ABCF->ab-angle b

Average Error: 52.0 → 43.8

Time: 56.0s

Precision: binary64

Cost: 96080

Math

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

↓

\[\begin{array}{l} t_0 := C - -1 \cdot C\\ t_1 := \left({B}^{2} + {C}^{2}\right) - {\left(-1 \cdot C\right)}^{2}\\ t_2 := \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\\ t_3 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := F \cdot {B}^{2}\\ \mathbf{if}\;C \leq -1.18 \cdot 10^{-201}:\\ \;\;\;\;-0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot -16}\right)\\ \mathbf{elif}\;C \leq 5.6 \cdot 10^{-290}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - \left(A \cdot C\right) \cdot 4\right)\right)\right) \cdot \left(\left(A - t_2\right) + C\right)}}{t_3}\\ \mathbf{elif}\;C \leq 4 \cdot 10^{-229}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-B\right)}\\ \mathbf{elif}\;C \leq 5.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \frac{-0.5 \cdot \left(t_1 \cdot t_4\right) + 2 \cdot \left(t_1 \cdot \left({C}^{2} \cdot F\right)\right)}{A} + \left(-8 \cdot \left(A \cdot \left(t_0 \cdot \left(C \cdot F\right)\right)\right) + 2 \cdot \left(2 \cdot \left(t_1 \cdot \left(C \cdot F\right)\right) + t_0 \cdot t_4\right)\right)}}{t_3}\\ \mathbf{elif}\;C \leq 4.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(A + \left(C - t_2\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{\left(C \cdot F\right) \cdot -16} \cdot \frac{1}{C}\right)\\ \end{array} \]

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 (* -1.0 C)))
        (t_1 (- (+ (pow B 2.0) (pow C 2.0)) (pow (* -1.0 C) 2.0)))
        (t_2 (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))
        (t_3 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_4 (* F (pow B 2.0))))
   (if (<= C -1.18e-201)
     (* -0.25 (* (/ 1.0 A) (sqrt (* (* A F) -16.0))))
     (if (<= C 5.6e-290)
       (/
        (-
         (sqrt
          (* (* 2.0 (* F (- (pow B 2.0) (* (* A C) 4.0)))) (+ (- A t_2) C))))
        t_3)
       (if (<= C 4e-229)
         (- (* (/ (sqrt 2.0) B) (sqrt (* F (- B)))))
         (if (<= C 5.8e-158)
           (/
            (-
             (sqrt
              (+
               (*
                2.0
                (/
                 (+ (* -0.5 (* t_1 t_4)) (* 2.0 (* t_1 (* (pow C 2.0) F))))
                 A))
               (+
                (* -8.0 (* A (* t_0 (* C F))))
                (* 2.0 (+ (* 2.0 (* t_1 (* C F))) (* t_0 t_4)))))))
            t_3)
           (if (<= C 4.5e-89)
             (/ (- (sqrt (* t_3 (* (+ A (- C t_2)) (* 2.0 F))))) t_3)
             (* -0.25 (* (sqrt (* (* C F) -16.0)) (/ 1.0 C))))))))))

