ABCF->ab-angle b

Average Error: 52.3 → 48.7

Time: 45.9s

Precision: binary64

Cost: 95376

Math

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

↓

\[\begin{array}{l} t_0 := {B}^{2} + {A}^{2}\\ t_1 := {\left(-A\right)}^{2}\\ t_2 := t_0 - t_1\\ t_3 := \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\\ t_4 := \frac{\sqrt{2}}{B}\\ t_5 := {B}^{2} - A \cdot \left(4 \cdot C\right)\\ t_6 := {B}^{2} - 4 \cdot \left(C \cdot A\right)\\ t_7 := A - \left(-A\right)\\ \mathbf{if}\;C \leq -9 \cdot 10^{+149}:\\ \;\;\;\;-t_4 \cdot \sqrt{\left(\frac{{B}^{2}}{A} \cdot -0.5\right) \cdot F}\\ \mathbf{elif}\;C \leq -5.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(t_5 \cdot F\right) \cdot \left(C + \left(A - t_3\right)\right)\right)}}{t_5}\\ \mathbf{elif}\;C \leq -3.4 \cdot 10^{-211}:\\ \;\;\;\;-\sqrt{F \cdot \left(C - B\right)} \cdot t_4\\ \mathbf{elif}\;C \leq 8 \cdot 10^{-46}:\\ \;\;\;\;\frac{-\sqrt{t_6 \cdot \left(\left(A + \left(C - t_3\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_6}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot t_7\right)\right) + 2 \cdot \left(F \cdot \left(2 \cdot \left(A \cdot t_2\right) + {B}^{2} \cdot t_7\right) + \frac{F \cdot \left({A}^{2} \cdot \left(-2 \cdot \left(t_1 - t_0\right)\right) + -0.5 \cdot \left({B}^{2} \cdot t_2\right)\right)}{C}\right)}}{t_6}\\ \end{array} \]

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 (+ (pow B 2.0) (pow A 2.0)))
        (t_1 (pow (- A) 2.0))
        (t_2 (- t_0 t_1))
        (t_3 (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))
        (t_4 (/ (sqrt 2.0) B))
        (t_5 (- (pow B 2.0) (* A (* 4.0 C))))
        (t_6 (- (pow B 2.0) (* 4.0 (* C A))))
        (t_7 (- A (- A))))
   (if (<= C -9e+149)
     (- (* t_4 (sqrt (* (* (/ (pow B 2.0) A) -0.5) F))))
     (if (<= C -5.5e-121)
       (/ (- (sqrt (* 2.0 (* (* t_5 F) (+ C (- A t_3)))))) t_5)
       (if (<= C -3.4e-211)
         (- (* (sqrt (* F (- C B))) t_4))
         (if (<= C 8e-46)
           (/ (- (sqrt (* t_6 (* (+ A (- C t_3)) (* 2.0 F))))) t_6)
           (/
            (-
             (sqrt
              (+
               (* -8.0 (* A (* (* F C) t_7)))
               (*
                2.0
                (+
                 (* F (+ (* 2.0 (* A t_2)) (* (pow B 2.0) t_7)))
                 (/
                  (*
                   F
                   (+
                    (* (pow A 2.0) (* -2.0 (- t_1 t_0)))
                    (* -0.5 (* (pow B 2.0) t_2))))
                  C))))))
            t_6)))))))

