ABCF->ab-angle b

Percentage Accurate: 18.9% → 37.8%

Time: 41.2s

Alternatives: 21

Speedup: 5.0×

• Unsound rule application detected in e-graph. Results may not be sound.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

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
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) - sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * 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]) / t$95$0), $MachinePrecision]]

Initial Program: 18.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

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
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) - sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * 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]) / t$95$0), $MachinePrecision]]

Alternative 1: 37.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \frac{B \cdot B}{C}\\ t_3 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -3.6 \cdot 10^{+109}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(\left(A - \mathsf{fma}\left(0.5, t_2, -A\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -1 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot t_0\right)\right)} \cdot \frac{-1}{t_3}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot t_0}}{t_1}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot t_2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- A (- (hypot B (- A C)) C)))
        (t_1 (fma B B (* C (* A -4.0))))
        (t_2 (/ (* B B) C))
        (t_3 (+ (* B B) (* -4.0 (* A C)))))
   (if (<= B -3.6e+109)
     (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
     (if (<= B -2e+37)
       (/
        (-
         (sqrt
          (*
           F
           (* (- A (fma 0.5 t_2 (- A))) (fma C (* A -8.0) (* 2.0 (* B B)))))))
        t_1)
       (if (<= B -1e-32)
         (* (sqrt (* 2.0 (* t_3 (* F t_0)))) (/ -1.0 t_3))
         (if (<= B 7.6e-90)
           (/ (- (sqrt (* 2.0 (* t_1 (* F (* 2.0 A)))))) t_1)
           (if (<= B 1.744e+61)
             (- (/ (sqrt (* (* 2.0 (* F t_1)) t_0)) t_1))
             (if (<= B 7e+121)
               (* (/ (sqrt 2.0) B) (- (sqrt (* F (* -0.5 t_2)))))
               (* (sqrt (* F (- A (hypot A B)))) (/ (- (sqrt 2.0)) B))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = A - (hypot(B, (A - C)) - C);
	double t_1 = fma(B, B, (C * (A * -4.0)));
	double t_2 = (B * B) / C;
	double t_3 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (B <= -3.6e+109) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= -2e+37) {
		tmp = -sqrt((F * ((A - fma(0.5, t_2, -A)) * fma(C, (A * -8.0), (2.0 * (B * B)))))) / t_1;
	} else if (B <= -1e-32) {
		tmp = sqrt((2.0 * (t_3 * (F * t_0)))) * (-1.0 / t_3);
	} else if (B <= 7.6e-90) {
		tmp = -sqrt((2.0 * (t_1 * (F * (2.0 * A))))) / t_1;
	} else if (B <= 1.744e+61) {
		tmp = -(sqrt(((2.0 * (F * t_1)) * t_0)) / t_1);
	} else if (B <= 7e+121) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (-0.5 * t_2)));
	} else {
		tmp = sqrt((F * (A - hypot(A, B)))) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(A - Float64(hypot(B, Float64(A - C)) - C))
	t_1 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_2 = Float64(Float64(B * B) / C)
	t_3 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -3.6e+109)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= -2e+37)
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(Float64(A - fma(0.5, t_2, Float64(-A))) * fma(C, Float64(A * -8.0), Float64(2.0 * Float64(B * B))))))) / t_1);
	elseif (B <= -1e-32)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_3 * Float64(F * t_0)))) * Float64(-1.0 / t_3));
	elseif (B <= 7.6e-90)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(F * Float64(2.0 * A)))))) / t_1);
	elseif (B <= 1.744e+61)
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * t_0)) / t_1));
	elseif (B <= 7e+121)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(-0.5 * t_2)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(A, B)))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A - N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]}, Block[{t$95$3 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -3.6e+109], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -2e+37], N[((-N[Sqrt[N[(F * N[(N[(A - N[(0.5 * t$95$2 + (-A)), $MachinePrecision]), $MachinePrecision] * N[(C * N[(A * -8.0), $MachinePrecision] + N[(2.0 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, -1e-32], N[(N[Sqrt[N[(2.0 * N[(t$95$3 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 7.6e-90], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 1.744e+61], (-N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), If[LessEqual[B, 7e+121], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -3.6e109

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -3.6e109 < B < -1.99999999999999991e37

   1. Initial program 16.3%

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

   3. Simplified 16.7%

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

   4. Taylor expanded in C around inf 20.2%

      \[\leadsto \frac{-\sqrt{F \cdot \left(\left(A - \color{blue}{\left(0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C} + -1 \cdot A\right)}\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 20.2%

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

      2. associate--l+ 20.3%

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

      3. unpow2 20.3%

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

      4. unpow2 20.3%

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

      5. unpow2 20.3%

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

      6. mul-1-neg 20.3%

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

      7. mul-1-neg 20.3%

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

      8. sqr-neg 20.3%

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

      9. mul-1-neg 20.3%

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

   6. Simplified 20.3%

      \[\leadsto \frac{-\sqrt{F \cdot \left(\left(A - \color{blue}{\mathsf{fma}\left(0.5, \frac{B \cdot B + \left(A \cdot A - A \cdot A\right)}{C}, -A\right)}\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 20.3%

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

   8. Applied egg-rr 20.4%

      \[\leadsto \color{blue}{1 \cdot \frac{-\sqrt{F \cdot \left(\left(A - \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(B, B, 0\right)}{C}, -A\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 20.4%

         \[\leadsto \color{blue}{\frac{-\sqrt{F \cdot \left(\left(A - \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(B, B, 0\right)}{C}, -A\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \]

   10. Simplified 20.4%

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

   if -1.99999999999999991e37 < B < -1.00000000000000006e-32

   1. Initial program 43.5%

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

   3. Simplified 43.5%

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

      1. div-inv 43.6%

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

   5. Applied egg-rr 50.7%

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

   if -1.00000000000000006e-32 < B < 7.6e-90

   1. Initial program 21.5%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in A around -inf 25.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 25.9%

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

   6. Simplified 25.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 25.9%

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

   8. Applied egg-rr 25.1%

      \[\leadsto \color{blue}{1 \cdot \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 25.1%

