left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\right)}^{3}}} \]

   6. Applied egg-rr 45.1%

      \[\leadsto \color{blue}{0 + \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]
      Proof

      [Start] 21.2	\[ \sqrt[3]{{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)}^{3}} \]
      add-log-exp [=>] 4.0	\[ \color{blue}{\log \left(e^{\sqrt[3]{{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)}^{3}}}\right)} \]
      rem-cbrt-cube [=>] 4.0	\[ \log \left(e^{\color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}}}\right) \]
      *-un-lft-identity [=>] 4.0	\[ \log \color{blue}{\left(1 \cdot e^{\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}}\right)} \]
      log-prod [=>] 4.0	\[ \color{blue}{\log 1 + \log \left(e^{\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}}\right)} \]
      metadata-eval [=>] 4.0	\[ \color{blue}{0} + \log \left(e^{\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}}\right) \]
      add-log-exp [<=] 45.0	\[ 0 + \color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
      associate-*l/ [=>] 45.0	\[ 0 + \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}}{B}} \]
      sqrt-unprod [=>] 45.1	\[ 0 + \frac{\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}}{B} \]
      hypot-udef [=>] 21.4	\[ 0 + \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \color{blue}{\sqrt{A \cdot A + B \cdot B}}\right)\right)}}{B} \]
      +-commutative [=>] 21.4	\[ 0 + \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\color{blue}{B \cdot B + A \cdot A}}\right)\right)}}{B} \]
      hypot-def [=>] 45.1	\[ 0 + \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \color{blue}{\mathsf{hypot}\left(B, A\right)}\right)\right)}}{B} \]

   7. Simplified 45.1%

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

      [Start] 45.1	\[ 0 + \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B} \]
      +-lft-identity [=>] 45.1	\[ \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]
      *-commutative [=>] 45.1	\[ \frac{\sqrt{\color{blue}{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right) \cdot 2}}}{B} \]
      *-commutative [=>] 45.1	\[ \frac{\sqrt{\color{blue}{\left(\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F\right)} \cdot 2}}{B} \]
      associate-*l* [=>] 45.1	\[ \frac{\sqrt{\color{blue}{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(F \cdot 2\right)}}}{B} \]

   if -1.0999999999999999e-70 < B < 2.29999999999999996e-151

   1. Initial program 17.3%

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

   2. Simplified 29.0%

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

      [Start] 17.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around inf 44.6%

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

   if 2.29999999999999996e-151 < B < 2.6e-47

   1. Initial program 25.3%

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

   2. Simplified 31.0%

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

      [Start] 25.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 29.9%

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

      [Start] 31.0	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      sqrt-prod [=>] 30.5	\[ \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-udef [=>] 30.5	\[ \frac{-\sqrt{\color{blue}{B \cdot B + A \cdot \left(-4 \cdot C\right)}} \cdot \sqrt{\left(2 \cdot F\right) \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      add-sqr-sqrt [=>] 27.5	\[ \frac{-\sqrt{B \cdot B + \color{blue}{\sqrt{A \cdot \left(-4 \cdot C\right)} \cdot \sqrt{A \cdot \left(-4 \cdot C\right)}}} \cdot \sqrt{\left(2 \cdot F\right) \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      hypot-def [=>] 27.5	\[ \frac{-\color{blue}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(-4 \cdot C\right)}\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 27.5	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(-4 \cdot C\right)}\right) \cdot \sqrt{\color{blue}{\left(F \cdot 2\right)} \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*l* [=>] 27.5	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(-4 \cdot C\right)}\right) \cdot \sqrt{\color{blue}{F \cdot \left(2 \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate--r- [=>] 29.9	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(-4 \cdot C\right)}\right) \cdot \sqrt{F \cdot \left(2 \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 29.9	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(-4 \cdot C\right)}\right) \cdot \sqrt{F \cdot \left(2 \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   4. Simplified 29.9%

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

      [Start] 29.9	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(-4 \cdot C\right)}\right) \cdot \sqrt{F \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 29.9	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \cdot \sqrt{F \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if 2.6e-47 < B < 4.5e13

   1. Initial program 33.7%

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

   2. Simplified 44.1%

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

      [Start] 33.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around inf 33.3%

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

   4. Simplified 33.3%

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

      [Start] 33.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 33.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{\left(\color{blue}{B \cdot B} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 33.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + \color{blue}{A \cdot A}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      mul-1-neg [=>] 33.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + A \cdot A\right) - {\color{blue}{\left(-A\right)}}^{2}}{C}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Applied egg-rr 26.1%

      \[\leadsto \frac{-\color{blue}{\sqrt[3]{{\left(\left(2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{hypot}\left(B, A\right)\right)}^{2} - A \cdot A}{C}, A\right)\right)\right)}^{1.5}}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      Proof