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 = pow(B, 2.0) + pow(A, 2.0);
	double t_1 = pow(-A, 2.0);
	double t_2 = t_0 - t_1;
	double t_3 = sqrt((pow(B, 2.0) + pow((A - C), 2.0)));
	double t_4 = sqrt(2.0) / B;
	double t_5 = pow(B, 2.0) - (A * (4.0 * C));
	double t_6 = pow(B, 2.0) - (4.0 * (C * A));
	double t_7 = A - -A;
	double tmp;
	if (C <= -9e+149) {
		tmp = -(t_4 * sqrt((((pow(B, 2.0) / A) * -0.5) * F)));
	} else if (C <= -5.5e-121) {
		tmp = -sqrt((2.0 * ((t_5 * F) * (C + (A - t_3))))) / t_5;
	} else if (C <= -3.4e-211) {
		tmp = -(sqrt((F * (C - B))) * t_4);
	} else if (C <= 8e-46) {
		tmp = -sqrt((t_6 * ((A + (C - t_3)) * (2.0 * F)))) / t_6;
	} else {
		tmp = -sqrt(((-8.0 * (A * ((F * C) * t_7))) + (2.0 * ((F * ((2.0 * (A * t_2)) + (pow(B, 2.0) * t_7))) + ((F * ((pow(A, 2.0) * (-2.0 * (t_1 - t_0))) + (-0.5 * (pow(B, 2.0) * t_2)))) / C))))) / t_6;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(((2.0d0 * (((b ** 2.0d0) - ((4.0d0 * a) * c)) * f)) * ((a + c) - sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / ((b ** 2.0d0) - ((4.0d0 * a) * c))
end function

↓

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_0 = (b ** 2.0d0) + (a ** 2.0d0)
    t_1 = -a ** 2.0d0
    t_2 = t_0 - t_1
    t_3 = sqrt(((b ** 2.0d0) + ((a - c) ** 2.0d0)))
    t_4 = sqrt(2.0d0) / b
    t_5 = (b ** 2.0d0) - (a * (4.0d0 * c))
    t_6 = (b ** 2.0d0) - (4.0d0 * (c * a))
    t_7 = a - -a
    if (c <= (-9d+149)) then
        tmp = -(t_4 * sqrt(((((b ** 2.0d0) / a) * (-0.5d0)) * f)))
    else if (c <= (-5.5d-121)) then
        tmp = -sqrt((2.0d0 * ((t_5 * f) * (c + (a - t_3))))) / t_5
    else if (c <= (-3.4d-211)) then
        tmp = -(sqrt((f * (c - b))) * t_4)
    else if (c <= 8d-46) then
        tmp = -sqrt((t_6 * ((a + (c - t_3)) * (2.0d0 * f)))) / t_6
    else
        tmp = -sqrt((((-8.0d0) * (a * ((f * c) * t_7))) + (2.0d0 * ((f * ((2.0d0 * (a * t_2)) + ((b ** 2.0d0) * t_7))) + ((f * (((a ** 2.0d0) * ((-2.0d0) * (t_1 - t_0))) + ((-0.5d0) * ((b ** 2.0d0) * t_2)))) / c))))) / t_6
    end if
    code = tmp
end function

Java

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

↓

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) + Math.pow(A, 2.0);
	double t_1 = Math.pow(-A, 2.0);
	double t_2 = t_0 - t_1;
	double t_3 = Math.sqrt((Math.pow(B, 2.0) + Math.pow((A - C), 2.0)));
	double t_4 = Math.sqrt(2.0) / B;
	double t_5 = Math.pow(B, 2.0) - (A * (4.0 * C));
	double t_6 = Math.pow(B, 2.0) - (4.0 * (C * A));
	double t_7 = A - -A;
	double tmp;
	if (C <= -9e+149) {
		tmp = -(t_4 * Math.sqrt((((Math.pow(B, 2.0) / A) * -0.5) * F)));
	} else if (C <= -5.5e-121) {
		tmp = -Math.sqrt((2.0 * ((t_5 * F) * (C + (A - t_3))))) / t_5;
	} else if (C <= -3.4e-211) {
		tmp = -(Math.sqrt((F * (C - B))) * t_4);
	} else if (C <= 8e-46) {
		tmp = -Math.sqrt((t_6 * ((A + (C - t_3)) * (2.0 * F)))) / t_6;
	} else {
		tmp = -Math.sqrt(((-8.0 * (A * ((F * C) * t_7))) + (2.0 * ((F * ((2.0 * (A * t_2)) + (Math.pow(B, 2.0) * t_7))) + ((F * ((Math.pow(A, 2.0) * (-2.0 * (t_1 - t_0))) + (-0.5 * (Math.pow(B, 2.0) * t_2)))) / C))))) / t_6;
	}
	return tmp;
}