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

      2. distribute-frac-neg 25.1%

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

   10. Simplified 25.1%

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

   if 7.6e-90 < B < 1.7439999999999999e61

   1. Initial program 44.6%

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

   3. Simplified 50.1%

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

   if 1.7439999999999999e61 < B < 6.9999999999999999e121

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

      1. div-inv 15.2%

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

   5. Applied egg-rr 19.7%

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

   6. Taylor expanded in A around 0 17.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 17.4%

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

      2. unpow2 17.4%

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

      3. unpow2 17.4%

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

      4. hypot-def 17.7%

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

   8. Simplified 17.7%

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

   9. Taylor expanded in C around inf 23.2%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot F} \]
   10. Step-by-step derivation

       1. unpow2 23.2%

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

   11. Simplified 23.2%

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

   if 6.9999999999999999e121 < B

   1. Initial program 5.8%

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

   3. Simplified 5.8%

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

   4. Taylor expanded in C around 0 7.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 7.9%

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

      2. *-commutative 7.9%

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

      3. +-commutative 7.9%

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

      4. unpow2 7.9%

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

      5. unpow2 7.9%

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

      6. hypot-def 45.0%

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

   6. Simplified 45.0%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.6 \cdot 10^{+109}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(\left(A - \mathsf{fma}\left(0.5, \frac{B \cdot B}{C}, -A\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq -1 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right) \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2: 42.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_2}\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(A - C, B\right) - C\right)\right)\right)} \cdot \left(-\sqrt{t_0}\right)}{t_1}\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)}}{t_0}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, B \cdot \left(2 \cdot B\right)\right)} \cdot \left(-\sqrt{F}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* C (* A -4.0))))
        (t_1 (fma B B (* A (* C -4.0))))
        (t_2 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
          t_2)))
   (if (<= t_3 -1e-181)
     (/
      (* (sqrt (* 2.0 (* F (- A (- (hypot (- A C) B) C))))) (- (sqrt t_0)))
      t_1)
     (if (<= t_3 0.0)
       (/
        (-
         (sqrt
          (*
           (* 2.0 (* F t_0))
           (- A (- (* -0.5 (/ (- (- (* A A) (* A A)) (* B B)) C)) A)))))
        t_0)
       (if (<= t_3 INFINITY)
         (/
          (*
           (sqrt
            (*
             (+ C (- A (hypot B (- A C))))
             (fma C (* A -8.0) (* B (* 2.0 B)))))
           (- (sqrt F)))
          t_1)
         (* (sqrt (* F (- A (hypot A B)))) (/ (- (sqrt 2.0)) B)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (C * (A * -4.0)));
	double t_1 = fma(B, B, (A * (C * -4.0)));
	double t_2 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -1e-181) {
		tmp = (sqrt((2.0 * (F * (A - (hypot((A - C), B) - C))))) * -sqrt(t_0)) / t_1;
	} else if (t_3 <= 0.0) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))) / t_0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt(((C + (A - hypot(B, (A - C)))) * fma(C, (A * -8.0), (B * (2.0 * B))))) * -sqrt(F)) / t_1;
	} else {
		tmp = sqrt((F * (A - hypot(A, B)))) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_1 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_2 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_2)
	tmp = 0.0
	if (t_3 <= -1e-181)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - Float64(hypot(Float64(A - C), B) - C))))) * Float64(-sqrt(t_0))) / t_1);
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A - Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(A * A) - Float64(A * A)) - Float64(B * B)) / C)) - A))))) / t_0);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(C + Float64(A - hypot(B, Float64(A - C)))) * fma(C, Float64(A * -8.0), Float64(B * Float64(2.0 * B))))) * Float64(-sqrt(F))) / t_1);
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(A, B)))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-181], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[t$95$0], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A - N[(N[(-0.5 * N[(N[(N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(N[(C + N[(A - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(C * N[(A * -8.0), $MachinePrecision] + N[(B * N[(2.0 * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -1.00000000000000005e-181

   1. Initial program 40.8%

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

   3. Simplified 51.1%

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

      1. sqrt-prod 64.1%

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

      2. associate-*r* 64.1%

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

      3. *-commutative 64.1%

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

      4. associate-*l* 64.1%

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

      5. associate--r- 63.1%

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

      6. +-commutative 63.1%

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

   5. Applied egg-rr 63.1%

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

      1. hypot-def 44.9%

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

      2. unpow2 44.9%

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

      3. unpow2 44.9%

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

      4. +-commutative 44.9%

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

      5. unpow2 44.9%

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

      6. unpow2 44.9%

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

      7. hypot-def 63.1%

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

   7. Simplified 63.1%

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

   if -1.00000000000000005e-181 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 0.0

   1. Initial program 3.6%

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

   3. Simplified 8.8%

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

   4. Taylor expanded in C around inf 42.1%

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

      1. associate--l+ 42.2%

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

      2. unpow2 42.2%

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

      3. unpow2 42.2%

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

      4. unpow2 42.2%

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

      5. mul-1-neg 42.2%

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

      6. mul-1-neg 42.2%

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

      7. sqr-neg 42.2%

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

   6. Simplified 42.5%

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

   if 0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 46.6%

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

   3. Simplified 61.5%

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

      1. sqrt-prod 69.3%

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

      2. associate--r- 69.3%

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

      3. associate-*r* 69.3%

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

   5. Applied egg-rr 69.3%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in C around 0 1.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 1.7%