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 - (-1.0 * C);
	double t_1 = (pow(B, 2.0) + pow(C, 2.0)) - pow((-1.0 * C), 2.0);
	double t_2 = sqrt((pow(B, 2.0) + pow((A - C), 2.0)));
	double t_3 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_4 = F * pow(B, 2.0);
	double tmp;
	if (C <= -1.18e-201) {
		tmp = -0.25 * ((1.0 / A) * sqrt(((A * F) * -16.0)));
	} else if (C <= 5.6e-290) {
		tmp = -sqrt(((2.0 * (F * (pow(B, 2.0) - ((A * C) * 4.0)))) * ((A - t_2) + C))) / t_3;
	} else if (C <= 4e-229) {
		tmp = -((sqrt(2.0) / B) * sqrt((F * -B)));
	} else if (C <= 5.8e-158) {
		tmp = -sqrt(((2.0 * (((-0.5 * (t_1 * t_4)) + (2.0 * (t_1 * (pow(C, 2.0) * F)))) / A)) + ((-8.0 * (A * (t_0 * (C * F)))) + (2.0 * ((2.0 * (t_1 * (C * F))) + (t_0 * t_4)))))) / t_3;
	} else if (C <= 4.5e-89) {
		tmp = -sqrt((t_3 * ((A + (C - t_2)) * (2.0 * F)))) / t_3;
	} else {
		tmp = -0.25 * (sqrt(((C * F) * -16.0)) * (1.0 / C));
	}
	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) :: tmp
    t_0 = c - ((-1.0d0) * c)
    t_1 = ((b ** 2.0d0) + (c ** 2.0d0)) - (((-1.0d0) * c) ** 2.0d0)
    t_2 = sqrt(((b ** 2.0d0) + ((a - c) ** 2.0d0)))
    t_3 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    t_4 = f * (b ** 2.0d0)
    if (c <= (-1.18d-201)) then
        tmp = (-0.25d0) * ((1.0d0 / a) * sqrt(((a * f) * (-16.0d0))))
    else if (c <= 5.6d-290) then
        tmp = -sqrt(((2.0d0 * (f * ((b ** 2.0d0) - ((a * c) * 4.0d0)))) * ((a - t_2) + c))) / t_3
    else if (c <= 4d-229) then
        tmp = -((sqrt(2.0d0) / b) * sqrt((f * -b)))
    else if (c <= 5.8d-158) then
        tmp = -sqrt(((2.0d0 * ((((-0.5d0) * (t_1 * t_4)) + (2.0d0 * (t_1 * ((c ** 2.0d0) * f)))) / a)) + (((-8.0d0) * (a * (t_0 * (c * f)))) + (2.0d0 * ((2.0d0 * (t_1 * (c * f))) + (t_0 * t_4)))))) / t_3
    else if (c <= 4.5d-89) then
        tmp = -sqrt((t_3 * ((a + (c - t_2)) * (2.0d0 * f)))) / t_3
    else
        tmp = (-0.25d0) * (sqrt(((c * f) * (-16.0d0))) * (1.0d0 / c))
    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 = C - (-1.0 * C);
	double t_1 = (Math.pow(B, 2.0) + Math.pow(C, 2.0)) - Math.pow((-1.0 * C), 2.0);
	double t_2 = Math.sqrt((Math.pow(B, 2.0) + Math.pow((A - C), 2.0)));
	double t_3 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	double t_4 = F * Math.pow(B, 2.0);
	double tmp;
	if (C <= -1.18e-201) {
		tmp = -0.25 * ((1.0 / A) * Math.sqrt(((A * F) * -16.0)));
	} else if (C <= 5.6e-290) {
		tmp = -Math.sqrt(((2.0 * (F * (Math.pow(B, 2.0) - ((A * C) * 4.0)))) * ((A - t_2) + C))) / t_3;
	} else if (C <= 4e-229) {
		tmp = -((Math.sqrt(2.0) / B) * Math.sqrt((F * -B)));
	} else if (C <= 5.8e-158) {
		tmp = -Math.sqrt(((2.0 * (((-0.5 * (t_1 * t_4)) + (2.0 * (t_1 * (Math.pow(C, 2.0) * F)))) / A)) + ((-8.0 * (A * (t_0 * (C * F)))) + (2.0 * ((2.0 * (t_1 * (C * F))) + (t_0 * t_4)))))) / t_3;
	} else if (C <= 4.5e-89) {
		tmp = -Math.sqrt((t_3 * ((A + (C - t_2)) * (2.0 * F)))) / t_3;
	} else {
		tmp = -0.25 * (Math.sqrt(((C * F) * -16.0)) * (1.0 / C));
	}
	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 = C - (-1.0 * C)
	t_1 = (math.pow(B, 2.0) + math.pow(C, 2.0)) - math.pow((-1.0 * C), 2.0)
	t_2 = math.sqrt((math.pow(B, 2.0) + math.pow((A - C), 2.0)))
	t_3 = math.pow(B, 2.0) - ((4.0 * A) * C)
	t_4 = F * math.pow(B, 2.0)
	tmp = 0
	if C <= -1.18e-201:
		tmp = -0.25 * ((1.0 / A) * math.sqrt(((A * F) * -16.0)))
	elif C <= 5.6e-290:
		tmp = -math.sqrt(((2.0 * (F * (math.pow(B, 2.0) - ((A * C) * 4.0)))) * ((A - t_2) + C))) / t_3
	elif C <= 4e-229:
		tmp = -((math.sqrt(2.0) / B) * math.sqrt((F * -B)))
	elif C <= 5.8e-158:
		tmp = -math.sqrt(((2.0 * (((-0.5 * (t_1 * t_4)) + (2.0 * (t_1 * (math.pow(C, 2.0) * F)))) / A)) + ((-8.0 * (A * (t_0 * (C * F)))) + (2.0 * ((2.0 * (t_1 * (C * F))) + (t_0 * t_4)))))) / t_3