Python

def code(A, B, C, F):
	return -math.sqrt(((2.0 * ((math.pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / (math.pow(B, 2.0) - ((4.0 * A) * C))

↓

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) + math.pow(A, 2.0)
	t_1 = math.pow(-A, 2.0)
	t_2 = t_0 - t_1
	t_3 = math.sqrt((math.pow(B, 2.0) + math.pow((A - C), 2.0)))
	t_4 = math.sqrt(2.0) / B
	t_5 = math.pow(B, 2.0) - (A * (4.0 * C))
	t_6 = math.pow(B, 2.0) - (4.0 * (C * A))
	t_7 = A - -A
	tmp = 0
	if C <= -9e+149:
		tmp = -(t_4 * math.sqrt((((math.pow(B, 2.0) / A) * -0.5) * F)))
	elif C <= -5.5e-121:
		tmp = -math.sqrt((2.0 * ((t_5 * F) * (C + (A - t_3))))) / t_5
	elif C <= -3.4e-211:
		tmp = -(math.sqrt((F * (C - B))) * t_4)
	elif C <= 8e-46:
		tmp = -math.sqrt((t_6 * ((A + (C - t_3)) * (2.0 * F)))) / t_6
	else:
		tmp = -math.sqrt(((-8.0 * (A * ((F * C) * t_7))) + (2.0 * ((F * ((2.0 * (A * t_2)) + (math.pow(B, 2.0) * t_7))) + ((F * ((math.pow(A, 2.0) * (-2.0 * (t_1 - t_0))) + (-0.5 * (math.pow(B, 2.0) * t_2)))) / C))))) / t_6
	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((B ^ 2.0) + (A ^ 2.0))
	t_1 = Float64(-A) ^ 2.0
	t_2 = Float64(t_0 - t_1)
	t_3 = sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))
	t_4 = Float64(sqrt(2.0) / B)
	t_5 = Float64((B ^ 2.0) - Float64(A * Float64(4.0 * C)))
	t_6 = Float64((B ^ 2.0) - Float64(4.0 * Float64(C * A)))
	t_7 = Float64(A - Float64(-A))
	tmp = 0.0
	if (C <= -9e+149)
		tmp = Float64(-Float64(t_4 * sqrt(Float64(Float64(Float64((B ^ 2.0) / A) * -0.5) * F))));
	elseif (C <= -5.5e-121)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_5 * F) * Float64(C + Float64(A - t_3)))))) / t_5);
	elseif (C <= -3.4e-211)
		tmp = Float64(-Float64(sqrt(Float64(F * Float64(C - B))) * t_4));
	elseif (C <= 8e-46)
		tmp = Float64(Float64(-sqrt(Float64(t_6 * Float64(Float64(A + Float64(C - t_3)) * Float64(2.0 * F))))) / t_6);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(-8.0 * Float64(A * Float64(Float64(F * C) * t_7))) + Float64(2.0 * Float64(Float64(F * Float64(Float64(2.0 * Float64(A * t_2)) + Float64((B ^ 2.0) * t_7))) + Float64(Float64(F * Float64(Float64((A ^ 2.0) * Float64(-2.0 * Float64(t_1 - t_0))) + Float64(-0.5 * Float64((B ^ 2.0) * t_2)))) / C)))))) / t_6);
	end
	return tmp
end

MATLAB

function tmp = code(A, B, C, F)
	tmp = -sqrt(((2.0 * (((B ^ 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / ((B ^ 2.0) - ((4.0 * A) * C));
end