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

      2. *-commutative 1.7%

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

      3. +-commutative 1.7%

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

      4. unpow2 1.7%

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

      5. unpow2 1.7%

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

      6. hypot-def 15.3%

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

   6. Simplified 15.3%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(A - C, B\right) - C\right)\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right) \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, B \cdot \left(2 \cdot B\right)\right)} \cdot \left(-\sqrt{F}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3: 37.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{+110}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -1.12 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot t_0\right)\right)} \cdot \frac{-1}{t_1}\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{-88}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(\mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right) \cdot t_0\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- A (- (hypot B (- A C)) C)))
        (t_1 (+ (* B B) (* -4.0 (* A C))))
        (t_2 (fma B B (* C (* A -4.0)))))
   (if (<= B -3.5e+110)
     (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
     (if (<= B -4.5e+36)
       (/
        (-
         (sqrt
          (*
           2.0
           (*
            t_1
            (*
             F
             (- A (- (* -0.5 (/ (- (- (* A A) (* A A)) (* B B)) C)) A)))))))
        t_1)
       (if (<= B -1.12e-32)
         (* (sqrt (* 2.0 (* t_1 (* F t_0)))) (/ -1.0 t_1))
         (if (<= B 4.3e-88)
           (/ (- (sqrt (* 2.0 (* t_2 (* F (* 2.0 A)))))) t_2)
           (if (<= B 1.744e+61)
             (/
              (- (sqrt (* F (* (fma C (* A -8.0) (* 2.0 (* B B))) t_0))))
              (fma B B (* A (* C -4.0))))
             (if (<= B 1.55e+122)
               (* (/ (sqrt 2.0) B) (- (sqrt (* F (* -0.5 (/ (* B B) C))))))
               (* (sqrt (* F (- A (hypot A B)))) (/ (- (sqrt 2.0)) B))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = A - (hypot(B, (A - C)) - C);
	double t_1 = (B * B) + (-4.0 * (A * C));
	double t_2 = fma(B, B, (C * (A * -4.0)));
	double tmp;
	if (B <= -3.5e+110) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= -4.5e+36) {
		tmp = -sqrt((2.0 * (t_1 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_1;
	} else if (B <= -1.12e-32) {
		tmp = sqrt((2.0 * (t_1 * (F * t_0)))) * (-1.0 / t_1);
	} else if (B <= 4.3e-88) {
		tmp = -sqrt((2.0 * (t_2 * (F * (2.0 * A))))) / t_2;
	} else if (B <= 1.744e+61) {
		tmp = -sqrt((F * (fma(C, (A * -8.0), (2.0 * (B * B))) * t_0))) / fma(B, B, (A * (C * -4.0)));
	} else if (B <= 1.55e+122) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (-0.5 * ((B * B) / C))));
	} else {
		tmp = sqrt((F * (A - hypot(A, B)))) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(A - Float64(hypot(B, Float64(A - C)) - C))
	t_1 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_2 = fma(B, B, Float64(C * Float64(A * -4.0)))
	tmp = 0.0
	if (B <= -3.5e+110)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= -4.5e+36)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(F * Float64(A - Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(A * A) - Float64(A * A)) - Float64(B * B)) / C)) - A))))))) / t_1);
	elseif (B <= -1.12e-32)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_1 * Float64(F * t_0)))) * Float64(-1.0 / t_1));
	elseif (B <= 4.3e-88)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_2 * Float64(F * Float64(2.0 * A)))))) / t_2);
	elseif (B <= 1.744e+61)
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(fma(C, Float64(A * -8.0), Float64(2.0 * Float64(B * B))) * t_0)))) / fma(B, B, Float64(A * Float64(C * -4.0))));
	elseif (B <= 1.55e+122)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B * B) / C))))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(A, B)))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A - N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -3.5e+110], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -4.5e+36], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(F * N[(A - N[(N[(-0.5 * N[(N[(N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, -1.12e-32], N[(N[Sqrt[N[(2.0 * N[(t$95$1 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 4.3e-88], N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 1.744e+61], N[((-N[Sqrt[N[(F * N[(N[(C * N[(A * -8.0), $MachinePrecision] + N[(2.0 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.55e+122], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -3.4999999999999999e110

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -3.4999999999999999e110 < B < -4.49999999999999997e36

   1. Initial program 16.3%

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

   3. Simplified 16.3%

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

      1. distribute-frac-neg 16.3%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in C around inf 15.7%

      \[\leadsto -\frac{\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 15.9%

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

      2. unpow2 15.9%

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

      3. unpow2 15.9%

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

      4. unpow2 15.9%

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

      5. mul-1-neg 15.9%

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

      6. mul-1-neg 15.9%

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

      7. sqr-neg 15.9%

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

   8. Simplified 15.9%

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

   if -4.49999999999999997e36 < B < -1.12e-32

   1. Initial program 43.5%

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

   3. Simplified 43.5%

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

      1. div-inv 43.6%

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

   5. Applied egg-rr 50.7%

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

   if -1.12e-32 < B < 4.2999999999999997e-88

   1. Initial program 21.5%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in A around -inf 25.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 25.9%

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

   6. Simplified 25.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 25.9%

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

   8. Applied egg-rr 25.1%

      \[\leadsto \color{blue}{1 \cdot \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 25.1%

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

      2. distribute-frac-neg 25.1%

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

   10. Simplified 25.1%

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

   if 4.2999999999999997e-88 < B < 1.7439999999999999e61

   1. Initial program 44.6%

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

   3. Simplified 50.1%

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

   if 1.7439999999999999e61 < B < 1.54999999999999999e122

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

      1. div-inv 15.2%

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

   5. Applied egg-rr 19.7%

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

   6. Taylor expanded in A around 0 17.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 17.4%

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

      2. unpow2 17.4%

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

      3. unpow2 17.4%

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

      4. hypot-def 17.7%

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

   8. Simplified 17.7%

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

   9. Taylor expanded in C around inf 23.2%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot F} \]
   10. Step-by-step derivation