      [Start] 33.3

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

      add-cbrt-cube [=>] 26.1

      \[ \frac{-\color{blue}{\sqrt[3]{\left(\sqrt{\left(2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + A \cdot A\right) - {\left(-A\right)}^{2}}{C}\right)\right)} \cdot \sqrt{\left(2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + A \cdot A\right) - {\left(-A\right)}^{2}}{C}\right)\right)}\right) \cdot \sqrt{\left(2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + A \cdot A\right) - {\left(-A\right)}^{2}}{C}\right)\right)}}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 4.5e13 < B < 6.9999999999999998e148

   1. Initial program 27.6%

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

   2. Simplified 22.3%

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

      [Start] 27.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 49.4%

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

      [Start] 22.3	\[ \frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [=>] 35.0	\[ \frac{-\sqrt{\color{blue}{\left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      sqrt-prod [=>] 50.0	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate--r- [=>] 49.4	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(\left(A - \mathsf{hypot}\left(B, A - C\right)\right) + C\right)}} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      +-commutative [=>] 49.4	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 49.4	\[ \frac{-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, \color{blue}{\left(B \cdot B\right) \cdot 2}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*l* [=>] 49.4	\[ \frac{-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, \color{blue}{B \cdot \left(B \cdot 2\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   4. Simplified 49.4%

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

      [Start] 49.4	\[ \frac{-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow1/2 [<=] 49.4	\[ \frac{-\color{blue}{{\left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.5}} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 49.4	\[ \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot {\left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.5}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 49.4	\[ \frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \color{blue}{\left(2 \cdot B\right)}\right)} \cdot {\left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.5}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow1/2 [=>] 49.4	\[ \frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(2 \cdot B\right)\right)} \cdot \color{blue}{\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 6.9999999999999998e148 < B

   1. Initial program 0.6%

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

   2. Simplified 0.7%

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

      [Start] 0.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around 0 1.2%

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

   4. Simplified 46.9%

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

      [Start] 1.2	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right) \]
      mul-1-neg [=>] 1.2	\[ \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]
      distribute-rgt-neg-in [=>] 1.2	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
      *-commutative [=>] 1.2	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]
      unpow2 [=>] 1.2	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)}\right) \]
      unpow2 [=>] 1.2	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}\right)}\right) \]
      hypot-def [=>] 46.9	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - \color{blue}{\mathsf{hypot}\left(B, C\right)}\right)}\right) \]

3. Recombined 6 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.1 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right) \cdot \left(A + A\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \left(-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 45000000000000:\\ \;\;\;\;\frac{-\sqrt[3]{{\left(\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{hypot}\left(B, A\right)\right)}^{2} - A \cdot A}{C}, A\right)\right)\right)}^{1.5}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	43.5%
Cost	34516

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_3 := 2 \cdot \left(F \cdot t_2\right)\\ t_4 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;B \leq -2.7 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-151}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(A + A\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(A + \left(C - t_4\right)\right)\right)} \cdot \left(-\mathsf{hypot}\left(B, \sqrt{t_0}\right)\right)}{t_1}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(A + \left(A + -0.5 \cdot \frac{\left(A \cdot A + B \cdot B\right) - {\left(-A\right)}^{2}}{C}\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \left(A - t_4\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 2
Accuracy	44.0%
Cost	34188

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ t_2 := A - \mathsf{hypot}\left(B, A\right)\\ t_3 := A \cdot \left(C \cdot -4\right)\\ \mathbf{if}\;B \leq -7.8 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{t_2 \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-151}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + A\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \left(-\mathsf{hypot}\left(B, \sqrt{t_3}\right)\right)}{\mathsf{fma}\left(B, B, t_3\right)}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \left(A + -0.5 \cdot \frac{\left(A \cdot A + B \cdot B\right) - {\left(-A\right)}^{2}}{C}\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_2 \cdot F} \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 3
Accuracy	45.0%
Cost	27980

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{B \cdot B}{C}\right)\right)}}{t_1}\\ t_3 := A - \mathsf{hypot}\left(B, A\right)\\ \mathbf{if}\;B \leq -2.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{t_3 \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 2.42 \cdot 10^{-14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\left(F \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right) \cdot t_0} \cdot \frac{1}{-t_0}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+78}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_3 \cdot F} \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 4
Accuracy	45.0%
Cost	27852