↓

function tmp_2 = code(A, B, C, F)
	t_0 = (B ^ 2.0) + (A ^ 2.0);
	t_1 = -A ^ 2.0;
	t_2 = t_0 - t_1;
	t_3 = sqrt(((B ^ 2.0) + ((A - C) ^ 2.0)));
	t_4 = sqrt(2.0) / B;
	t_5 = (B ^ 2.0) - (A * (4.0 * C));
	t_6 = (B ^ 2.0) - (4.0 * (C * A));
	t_7 = A - -A;
	tmp = 0.0;
	if (C <= -9e+149)
		tmp = -(t_4 * sqrt(((((B ^ 2.0) / A) * -0.5) * F)));
	elseif (C <= -5.5e-121)
		tmp = -sqrt((2.0 * ((t_5 * F) * (C + (A - t_3))))) / t_5;
	elseif (C <= -3.4e-211)
		tmp = -(sqrt((F * (C - B))) * t_4);
	elseif (C <= 8e-46)
		tmp = -sqrt((t_6 * ((A + (C - t_3)) * (2.0 * F)))) / t_6;
	else
		tmp = -sqrt(((-8.0 * (A * ((F * C) * t_7))) + (2.0 * ((F * ((2.0 * (A * t_2)) + ((B ^ 2.0) * t_7))) + ((F * (((A ^ 2.0) * (-2.0 * (t_1 - t_0))) + (-0.5 * ((B ^ 2.0) * t_2)))) / C))))) / t_6;
	end
	tmp_2 = 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[(N[Power[B, 2.0], $MachinePrecision] + N[Power[A, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[(-A), 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(A - (-A)), $MachinePrecision]}, If[LessEqual[C, -9e+149], (-N[(t$95$4 * N[Sqrt[N[(N[(N[(N[Power[B, 2.0], $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), If[LessEqual[C, -5.5e-121], N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$5 * F), $MachinePrecision] * N[(C + N[(A - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$5), $MachinePrecision], If[LessEqual[C, -3.4e-211], (-N[(N[Sqrt[N[(F * N[(C - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision]), If[LessEqual[C, 8e-46], N[((-N[Sqrt[N[(t$95$6 * N[(N[(A + N[(C - t$95$3), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision], N[((-N[Sqrt[N[(N[(-8.0 * N[(A * N[(N[(F * C), $MachinePrecision] * t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(F * N[(N[(2.0 * N[(A * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[Power[B, 2.0], $MachinePrecision] * t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F * N[(N[(N[Power[A, 2.0], $MachinePrecision] * N[(-2.0 * N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[Power[B, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if C < -8.99999999999999965e149

   1. Initial program 63.8

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

   2. Simplified 63.8

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

      [Start] 63.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 63.8

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

   4. Taylor expanded in C around 0 63.2

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

   5. Simplified 63.2

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

      [Start] 63.2	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 63.2	\[ \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \cdot -1} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 63.2	\[ \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}} \]

   6. Taylor expanded in A around inf 54.3

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

   7. Simplified 54.3

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

      [Start] 54.3	\[ -\frac{\sqrt{2}}{B} \cdot \sqrt{\left(-0.5 \cdot \frac{{B}^{2}}{A}\right) \cdot F} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 54.3	\[ -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(\frac{{B}^{2}}{A} \cdot -0.5\right)} \cdot F} \]

   if -8.99999999999999965e149 < C < -5.50000000000000031e-121

   1. Initial program 42.7

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

   2. Simplified 42.3

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

      [Start] 42.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if -5.50000000000000031e-121 < C < -3.4000000000000001e-211

   1. Initial program 46.5

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

   2. Simplified 46.5

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

      [Start] 46.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around inf 57.3

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

   4. Taylor expanded in A around 0 52.7

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

   5. Simplified 52.7

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

      [Start] 52.7	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - B\right) \cdot F}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 52.7	\[ \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - B\right) \cdot F}\right) \cdot -1} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 52.7	\[ \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - B\right) \cdot F}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 52.7	\[ -\color{blue}{\sqrt{\left(C - B\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 52.7	\[ -\sqrt{\color{blue}{F \cdot \left(C - B\right)}} \cdot \frac{\sqrt{2}}{B} \]

   if -3.4000000000000001e-211 < C < 8.00000000000000018e-46

   1. Initial program 48.8

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

   2. Simplified 48.8

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

      [Start] 48.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if 8.00000000000000018e-46 < C