       1. unpow2 23.2%

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

   11. Simplified 23.2%

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

   if 1.54999999999999999e122 < B

   1. Initial program 5.8%

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

   3. Simplified 5.8%

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

   4. Taylor expanded in C around 0 7.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 7.9%

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

      2. *-commutative 7.9%

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

      3. +-commutative 7.9%

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

      4. unpow2 7.9%

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

      5. unpow2 7.9%

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

      6. hypot-def 45.0%

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

   6. Simplified 45.0%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.5 \cdot 10^{+110}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -1.12 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{-88}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(\mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right) \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4: 37.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.65 \cdot 10^{+109}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -4.9 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-36}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot t_0\right)\right)} \cdot \frac{-1}{t_1}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_2\right)\right) \cdot t_0}}{t_2}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- A (- (hypot B (- A C)) C)))
        (t_1 (+ (* B B) (* -4.0 (* A C))))
        (t_2 (fma B B (* C (* A -4.0)))))
   (if (<= B -1.65e+109)
     (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
     (if (<= B -4.9e+36)
       (/
        (-
         (sqrt
          (*
           2.0
           (*
            t_1
            (*
             F
             (- A (- (* -0.5 (/ (- (- (* A A) (* A A)) (* B B)) C)) A)))))))
        t_1)
       (if (<= B -2.3e-36)
         (* (sqrt (* 2.0 (* t_1 (* F t_0)))) (/ -1.0 t_1))
         (if (<= B 2.5e-90)
           (/ (- (sqrt (* 2.0 (* t_2 (* F (* 2.0 A)))))) t_2)
           (if (<= B 1.744e+61)
             (- (/ (sqrt (* (* 2.0 (* F t_2)) t_0)) t_2))
             (if (<= B 1.3e+122)
               (* (/ (sqrt 2.0) B) (- (sqrt (* F (* -0.5 (/ (* B B) C))))))
               (* (sqrt (* F (- A (hypot A B)))) (/ (- (sqrt 2.0)) B))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = A - (hypot(B, (A - C)) - C);
	double t_1 = (B * B) + (-4.0 * (A * C));
	double t_2 = fma(B, B, (C * (A * -4.0)));
	double tmp;
	if (B <= -1.65e+109) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= -4.9e+36) {
		tmp = -sqrt((2.0 * (t_1 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_1;
	} else if (B <= -2.3e-36) {
		tmp = sqrt((2.0 * (t_1 * (F * t_0)))) * (-1.0 / t_1);
	} else if (B <= 2.5e-90) {
		tmp = -sqrt((2.0 * (t_2 * (F * (2.0 * A))))) / t_2;
	} else if (B <= 1.744e+61) {
		tmp = -(sqrt(((2.0 * (F * t_2)) * t_0)) / t_2);
	} else if (B <= 1.3e+122) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (-0.5 * ((B * B) / C))));
	} else {
		tmp = sqrt((F * (A - hypot(A, B)))) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(A - Float64(hypot(B, Float64(A - C)) - C))
	t_1 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_2 = fma(B, B, Float64(C * Float64(A * -4.0)))
	tmp = 0.0
	if (B <= -1.65e+109)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= -4.9e+36)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(F * Float64(A - Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(A * A) - Float64(A * A)) - Float64(B * B)) / C)) - A))))))) / t_1);
	elseif (B <= -2.3e-36)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_1 * Float64(F * t_0)))) * Float64(-1.0 / t_1));
	elseif (B <= 2.5e-90)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_2 * Float64(F * Float64(2.0 * A)))))) / t_2);
	elseif (B <= 1.744e+61)
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_2)) * t_0)) / t_2));
	elseif (B <= 1.3e+122)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B * B) / C))))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(A, B)))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A - N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.65e+109], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -4.9e+36], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(F * N[(A - N[(N[(-0.5 * N[(N[(N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, -2.3e-36], N[(N[Sqrt[N[(2.0 * N[(t$95$1 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 2.5e-90], N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 1.744e+61], (-N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), If[LessEqual[B, 1.3e+122], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -1.6499999999999999e109

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -1.6499999999999999e109 < B < -4.89999999999999981e36

   1. Initial program 16.3%

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

   3. Simplified 16.3%

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

      1. distribute-frac-neg 16.3%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in C around inf 15.7%

      \[\leadsto -\frac{\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 15.9%

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

      2. unpow2 15.9%

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

      3. unpow2 15.9%

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

      4. unpow2 15.9%

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

      5. mul-1-neg 15.9%

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

      6. mul-1-neg 15.9%

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

      7. sqr-neg 15.9%

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

   8. Simplified 15.9%

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

   if -4.89999999999999981e36 < B < -2.29999999999999996e-36

   1. Initial program 43.5%

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

   3. Simplified 43.5%

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

      1. div-inv 43.6%

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

   5. Applied egg-rr 50.7%

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

   if -2.29999999999999996e-36 < B < 2.5000000000000001e-90

   1. Initial program 21.5%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in A around -inf 25.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 25.9%

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

   6. Simplified 25.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 25.9%

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

   8. Applied egg-rr 25.1%

      \[\leadsto \color{blue}{1 \cdot \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 25.1%

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

      2. distribute-frac-neg 25.1%

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

   10. Simplified 25.1%

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

   if 2.5000000000000001e-90 < B < 1.7439999999999999e61

   1. Initial program 44.6%

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

   3. Simplified 50.1%

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

   if 1.7439999999999999e61 < B < 1.30000000000000004e122

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

      1. div-inv 15.2%

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

   5. Applied egg-rr 19.7%

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

   6. Taylor expanded in A around 0 17.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 17.4%

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

      2. unpow2 17.4%

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

      3. unpow2 17.4%

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

      4. hypot-def 17.7%

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

   8. Simplified 17.7%

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

   9. Taylor expanded in C around inf 23.2%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot F} \]
   10. Step-by-step derivation