	elif C <= 4.5e-89:
		tmp = -math.sqrt((t_3 * ((A + (C - t_2)) * (2.0 * F)))) / t_3
	else:
		tmp = -0.25 * (math.sqrt(((C * F) * -16.0)) * (1.0 / C))
	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(-1.0 * C))
	t_1 = Float64(Float64((B ^ 2.0) + (C ^ 2.0)) - (Float64(-1.0 * C) ^ 2.0))
	t_2 = sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))
	t_3 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(F * (B ^ 2.0))
	tmp = 0.0
	if (C <= -1.18e-201)
		tmp = Float64(-0.25 * Float64(Float64(1.0 / A) * sqrt(Float64(Float64(A * F) * -16.0))));
	elseif (C <= 5.6e-290)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * Float64((B ^ 2.0) - Float64(Float64(A * C) * 4.0)))) * Float64(Float64(A - t_2) + C)))) / t_3);
	elseif (C <= 4e-229)
		tmp = Float64(-Float64(Float64(sqrt(2.0) / B) * sqrt(Float64(F * Float64(-B)))));
	elseif (C <= 5.8e-158)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(Float64(Float64(-0.5 * Float64(t_1 * t_4)) + Float64(2.0 * Float64(t_1 * Float64((C ^ 2.0) * F)))) / A)) + Float64(Float64(-8.0 * Float64(A * Float64(t_0 * Float64(C * F)))) + Float64(2.0 * Float64(Float64(2.0 * Float64(t_1 * Float64(C * F))) + Float64(t_0 * t_4))))))) / t_3);
	elseif (C <= 4.5e-89)
		tmp = Float64(Float64(-sqrt(Float64(t_3 * Float64(Float64(A + Float64(C - t_2)) * Float64(2.0 * F))))) / t_3);
	else
		tmp = Float64(-0.25 * Float64(sqrt(Float64(Float64(C * F) * -16.0)) * Float64(1.0 / C)));
	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 = C - (-1.0 * C);
	t_1 = ((B ^ 2.0) + (C ^ 2.0)) - ((-1.0 * C) ^ 2.0);
	t_2 = sqrt(((B ^ 2.0) + ((A - C) ^ 2.0)));
	t_3 = (B ^ 2.0) - ((4.0 * A) * C);
	t_4 = F * (B ^ 2.0);
	tmp = 0.0;
	if (C <= -1.18e-201)
		tmp = -0.25 * ((1.0 / A) * sqrt(((A * F) * -16.0)));
	elseif (C <= 5.6e-290)
		tmp = -sqrt(((2.0 * (F * ((B ^ 2.0) - ((A * C) * 4.0)))) * ((A - t_2) + C))) / t_3;
	elseif (C <= 4e-229)
		tmp = -((sqrt(2.0) / B) * sqrt((F * -B)));
	elseif (C <= 5.8e-158)
		tmp = -sqrt(((2.0 * (((-0.5 * (t_1 * t_4)) + (2.0 * (t_1 * ((C ^ 2.0) * F)))) / A)) + ((-8.0 * (A * (t_0 * (C * F)))) + (2.0 * ((2.0 * (t_1 * (C * F))) + (t_0 * t_4)))))) / t_3;
	elseif (C <= 4.5e-89)
		tmp = -sqrt((t_3 * ((A + (C - t_2)) * (2.0 * F)))) / t_3;
	else
		tmp = -0.25 * (sqrt(((C * F) * -16.0)) * (1.0 / C));
	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[(C - N[(-1.0 * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[C, 2.0], $MachinePrecision]), $MachinePrecision] - N[Power[N[(-1.0 * C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(F * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -1.18e-201], N[(-0.25 * N[(N[(1.0 / A), $MachinePrecision] * N[Sqrt[N[(N[(A * F), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 5.6e-290], N[((-N[Sqrt[N[(N[(2.0 * N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(A - t$95$2), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[C, 4e-229], (-N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[Sqrt[N[(F * (-B)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), If[LessEqual[C, 5.8e-158], N[((-N[Sqrt[N[(N[(2.0 * N[(N[(N[(-0.5 * N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$1 * N[(N[Power[C, 2.0], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(N[(-8.0 * N[(A * N[(t$95$0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(2.0 * N[(t$95$1 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[C, 4.5e-89], N[((-N[Sqrt[N[(t$95$3 * N[(N[(A + N[(C - t$95$2), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], N[(-0.25 * N[(N[Sqrt[N[(N[(C * F), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] * N[(1.0 / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if C < -1.18e-201