   1. Initial program 59.4

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

   2. Simplified 58.5

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

      [Start] 59.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 58.6

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

   4. Taylor expanded in C around inf 50.0

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

   5. Simplified 50.0

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

      [Start] 50.0	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right)\right) + \left(2 \cdot \left(\left(2 \cdot \left(A \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)\right) + \left(A - -1 \cdot A\right) \cdot {B}^{2}\right) \cdot F\right) + 2 \cdot \frac{F \cdot \left(2 \cdot \left({A}^{2} \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)\right) + -0.5 \cdot \left(\left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right) \cdot {B}^{2}\right)\right)}{C}\right)}}{{B}^{2} - 4 \cdot \left(C \cdot A\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 50.0	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(A - -1 \cdot A\right)\right)}\right) + \left(2 \cdot \left(\left(2 \cdot \left(A \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)\right) + \left(A - -1 \cdot A\right) \cdot {B}^{2}\right) \cdot F\right) + 2 \cdot \frac{F \cdot \left(2 \cdot \left({A}^{2} \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)\right) + -0.5 \cdot \left(\left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right) \cdot {B}^{2}\right)\right)}{C}\right)}}{{B}^{2} - 4 \cdot \left(C \cdot A\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 50.0	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\color{blue}{\left(F \cdot C\right)} \cdot \left(A - -1 \cdot A\right)\right)\right) + \left(2 \cdot \left(\left(2 \cdot \left(A \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)\right) + \left(A - -1 \cdot A\right) \cdot {B}^{2}\right) \cdot F\right) + 2 \cdot \frac{F \cdot \left(2 \cdot \left({A}^{2} \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)\right) + -0.5 \cdot \left(\left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right) \cdot {B}^{2}\right)\right)}{C}\right)}}{{B}^{2} - 4 \cdot \left(C \cdot A\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 50.0	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A - \color{blue}{A \cdot -1}\right)\right)\right) + \left(2 \cdot \left(\left(2 \cdot \left(A \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)\right) + \left(A - -1 \cdot A\right) \cdot {B}^{2}\right) \cdot F\right) + 2 \cdot \frac{F \cdot \left(2 \cdot \left({A}^{2} \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)\right) + -0.5 \cdot \left(\left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right) \cdot {B}^{2}\right)\right)}{C}\right)}}{{B}^{2} - 4 \cdot \left(C \cdot A\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 50.0	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A - \color{blue}{\left(-A\right)}\right)\right)\right) + \left(2 \cdot \left(\left(2 \cdot \left(A \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)\right) + \left(A - -1 \cdot A\right) \cdot {B}^{2}\right) \cdot F\right) + 2 \cdot \frac{F \cdot \left(2 \cdot \left({A}^{2} \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)\right) + -0.5 \cdot \left(\left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right) \cdot {B}^{2}\right)\right)}{C}\right)}}{{B}^{2} - 4 \cdot \left(C \cdot A\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 50.0	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A - \left(-A\right)\right)\right)\right) + \left(\color{blue}{\left(\left(2 \cdot \left(A \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)\right) + \left(A - -1 \cdot A\right) \cdot {B}^{2}\right) \cdot F\right) \cdot 2} + 2 \cdot \frac{F \cdot \left(2 \cdot \left({A}^{2} \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)\right) + -0.5 \cdot \left(\left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right) \cdot {B}^{2}\right)\right)}{C}\right)}}{{B}^{2} - 4 \cdot \left(C \cdot A\right)} \]

3. Recombined 5 regimes into one program.

4. Final simplification 48.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -9 \cdot 10^{+149}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(\frac{{B}^{2}}{A} \cdot -0.5\right) \cdot F}\\ \mathbf{elif}\;C \leq -5.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(C + \left(A - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{{B}^{2} - A \cdot \left(4 \cdot C\right)}\\ \mathbf{elif}\;C \leq -3.4 \cdot 10^{-211}:\\ \;\;\;\;-\sqrt{F \cdot \left(C - B\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;C \leq 8 \cdot 10^{-46}:\\ \;\;\;\;\frac{-\sqrt{\left({B}^{2} - 4 \cdot \left(C \cdot A\right)\right) \cdot \left(\left(A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{{B}^{2} - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A - \left(-A\right)\right)\right)\right) + 2 \cdot \left(F \cdot \left(2 \cdot \left(A \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-A\right)}^{2}\right)\right) + {B}^{2} \cdot \left(A - \left(-A\right)\right)\right) + \frac{F \cdot \left({A}^{2} \cdot \left(-2 \cdot \left({\left(-A\right)}^{2} - \left({B}^{2} + {A}^{2}\right)\right)\right) + -0.5 \cdot \left({B}^{2} \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-A\right)}^{2}\right)\right)\right)}{C}\right)}}{{B}^{2} - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	46.3
Cost	183632