       1. unpow2 23.2%

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

   11. Simplified 23.2%

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

   if 1.30000000000000004e122 < B

   1. Initial program 5.8%

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

   3. Simplified 5.8%

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

   4. Taylor expanded in C around 0 7.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 7.9%

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

      2. *-commutative 7.9%

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

      3. +-commutative 7.9%

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

      4. unpow2 7.9%

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

      5. unpow2 7.9%

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

      6. hypot-def 45.0%

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

   6. Simplified 45.0%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.65 \cdot 10^{+109}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -4.9 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-36}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right) \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5: 38.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := -\sqrt{2}\\ t_2 := t_0 \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\\ t_3 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -7.2 \cdot 10^{+108}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -9.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{2 \cdot t_2} \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt{t_2} \cdot t_1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{t_1}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C))))
        (t_1 (- (sqrt 2.0)))
        (t_2 (* t_0 (* F (- A (- (hypot B (- A C)) C)))))
        (t_3 (fma B B (* C (* A -4.0)))))
   (if (<= B -7.2e+108)
     (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
     (if (<= B -9.8e+35)
       (/
        (-
         (sqrt
          (*
           2.0
           (*
            t_0
            (*
             F
             (- A (- (* -0.5 (/ (- (- (* A A) (* A A)) (* B B)) C)) A)))))))
        t_0)
       (if (<= B -1.15e-38)
         (* (sqrt (* 2.0 t_2)) (/ -1.0 t_0))
         (if (<= B 6.5e-29)
           (/ (- (sqrt (* 2.0 (* t_3 (* F (* 2.0 A)))))) t_3)
           (if (<= B 1.744e+61)
             (/ (* (sqrt t_2) t_1) (- (* B B) (* 4.0 (* A C))))
             (if (<= B 6.6e+121)
               (* (/ (sqrt 2.0) B) (- (sqrt (* F (* -0.5 (/ (* B B) C))))))
               (* (sqrt (* F (- A (hypot A B)))) (/ t_1 B))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = -sqrt(2.0);
	double t_2 = t_0 * (F * (A - (hypot(B, (A - C)) - C)));
	double t_3 = fma(B, B, (C * (A * -4.0)));
	double tmp;
	if (B <= -7.2e+108) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= -9.8e+35) {
		tmp = -sqrt((2.0 * (t_0 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_0;
	} else if (B <= -1.15e-38) {
		tmp = sqrt((2.0 * t_2)) * (-1.0 / t_0);
	} else if (B <= 6.5e-29) {
		tmp = -sqrt((2.0 * (t_3 * (F * (2.0 * A))))) / t_3;
	} else if (B <= 1.744e+61) {
		tmp = (sqrt(t_2) * t_1) / ((B * B) - (4.0 * (A * C)));
	} else if (B <= 6.6e+121) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (-0.5 * ((B * B) / C))));
	} else {
		tmp = sqrt((F * (A - hypot(A, B)))) * (t_1 / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(-sqrt(2.0))
	t_2 = Float64(t_0 * Float64(F * Float64(A - Float64(hypot(B, Float64(A - C)) - C))))
	t_3 = fma(B, B, Float64(C * Float64(A * -4.0)))
	tmp = 0.0
	if (B <= -7.2e+108)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= -9.8e+35)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A - Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(A * A) - Float64(A * A)) - Float64(B * B)) / C)) - A))))))) / t_0);
	elseif (B <= -1.15e-38)
		tmp = Float64(sqrt(Float64(2.0 * t_2)) * Float64(-1.0 / t_0));
	elseif (B <= 6.5e-29)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_3 * Float64(F * Float64(2.0 * A)))))) / t_3);
	elseif (B <= 1.744e+61)
		tmp = Float64(Float64(sqrt(t_2) * t_1) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	elseif (B <= 6.6e+121)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B * B) / C))))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(A, B)))) * Float64(t_1 / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$2 = N[(t$95$0 * N[(F * N[(A - N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -7.2e+108], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -9.8e+35], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A - N[(N[(-0.5 * N[(N[(N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, -1.15e-38], N[(N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 6.5e-29], N[((-N[Sqrt[N[(2.0 * N[(t$95$3 * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[B, 1.744e+61], N[(N[(N[Sqrt[t$95$2], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 6.6e+121], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -7.2e108

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -7.2e108 < B < -9.8000000000000005e35

   1. Initial program 16.3%

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

   3. Simplified 16.3%

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

      1. distribute-frac-neg 16.3%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in C around inf 15.7%

      \[\leadsto -\frac{\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 15.9%

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

      2. unpow2 15.9%

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

      3. unpow2 15.9%

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

      4. unpow2 15.9%

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

      5. mul-1-neg 15.9%

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

      6. mul-1-neg 15.9%

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

      7. sqr-neg 15.9%

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

   8. Simplified 15.9%

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

   if -9.8000000000000005e35 < B < -1.15000000000000001e-38

   1. Initial program 43.5%

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

   3. Simplified 43.5%

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

      1. div-inv 43.6%

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

   5. Applied egg-rr 50.7%

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

   if -1.15000000000000001e-38 < B < 6.5e-29

   1. Initial program 22.3%

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

   3. Simplified 22.1%

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

   4. Taylor expanded in A around -inf 26.5%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 26.5%

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

   6. Simplified 26.5%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 26.5%

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

   8. Applied egg-rr 25.7%

      \[\leadsto \color{blue}{1 \cdot \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 25.7%

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

      2. distribute-frac-neg 25.7%

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

   10. Simplified 25.7%

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

   if 6.5e-29 < B < 1.7439999999999999e61

   1. Initial program 47.8%

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

   3. Simplified 47.8%

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

      1. sqrt-prod 47.7%

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

      2. associate-*l* 47.8%

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

      3. cancel-sign-sub-inv 47.8%

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

      4. metadata-eval 47.8%

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

      5. associate--l+ 48.7%

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

      6. unpow2 48.7%

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

      7. hypot-udef 55.4%

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

   5. Applied egg-rr 55.4%

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

   if 1.7439999999999999e61 < B < 6.59999999999999958e121

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

      1. div-inv 15.2%

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

   5. Applied egg-rr 19.7%

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

   6. Taylor expanded in A around 0 17.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 17.4%

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

      2. unpow2 17.4%

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

      3. unpow2 17.4%

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

      4. hypot-def 17.7%

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

   8. Simplified 17.7%

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

   9. Taylor expanded in C around inf 23.2%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot F} \]
   10. Step-by-step derivation