   1. Initial program 49.7

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

   2. Simplified 49.7

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

      [Start] 49.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 49.8

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

   4. Simplified 49.8

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

      [Start] 49.8	\[ \frac{-\sqrt{\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + C\right)\right)\right) - \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} \cdot \left(F + F\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best_45_simplify-1 [=>] 49.8	\[ \frac{-\sqrt{\left({B}^{2} - \color{blue}{A \cdot \left(4 \cdot C\right)}\right) \cdot \left(2 \cdot \left(F \cdot \left(A + C\right)\right)\right) - \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} \cdot \left(F + F\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best_45_simplify-1 [=>] 49.8	\[ \frac{-\sqrt{\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + C\right)\right)\right) - \left({B}^{2} - \color{blue}{A \cdot \left(4 \cdot C\right)}\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} \cdot \left(F + F\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in C around -inf 40.5

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{-8 \cdot \left(A \cdot F\right) - 8 \cdot \left(A \cdot F\right)}\right)} \]

   6. Simplified 40.5

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

      [Start] 40.5	\[ -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{-8 \cdot \left(A \cdot F\right) - 8 \cdot \left(A \cdot F\right)}\right) \]
      rational_best_45_simplify-18 [=>] 40.5	\[ -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{-8 \cdot \left(A \cdot F\right) - \color{blue}{\left(A \cdot F\right) \cdot 8}}\right) \]
      rational_best_45_simplify-12 [=>] 40.5	\[ -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left(-8 - 8\right)}}\right) \]
      metadata-eval [=>] 40.5	\[ -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{-16}}\right) \]

   if -1.18e-201 < C < 5.59999999999999993e-290

   1. Initial program 49.1

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

   2. Simplified 49.0

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

      [Start] 49.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in F around 0 51.0

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

   4. Simplified 48.8

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

      [Start] 51.0	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best_45_simplify-1 [=>] 49.0	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best_45_simplify-20 [=>] 49.0	\[ \frac{-\sqrt{2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best_45_simplify-20 [=>] 49.0	\[ \frac{-\sqrt{2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(\color{blue}{\left(C + A\right)} - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best_45_simplify-109 [<=] 49.0	\[ \frac{-\sqrt{2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \color{blue}{\left(A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best_45_simplify-18 [<=] 49.0	\[ \frac{-\sqrt{2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(\left(A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot F\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best_45_simplify-1 [=>] 49.1	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left(A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best_45_simplify-1 [<=] 49.1	\[ \frac{-\sqrt{\color{blue}{\left(A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best_45_simplify-18 [=>] 49.1	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best_45_simplify-109 [=>] 49.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(C + A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if 5.59999999999999993e-290 < C < 4.00000000000000028e-229

   1. Initial program 48.0

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

   2. Simplified 48.0

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

      [Start] 48.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around 0 57.1

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

   4. Simplified 57.1

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

      [Start] 57.1	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right) \]
      rational_best_45_simplify-18 [=>] 57.1	\[ \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right) \cdot -1} \]
      rational_best_45_simplify-63 [=>] 57.1	\[ \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]
      rational_best_45_simplify-18 [=>] 57.1	\[ -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

   5. Taylor expanded in C around 0 53.6

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

   6. Simplified 53.6

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

      [Start] 53.6	\[ -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-1 \cdot B\right)} \]
      rational_best_45_simplify-18 [=>] 53.6	\[ -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(B \cdot -1\right)}} \]
      rational_best_45_simplify-63 [=>] 53.6	\[ -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(-B\right)}} \]

   if 4.00000000000000028e-229 < C < 5.79999999999999961e-158

   1. Initial program 47.7

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

   2. Simplified 47.8

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

      [Start] 47.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around inf 56.1

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

   if 5.79999999999999961e-158 < C < 4.4999999999999999e-89

   1. Initial program 46.7

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

   2. Simplified 46.8

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

      [Start] 46.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if 4.4999999999999999e-89 < C