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ t_2 := \frac{-\sqrt{t_1 \cdot \left(A + \left(A + -0.5 \cdot \frac{B \cdot B}{C}\right)\right)}}{t_0}\\ t_3 := A - \mathsf{hypot}\left(B, A\right)\\ \mathbf{if}\;B \leq -1.55 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{t_3 \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{+81}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_3 \cdot F} \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 5
Accuracy	43.8%
Cost	21000

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := A - \mathsf{hypot}\left(B, A\right)\\ \mathbf{if}\;B \leq -1.95 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{t_1 \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-123}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot F} \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 6
Accuracy	40.4%
Cost	20232

\[\begin{array}{l} t_0 := A - \mathsf{hypot}\left(B, A\right)\\ \mathbf{if}\;B \leq -1.75 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{t_0 \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-123}:\\ \;\;\;\;-\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_0 \cdot F} \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 7
Accuracy	40.5%
Cost	20168

\[\begin{array}{l} t_0 := A - \mathsf{hypot}\left(B, A\right)\\ \mathbf{if}\;B \leq -1.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{t_0 \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-124}:\\ \;\;\;\;-\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_0 \cdot F} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 8
Accuracy	38.1%
Cost	14344

\[\begin{array}{l} \mathbf{if}\;B \leq -9.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-123}:\\ \;\;\;\;-\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 9
Accuracy	35.0%
Cost	14216

\[\begin{array}{l} \mathbf{if}\;B \leq -1.66 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-154}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot C\right) \cdot \left(\left(A \cdot A\right) \cdot -16\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 10
Accuracy	24.7%
Cost	13704

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -1.42 \cdot 10^{-33}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-152}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - \left(C - A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 11
Accuracy	33.7%
Cost	13704

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -7.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 8.6 \cdot 10^{-155}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) - \left(C - A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 12
Accuracy	24.5%
Cost	13640

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -1.12 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-152}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - \left(C - A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{B \cdot \left(-F\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 13
Accuracy	8.1%
Cost	8716

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(t_0 - B \cdot B\right)\right) \cdot \left(B - \left(A + C\right)\right)\right)}}{B \cdot B - t_0}\\ \mathbf{if}\;C \leq -1.26 \cdot 10^{-232}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 2.15 \cdot 10^{-254}:\\ \;\;\;\;2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \mathbf{elif}\;C \leq 3.3:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]

Alternative 14
Accuracy	18.2%
Cost	8712

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B - t_0\\ t_2 := F \cdot t_1\\ \mathbf{if}\;B \leq -4.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-72}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(\left(A + C\right) - \left(C - A\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(t_0 - B \cdot B\right)\right) \cdot \left(B - \left(A + C\right)\right)\right)}}{t_1}\\ \end{array} \]

Alternative 15
Accuracy	17.0%
Cost	8584

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B - t_0\\ \mathbf{if}\;B \leq -7.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C \cdot -4 + A \cdot 4\right)}}{B}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(t_0 - B \cdot B\right)\right) \cdot \left(B - \left(A + C\right)\right)\right)}}{t_1}\\ \end{array} \]

Alternative 16
Accuracy	17.7%
Cost	8584

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B - t_0\\ t_2 := F \cdot t_1\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(t_0 - B \cdot B\right)\right) \cdot \left(B - \left(A + C\right)\right)\right)}}{t_1}\\ \end{array} \]

Alternative 17
Accuracy	8.9%
Cost	7300

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(C \cdot -4 + A \cdot 4\right)}\\ \mathbf{if}\;B \leq 9 \cdot 10^{-247}:\\ \;\;\;\;\frac{t_0}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t_0}{B}\\ \end{array} \]

Alternative 18
Accuracy	8.7%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;B \leq -6.2 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C \cdot -4 + A \cdot 4\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A \cdot F}}{B} \cdot -2\\ \end{array} \]

Alternative 19
Accuracy	8.5%
Cost	6980

\[\begin{array}{l} t_0 := \frac{\sqrt{A \cdot F}}{B}\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot -2\\ \end{array} \]

Alternative 20
Accuracy	5.3%
Cost	6848

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

Alternative 21
Accuracy	3.8%
Cost	6464

\[\sqrt{0} \]

Reproduce

herbie shell --seed 2023143 
(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))))