       1. unpow2 23.2%

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

   11. Simplified 23.2%

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

   if 6.59999999999999958e121 < B

   1. Initial program 5.8%

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

   3. Simplified 5.8%

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

   4. Taylor expanded in C around 0 7.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 7.9%

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

      2. *-commutative 7.9%

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

      3. +-commutative 7.9%

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

      4. unpow2 7.9%

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

      5. unpow2 7.9%

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

      6. hypot-def 45.0%

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

   6. Simplified 45.0%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -7.2 \cdot 10^{+108}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -9.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)} \cdot \left(-\sqrt{2}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 6: 37.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)}\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -6.6 \cdot 10^{+109}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.05 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-36}:\\ \;\;\;\;t_1 \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{-t_1}{t_0}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C))))
        (t_1 (sqrt (* 2.0 (* t_0 (* F (- A (- (hypot B (- A C)) C)))))))
        (t_2 (fma B B (* C (* A -4.0)))))
   (if (<= B -6.6e+109)
     (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
     (if (<= B -2.05e+37)
       (/
        (-
         (sqrt
          (*
           2.0
           (*
            t_0
            (*
             F
             (- A (- (* -0.5 (/ (- (- (* A A) (* A A)) (* B B)) C)) A)))))))
        t_0)
       (if (<= B -1.1e-36)
         (* t_1 (/ -1.0 t_0))
         (if (<= B 3.8e-92)
           (/ (- (sqrt (* 2.0 (* t_2 (* F (* 2.0 A)))))) t_2)
           (if (<= B 1.744e+61)
             (/ (- t_1) t_0)
             (if (<= B 6.8e+121)
               (* (/ (sqrt 2.0) B) (- (sqrt (* F (* -0.5 (/ (* B B) C))))))
               (* (sqrt (* F (- A (hypot A B)))) (/ (- (sqrt 2.0)) B))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = sqrt((2.0 * (t_0 * (F * (A - (hypot(B, (A - C)) - C))))));
	double t_2 = fma(B, B, (C * (A * -4.0)));
	double tmp;
	if (B <= -6.6e+109) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= -2.05e+37) {
		tmp = -sqrt((2.0 * (t_0 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_0;
	} else if (B <= -1.1e-36) {
		tmp = t_1 * (-1.0 / t_0);
	} else if (B <= 3.8e-92) {
		tmp = -sqrt((2.0 * (t_2 * (F * (2.0 * A))))) / t_2;
	} else if (B <= 1.744e+61) {
		tmp = -t_1 / t_0;
	} else if (B <= 6.8e+121) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (-0.5 * ((B * B) / C))));
	} else {
		tmp = sqrt((F * (A - hypot(A, B)))) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_1 = sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A - Float64(hypot(B, Float64(A - C)) - C))))))
	t_2 = fma(B, B, Float64(C * Float64(A * -4.0)))
	tmp = 0.0
	if (B <= -6.6e+109)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= -2.05e+37)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A - Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(A * A) - Float64(A * A)) - Float64(B * B)) / C)) - A))))))) / t_0);
	elseif (B <= -1.1e-36)
		tmp = Float64(t_1 * Float64(-1.0 / t_0));
	elseif (B <= 3.8e-92)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_2 * Float64(F * Float64(2.0 * A)))))) / t_2);
	elseif (B <= 1.744e+61)
		tmp = Float64(Float64(-t_1) / t_0);
	elseif (B <= 6.8e+121)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B * B) / C))))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(A, B)))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A - N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -6.6e+109], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -2.05e+37], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A - N[(N[(-0.5 * N[(N[(N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, -1.1e-36], N[(t$95$1 * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 3.8e-92], N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 1.744e+61], N[((-t$95$1) / t$95$0), $MachinePrecision], If[LessEqual[B, 6.8e+121], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -6.5999999999999998e109

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -6.5999999999999998e109 < B < -2.0499999999999999e37

   1. Initial program 16.3%

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

   3. Simplified 16.3%

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

      1. distribute-frac-neg 16.3%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in C around inf 15.7%

      \[\leadsto -\frac{\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 15.9%

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

      2. unpow2 15.9%

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

      3. unpow2 15.9%

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

      4. unpow2 15.9%

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

      5. mul-1-neg 15.9%

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

      6. mul-1-neg 15.9%

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

      7. sqr-neg 15.9%

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

   8. Simplified 15.9%

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

   if -2.0499999999999999e37 < B < -1.1e-36

   1. Initial program 43.5%

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

   3. Simplified 43.5%

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

      1. div-inv 43.6%

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

   5. Applied egg-rr 50.7%

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

   if -1.1e-36 < B < 3.8000000000000001e-92

   1. Initial program 21.5%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in A around -inf 25.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 25.9%

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

   6. Simplified 25.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 25.9%

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

   8. Applied egg-rr 25.1%

      \[\leadsto \color{blue}{1 \cdot \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 25.1%

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

      2. distribute-frac-neg 25.1%

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

   10. Simplified 25.1%

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

   if 3.8000000000000001e-92 < B < 1.7439999999999999e61

   1. Initial program 44.6%

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

   3. Simplified 44.6%

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

      1. distribute-frac-neg 44.6%

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

   5. Applied egg-rr 50.3%

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

   if 1.7439999999999999e61 < B < 6.80000000000000021e121

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

      1. div-inv 15.2%

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

   5. Applied egg-rr 19.7%

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

   6. Taylor expanded in A around 0 17.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 17.4%

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

      2. unpow2 17.4%

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

      3. unpow2 17.4%

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

      4. hypot-def 17.7%

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

   8. Simplified 17.7%

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

   9. Taylor expanded in C around inf 23.2%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot F} \]
   10. Step-by-step derivation