   1. Initial program 58.4

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

   2. Simplified 57.4

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

      [Start] 58.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 58.4

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

   4. Simplified 58.4

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

      [Start] 58.4	\[ \frac{-\sqrt{\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + C\right)\right)\right) - \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} \cdot \left(F + F\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best_45_simplify-1 [=>] 58.4	\[ \frac{-\sqrt{\left({B}^{2} - \color{blue}{A \cdot \left(4 \cdot C\right)}\right) \cdot \left(2 \cdot \left(F \cdot \left(A + C\right)\right)\right) - \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} \cdot \left(F + F\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best_45_simplify-1 [=>] 58.4	\[ \frac{-\sqrt{\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + C\right)\right)\right) - \left({B}^{2} - \color{blue}{A \cdot \left(4 \cdot C\right)}\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} \cdot \left(F + F\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in A around -inf 42.1

      \[\leadsto \color{blue}{-0.25 \cdot \left(\sqrt{-8 \cdot \left(C \cdot F\right) - 8 \cdot \left(C \cdot F\right)} \cdot \frac{1}{C}\right)} \]

   6. Simplified 42.1

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

      [Start] 42.1	\[ -0.25 \cdot \left(\sqrt{-8 \cdot \left(C \cdot F\right) - 8 \cdot \left(C \cdot F\right)} \cdot \frac{1}{C}\right) \]
      rational_best_45_simplify-18 [=>] 42.1	\[ -0.25 \cdot \left(\sqrt{-8 \cdot \left(C \cdot F\right) - \color{blue}{\left(C \cdot F\right) \cdot 8}} \cdot \frac{1}{C}\right) \]
      rational_best_45_simplify-12 [=>] 42.1	\[ -0.25 \cdot \left(\sqrt{\color{blue}{\left(C \cdot F\right) \cdot \left(-8 - 8\right)}} \cdot \frac{1}{C}\right) \]
      metadata-eval [=>] 42.1	\[ -0.25 \cdot \left(\sqrt{\left(C \cdot F\right) \cdot \color{blue}{-16}} \cdot \frac{1}{C}\right) \]

3. Recombined 6 regimes into one program.

4. Final simplification 43.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.18 \cdot 10^{-201}:\\ \;\;\;\;-0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot -16}\right)\\ \mathbf{elif}\;C \leq 5.6 \cdot 10^{-290}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - \left(A \cdot C\right) \cdot 4\right)\right)\right) \cdot \left(\left(A - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;C \leq 4 \cdot 10^{-229}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-B\right)}\\ \mathbf{elif}\;C \leq 5.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \frac{-0.5 \cdot \left(\left(\left({B}^{2} + {C}^{2}\right) - {\left(-1 \cdot C\right)}^{2}\right) \cdot \left(F \cdot {B}^{2}\right)\right) + 2 \cdot \left(\left(\left({B}^{2} + {C}^{2}\right) - {\left(-1 \cdot C\right)}^{2}\right) \cdot \left({C}^{2} \cdot F\right)\right)}{A} + \left(-8 \cdot \left(A \cdot \left(\left(C - -1 \cdot C\right) \cdot \left(C \cdot F\right)\right)\right) + 2 \cdot \left(2 \cdot \left(\left(\left({B}^{2} + {C}^{2}\right) - {\left(-1 \cdot C\right)}^{2}\right) \cdot \left(C \cdot F\right)\right) + \left(C - -1 \cdot C\right) \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;C \leq 4.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{-\sqrt{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{\left(C \cdot F\right) \cdot -16} \cdot \frac{1}{C}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	43.0
Cost	135308

\[\begin{array}{l} t_0 := {\left(A - C\right)}^{2}\\ t_1 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{t_0 + {B}^{2}}\right)}}{t_1}\\ t_3 := -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot -16}\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-205}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - \left(A \cdot C\right) \cdot 4\right)\right)\right) \cdot \left(\left(A - \sqrt{{B}^{2} + t_0}\right) + C\right)}}{t_1}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-B\right)}\\ \end{array} \]