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := {B}^{2} - 4 \cdot \left(C \cdot A\right)\\ t_2 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t_2}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;-t_0 \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\\ \mathbf{elif}\;t_3 \leq -2 \cdot 10^{-191}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq 10^{-121}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A - \left(-A\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + \left(C - \left(-C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \left(-B\right)} \cdot t_0\\ \end{array} \]

Alternative 2
Error	46.5
Cost	183632

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := {\left(A - C\right)}^{2}\\ t_2 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{t_1 + {B}^{2}}\right)}}{t_2}\\ t_4 := {B}^{2} - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;-t_0 \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\\ \mathbf{elif}\;t_3 \leq -2 \cdot 10^{-191}:\\ \;\;\;\;\frac{-\sqrt{t_4 \cdot \left(\left(A + \left(C - \sqrt{{B}^{2} + t_1}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_4}\\ \mathbf{elif}\;t_3 \leq 10^{-121}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A - \left(-A\right)\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\frac{-\sqrt{t_4 \cdot \left(\left(A + \left(C - \left(-C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \left(-B\right)} \cdot t_0\\ \end{array} \]

Alternative 3
Error	46.5
Cost	183632

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := {\left(A - C\right)}^{2}\\ t_2 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := {B}^{2} - 4 \cdot \left(C \cdot A\right)\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{t_1 + {B}^{2}}\right)}}{t_2}\\ t_5 := \sqrt{{B}^{2} + t_1}\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;-t_0 \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\\ \mathbf{elif}\;t_4 \leq -2 \cdot 10^{-191}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(A + \left(C - t_5 \cdot \left(t_5 \cdot \frac{1}{t_5}\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_3}\\ \mathbf{elif}\;t_4 \leq 10^{-121}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A - \left(-A\right)\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(A + \left(C - \left(-C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \left(-B\right)} \cdot t_0\\ \end{array} \]

Alternative 4
Error	48.6
Cost	41104

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := {B}^{2} - A \cdot \left(4 \cdot C\right)\\ t_2 := \frac{-\sqrt{2 \cdot \left(\left(t_1 \cdot F\right) \cdot \left(C + \left(A - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{t_1}\\ \mathbf{if}\;C \leq -7.5 \cdot 10^{+149}:\\ \;\;\;\;-t_0 \cdot \sqrt{\left(\frac{{B}^{2}}{A} \cdot -0.5\right) \cdot F}\\ \mathbf{elif}\;C \leq -6.1 \cdot 10^{-120}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -7.5 \cdot 10^{-198}:\\ \;\;\;\;-\sqrt{F \cdot \left(C - B\right)} \cdot t_0\\ \mathbf{elif}\;C \leq 5.2 \cdot 10^{-45}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A - \left(-A\right)\right)\right)\right)}}{{B}^{2} - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 5
Error	51.5
Cost	21520

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;C \leq -1.25 \cdot 10^{+208}:\\ \;\;\;\;-t_0 \cdot \sqrt{\left(\frac{{B}^{2}}{A} \cdot -0.5\right) \cdot F}\\ \mathbf{elif}\;C \leq -5.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(-8 \cdot \left(C \cdot \left(\left(C - -1 \cdot C\right) \cdot F\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{elif}\;C \leq -2.75 \cdot 10^{-216}:\\ \;\;\;\;-\sqrt{F \cdot \left(C - B\right)} \cdot t_0\\ \mathbf{elif}\;C \leq -6.5 \cdot 10^{-261}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C + A \cdot 2\right) \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{{B}^{2} - A \cdot \left(4 \cdot C\right)}\\ \mathbf{elif}\;C \leq 2.25 \cdot 10^{-141}:\\ \;\;\;\;-\sqrt{F \cdot \left(-B\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A - \left(-A\right)\right)\right)\right)}}{{B}^{2} - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 6
Error	51.4
Cost	21388