       1. unpow2 23.2%

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

   11. Simplified 23.2%

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

   if 6.80000000000000021e121 < B

   1. Initial program 5.8%

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

   3. Simplified 5.8%

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

   4. Taylor expanded in C around 0 7.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 7.9%

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

      2. *-commutative 7.9%

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

      3. +-commutative 7.9%

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

      4. unpow2 7.9%

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

      5. unpow2 7.9%

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

      6. hypot-def 45.0%

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

   6. Simplified 45.0%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -6.6 \cdot 10^{+109}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.05 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-36}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 7: 37.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)}\\ \mathbf{if}\;B \leq -2.65 \cdot 10^{+110}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -8.5 \cdot 10^{-36}:\\ \;\;\;\;t_1 \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{-88}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{-t_1}{t_0}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C))))
        (t_1 (sqrt (* 2.0 (* t_0 (* F (- A (- (hypot B (- A C)) C))))))))
   (if (<= B -2.65e+110)
     (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
     (if (<= B -1.3e+37)
       (/
        (-
         (sqrt
          (*
           2.0
           (*
            t_0
            (*
             F
             (- A (- (* -0.5 (/ (- (- (* A A) (* A A)) (* B B)) C)) A)))))))
        t_0)
       (if (<= B -8.5e-36)
         (* t_1 (/ -1.0 t_0))
         (if (<= B 4.3e-88)
           (/
            (- (sqrt (* 2.0 (* (fma B B (* C (* A -4.0))) (* F (* 2.0 A))))))
            (- (* B B) (* 4.0 (* A C))))
           (if (<= B 1.744e+61)
             (/ (- t_1) t_0)
             (if (<= B 7e+121)
               (* (/ (sqrt 2.0) B) (- (sqrt (* F (* -0.5 (/ (* B B) C))))))
               (* (sqrt (* F (- A (hypot A B)))) (/ (- (sqrt 2.0)) B))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = sqrt((2.0 * (t_0 * (F * (A - (hypot(B, (A - C)) - C))))));
	double tmp;
	if (B <= -2.65e+110) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= -1.3e+37) {
		tmp = -sqrt((2.0 * (t_0 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_0;
	} else if (B <= -8.5e-36) {
		tmp = t_1 * (-1.0 / t_0);
	} else if (B <= 4.3e-88) {
		tmp = -sqrt((2.0 * (fma(B, B, (C * (A * -4.0))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	} else if (B <= 1.744e+61) {
		tmp = -t_1 / t_0;
	} else if (B <= 7e+121) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (-0.5 * ((B * B) / C))));
	} else {
		tmp = sqrt((F * (A - hypot(A, B)))) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_1 = sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A - Float64(hypot(B, Float64(A - C)) - C))))))
	tmp = 0.0
	if (B <= -2.65e+110)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= -1.3e+37)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A - Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(A * A) - Float64(A * A)) - Float64(B * B)) / C)) - A))))))) / t_0);
	elseif (B <= -8.5e-36)
		tmp = Float64(t_1 * Float64(-1.0 / t_0));
	elseif (B <= 4.3e-88)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(fma(B, B, Float64(C * Float64(A * -4.0))) * Float64(F * Float64(2.0 * A)))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	elseif (B <= 1.744e+61)
		tmp = Float64(Float64(-t_1) / t_0);
	elseif (B <= 7e+121)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B * B) / C))))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(A, B)))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A - N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, -2.65e+110], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.3e+37], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A - N[(N[(-0.5 * N[(N[(N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, -8.5e-36], N[(t$95$1 * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 4.3e-88], N[((-N[Sqrt[N[(2.0 * N[(N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.744e+61], N[((-t$95$1) / t$95$0), $MachinePrecision], If[LessEqual[B, 7e+121], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -2.6499999999999999e110

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -2.6499999999999999e110 < B < -1.3e37

   1. Initial program 16.3%

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

   3. Simplified 16.3%

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

      1. distribute-frac-neg 16.3%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in C around inf 15.7%

      \[\leadsto -\frac{\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 15.9%

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

      2. unpow2 15.9%

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

      3. unpow2 15.9%

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

      4. unpow2 15.9%

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

      5. mul-1-neg 15.9%

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

      6. mul-1-neg 15.9%

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

      7. sqr-neg 15.9%

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

   8. Simplified 15.9%

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

   if -1.3e37 < B < -8.5000000000000007e-36

   1. Initial program 43.5%

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

   3. Simplified 43.5%

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

      1. div-inv 43.6%

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

   5. Applied egg-rr 50.7%

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

   if -8.5000000000000007e-36 < B < 4.2999999999999997e-88

   1. Initial program 21.5%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in A around -inf 25.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 25.9%

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

   6. Simplified 25.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 25.9%

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

      2. associate-*l* 25.1%

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

      3. cancel-sign-sub-inv 25.1%

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

      4. metadata-eval 25.1%

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

      5. *-commutative 25.1%

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

      6. *-commutative 25.1%

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

      7. associate-*r* 25.1%

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

      8. fma-udef 25.1%

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

      9. *-commutative 25.1%

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

   8. Applied egg-rr 25.1%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 25.1%

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

   10. Simplified 25.1%

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

   if 4.2999999999999997e-88 < B < 1.7439999999999999e61

   1. Initial program 44.6%

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

   3. Simplified 44.6%

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

      1. distribute-frac-neg 44.6%

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

   5. Applied egg-rr 50.3%

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

   if 1.7439999999999999e61 < B < 6.9999999999999999e121

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

      1. div-inv 15.2%

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

   5. Applied egg-rr 19.7%

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

   6. Taylor expanded in A around 0 17.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 17.4%

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

      2. unpow2 17.4%

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

      3. unpow2 17.4%

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

      4. hypot-def 17.7%

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

   8. Simplified 17.7%

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

   9. Taylor expanded in C around inf 23.2%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot F} \]
   10. Step-by-step derivation