Alternative 2
Error	45.5
Cost	40840

\[\begin{array}{l} t_0 := -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot -16}\right)\\ t_1 := {B}^{2} - A \cdot \left(4 \cdot C\right)\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(t_1 \cdot F\right) \cdot \left(A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -9.8 \cdot 10^{-256}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-294}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{\left(C \cdot F\right) \cdot -16} \cdot \frac{1}{C}\right)\\ \mathbf{elif}\;B \leq 1.26 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 3
Error	45.6
Cost	40840

\[\begin{array}{l} t_0 := -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot -16}\right)\\ t_1 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B \leq -2.5 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -9.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -5.7 \cdot 10^{-260}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-298}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{\left(C \cdot F\right) \cdot -16} \cdot \frac{1}{C}\right)\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 4
Error	44.1
Cost	40200

\[\begin{array}{l} \mathbf{if}\;C \leq -6.4 \cdot 10^{-233}:\\ \;\;\;\;-0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot -16}\right)\\ \mathbf{elif}\;C \leq 4.8 \cdot 10^{-289}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot \left(2 \cdot \left(F \cdot {B}^{2}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;C \leq 1.8 \cdot 10^{-102}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-B\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{\left(C \cdot F\right) \cdot -16} \cdot \frac{1}{C}\right)\\ \end{array} \]

Alternative 5
Error	46.3
Cost	21320

\[\begin{array}{l} t_0 := -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot -16}\right)\\ t_1 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B \leq -1.55 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + \left(C - \left(-B\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -6.2 \cdot 10^{-256}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-296}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{\left(C \cdot F\right) \cdot -16} \cdot \frac{1}{C}\right)\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 6
Error	46.3
Cost	21256

\[\begin{array}{l} t_0 := -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot -16}\right)\\ \mathbf{if}\;B \leq -1.8 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B + \left(A + C\right)\right) \cdot \left(F \cdot \left({B}^{2} - A \cdot \left(4 \cdot C\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq -4.7 \cdot 10^{-265}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-296}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{\left(C \cdot F\right) \cdot -16} \cdot \frac{1}{C}\right)\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 7
Error	46.4
Cost	20936

\[\begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot -16}\right)\\ \mathbf{if}\;B \leq -1.15 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-27}:\\ \;\;\;\;\frac{-\sqrt{{B}^{2} \cdot \left(\left(A + \left(C - \left(-B\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot 1}{t_0}\\ \mathbf{elif}\;B \leq -1.52 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(-8 \cdot \left(\left(F \cdot C\right) \cdot \left(C - -1 \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-261}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-293}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{\left(C \cdot F\right) \cdot -16} \cdot \frac{1}{C}\right)\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 8
Error	46.4
Cost	20744

\[\begin{array}{l} t_0 := -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot -16}\right)\\ \mathbf{if}\;B \leq -3.9 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -2.65 \cdot 10^{-48}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C + B\right) \cdot \left(F \cdot {B}^{2}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq -1.05 \cdot 10^{-264}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-299}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{\left(C \cdot F\right) \cdot -16} \cdot \frac{1}{C}\right)\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 9
Error	46.4
Cost	20744

\[\begin{array}{l} t_0 := -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot -16}\right)\\ \mathbf{if}\;B \leq -1.85 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.38 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\sqrt{\left(A + B\right) \cdot \left(2 \cdot \left(F \cdot {B}^{2}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq -5.4 \cdot 10^{-265}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 9.8 \cdot 10^{-294}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{\left(C \cdot F\right) \cdot -16} \cdot \frac{1}{C}\right)\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 10
Error	46.4
Cost	20488

\[\begin{array}{l} t_0 := -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot -16}\right)\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -2.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot {B}^{3}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq -1.2 \cdot 10^{-257}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-298}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{\left(C \cdot F\right) \cdot -16} \cdot \frac{1}{C}\right)\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 11
Error	44.2
Cost	13640

\[\begin{array}{l} \mathbf{if}\;C \leq -1 \cdot 10^{-207}:\\ \;\;\;\;-0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot -16}\right)\\ \mathbf{elif}\;C \leq 2.4 \cdot 10^{-107}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-B\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{\left(C \cdot F\right) \cdot -16} \cdot \frac{1}{C}\right)\\ \end{array} \]

Alternative 12
Error	44.4
Cost	7236

\[\begin{array}{l} \mathbf{if}\;A \leq 6 \cdot 10^{-302}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{\left(C \cdot F\right) \cdot -16} \cdot \frac{1}{C}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{\left(A \cdot F\right) \cdot -16}\right)\\ \end{array} \]

Alternative 13
Error	51.7
Cost	7104

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

Alternative 14
Error	62.0
Cost	6976

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

Reproduce

herbie shell --seed 2023102 
(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))))