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := {B}^{2} - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;C \leq -2.1 \cdot 10^{-75}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + \left(C - \left(-C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;C \leq -1.32 \cdot 10^{-216}:\\ \;\;\;\;-\sqrt{F \cdot \left(C - B\right)} \cdot t_0\\ \mathbf{elif}\;C \leq -3.05 \cdot 10^{-261}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C + A \cdot 2\right) \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{{B}^{2} - A \cdot \left(4 \cdot C\right)}\\ \mathbf{elif}\;C \leq 1.6 \cdot 10^{-141}:\\ \;\;\;\;-\sqrt{F \cdot \left(-B\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A - \left(-A\right)\right)\right)\right)}}{t_1}\\ \end{array} \]

Alternative 7
Error	51.3
Cost	20684

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;C \leq -2.2 \cdot 10^{+209}:\\ \;\;\;\;-t_0 \cdot \sqrt{\left(\frac{{B}^{2}}{A} \cdot -0.5\right) \cdot F}\\ \mathbf{elif}\;C \leq -1.85 \cdot 10^{-132}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(-8 \cdot \left(C \cdot \left(\left(C - -1 \cdot C\right) \cdot F\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{elif}\;C \leq 1.85 \cdot 10^{-141}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{F \cdot \left(C - \left(0.5 \cdot \frac{{C}^{2}}{B} + B\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A - \left(-A\right)\right)\right)\right)}}{{B}^{2} - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 8
Error	49.6
Cost	20616

\[\begin{array}{l} t_0 := {B}^{2} - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -5.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot {B}^{3}\right)}}{t_0}\\ \mathbf{elif}\;B \leq -5.1 \cdot 10^{-173}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(C \cdot \left({A}^{2} \cdot F\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(-8 \cdot \left(C \cdot \left(\left(C - -1 \cdot C\right) \cdot F\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \left(-B\right)} \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]

Alternative 9
Error	51.3
Cost	20164

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;C \leq -4.8 \cdot 10^{+210}:\\ \;\;\;\;-t_0 \cdot \sqrt{\left(\frac{{B}^{2}}{A} \cdot -0.5\right) \cdot F}\\ \mathbf{elif}\;C \leq -1.02 \cdot 10^{-135}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(-8 \cdot \left(C \cdot \left(\left(C - -1 \cdot C\right) \cdot F\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{elif}\;C \leq 1.9 \cdot 10^{-141}:\\ \;\;\;\;-\sqrt{F \cdot \left(C - B\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A - \left(-A\right)\right)\right)\right)}}{{B}^{2} - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 10
Error	51.9
Cost	14472

\[\begin{array}{l} \mathbf{if}\;C \leq -1.95 \cdot 10^{-133}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(-8 \cdot \left(C \cdot \left(\left(C - -1 \cdot C\right) \cdot F\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{elif}\;C \leq 9 \cdot 10^{-142}:\\ \;\;\;\;-\sqrt{F \cdot \left(C - B\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A - \left(-A\right)\right)\right)\right)}}{{B}^{2} - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 11
Error	50.7
Cost	13508

\[\begin{array}{l} \mathbf{if}\;B \leq 1.4 \cdot 10^{-111}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(-8 \cdot \left(C \cdot \left(\left(C - -1 \cdot C\right) \cdot F\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \left(-B\right)} \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]

Alternative 12
Error	57.0
Cost	7680

\[\frac{-\sqrt{A \cdot \left(-8 \cdot \left(C \cdot \left(\left(C - -1 \cdot C\right) \cdot F\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)} \]

Alternative 13
Error	62.1
Cost	6976

\[2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right) \]

Reproduce

herbie shell --seed 2023090 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))