       1. unpow2 23.2%

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

   11. Simplified 23.2%

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

   if 6.9999999999999999e121 < B

   1. Initial program 5.8%

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

   3. Simplified 5.8%

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

   4. Taylor expanded in C around 0 7.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 7.9%

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

      2. *-commutative 7.9%

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

      3. +-commutative 7.9%

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

      4. unpow2 7.9%

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

      5. unpow2 7.9%

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

      6. hypot-def 45.0%

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

   6. Simplified 45.0%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.65 \cdot 10^{+110}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -8.5 \cdot 10^{-36}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{-88}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 8: 37.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)}}{t_0}\\ t_2 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.15 \cdot 10^{+110}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.5 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.65 \cdot 10^{+122}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C))))
        (t_1
         (/
          (- (sqrt (* 2.0 (* t_0 (* F (- A (- (hypot B (- A C)) C)))))))
          t_0))
        (t_2 (/ (sqrt 2.0) B)))
   (if (<= B -1.15e+110)
     (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
     (if (<= B -4.5e+36)
       (/
        (-
         (sqrt
          (*
           2.0
           (*
            t_0
            (*
             F
             (- A (- (* -0.5 (/ (- (- (* A A) (* A A)) (* B B)) C)) A)))))))
        t_0)
       (if (<= B -1.5e-35)
         t_1
         (if (<= B 6.2e-89)
           (/
            (- (sqrt (* 2.0 (* (fma B B (* C (* A -4.0))) (* F (* 2.0 A))))))
            (- (* B B) (* 4.0 (* A C))))
           (if (<= B 1.744e+61)
             t_1
             (if (<= B 2.65e+122)
               (* t_2 (- (sqrt (* F (* -0.5 (/ (* B B) C))))))
               (* t_2 (- (sqrt (* B (- F)))))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = -sqrt((2.0 * (t_0 * (F * (A - (hypot(B, (A - C)) - C)))))) / t_0;
	double t_2 = sqrt(2.0) / B;
	double tmp;
	if (B <= -1.15e+110) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= -4.5e+36) {
		tmp = -sqrt((2.0 * (t_0 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_0;
	} else if (B <= -1.5e-35) {
		tmp = t_1;
	} else if (B <= 6.2e-89) {
		tmp = -sqrt((2.0 * (fma(B, B, (C * (A * -4.0))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	} else if (B <= 1.744e+61) {
		tmp = t_1;
	} else if (B <= 2.65e+122) {
		tmp = t_2 * -sqrt((F * (-0.5 * ((B * B) / C))));
	} else {
		tmp = t_2 * -sqrt((B * -F));
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A - Float64(hypot(B, Float64(A - C)) - C))))))) / t_0)
	t_2 = Float64(sqrt(2.0) / B)
	tmp = 0.0
	if (B <= -1.15e+110)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= -4.5e+36)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A - Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(A * A) - Float64(A * A)) - Float64(B * B)) / C)) - A))))))) / t_0);
	elseif (B <= -1.5e-35)
		tmp = t_1;
	elseif (B <= 6.2e-89)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(fma(B, B, Float64(C * Float64(A * -4.0))) * Float64(F * Float64(2.0 * A)))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	elseif (B <= 1.744e+61)
		tmp = t_1;
	elseif (B <= 2.65e+122)
		tmp = Float64(t_2 * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B * B) / C))))));
	else
		tmp = Float64(t_2 * Float64(-sqrt(Float64(B * Float64(-F)))));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A - N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[B, -1.15e+110], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -4.5e+36], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A - N[(N[(-0.5 * N[(N[(N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, -1.5e-35], t$95$1, If[LessEqual[B, 6.2e-89], N[((-N[Sqrt[N[(2.0 * N[(N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.744e+61], t$95$1, If[LessEqual[B, 2.65e+122], N[(t$95$2 * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(t$95$2 * (-N[Sqrt[N[(B * (-F)), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if B < -1.15e110

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -1.15e110 < B < -4.49999999999999997e36

   1. Initial program 16.3%

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

   3. Simplified 16.3%

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

      1. distribute-frac-neg 16.3%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in C around inf 15.7%

      \[\leadsto -\frac{\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 15.9%

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

      2. unpow2 15.9%

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

      3. unpow2 15.9%

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

      4. unpow2 15.9%

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

      5. mul-1-neg 15.9%

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

      6. mul-1-neg 15.9%

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

      7. sqr-neg 15.9%

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

   8. Simplified 15.9%

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

   if -4.49999999999999997e36 < B < -1.49999999999999994e-35 or 6.19999999999999993e-89 < B < 1.7439999999999999e61

   1. Initial program 44.2%

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

   3. Simplified 44.2%

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

      1. distribute-frac-neg 44.2%

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

   5. Applied egg-rr 50.4%

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

   if -1.49999999999999994e-35 < B < 6.19999999999999993e-89

   1. Initial program 21.5%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in A around -inf 25.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 25.9%

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

   6. Simplified 25.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 25.9%

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

      2. associate-*l* 25.1%

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

      3. cancel-sign-sub-inv 25.1%

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

      4. metadata-eval 25.1%

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

      5. *-commutative 25.1%

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

      6. *-commutative 25.1%

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

      7. associate-*r* 25.1%

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

      8. fma-udef 25.1%

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

      9. *-commutative 25.1%

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

   8. Applied egg-rr 25.1%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 25.1%

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

   10. Simplified 25.1%

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

   if 1.7439999999999999e61 < B < 2.65e122

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

      1. div-inv 15.2%

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

   5. Applied egg-rr 19.7%

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

   6. Taylor expanded in A around 0 17.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 17.4%

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

      2. unpow2 17.4%

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

      3. unpow2 17.4%

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

      4. hypot-def 17.7%

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

   8. Simplified 17.7%

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

   9. Taylor expanded in C around inf 23.2%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot F} \]
   10. Step-by-step derivation

       1. unpow2 23.2%

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

   11. Simplified 23.2%

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

   if 2.65e122 < B

   1. Initial program 5.8%

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

   3. Simplified 5.8%

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

      1. div-inv 5.8%

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

   5. Applied egg-rr 5.8%

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

   6. Taylor expanded in A around 0 7.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 7.7%

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

      2. unpow2 7.7%

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

      3. unpow2 7.7%

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

      4. hypot-def 50.1%

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

   8. Simplified 50.1%

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

   9. Taylor expanded in C around 0 41.6%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{-1 \cdot \left(F \cdot B\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 41.6%

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

       2. mul-1-neg 41.6%

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

   11. Simplified 41.6%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-F\right) \cdot B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.15 \cdot 10^{+110}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -1.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.65 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \]

Alternative 9: 37.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)}\\ t_2 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -3.9 \cdot 10^{+110}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -4.9 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -9.6 \cdot 10^{-39}:\\ \;\;\;\;t_1 \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{-88}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{-t_1}{t_0}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{+121}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C))))
        (t_1 (sqrt (* 2.0 (* t_0 (* F (- A (- (hypot B (- A C)) C)))))))
        (t_2 (/ (sqrt 2.0) B)))
   (if (<= B -3.9e+110)
     (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
     (if (<= B -4.9e+36)
       (/
        (-
         (sqrt
          (*
           2.0
           (*
            t_0
            (*
             F
             (- A (- (* -0.5 (/ (- (- (* A A) (* A A)) (* B B)) C)) A)))))))
        t_0)
       (if (<= B -9.6e-39)
         (* t_1 (/ -1.0 t_0))
         (if (<= B 4.3e-88)
           (/
            (- (sqrt (* 2.0 (* (fma B B (* C (* A -4.0))) (* F (* 2.0 A))))))
            (- (* B B) (* 4.0 (* A C))))
           (if (<= B 1.744e+61)
             (/ (- t_1) t_0)
             (if (<= B 6.8e+121)
               (* t_2 (- (sqrt (* F (* -0.5 (/ (* B B) C))))))
               (* t_2 (- (sqrt (* B (- F)))))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = sqrt((2.0 * (t_0 * (F * (A - (hypot(B, (A - C)) - C))))));
	double t_2 = sqrt(2.0) / B;
	double tmp;
	if (B <= -3.9e+110) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= -4.9e+36) {
		tmp = -sqrt((2.0 * (t_0 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_0;
	} else if (B <= -9.6e-39) {
		tmp = t_1 * (-1.0 / t_0);
	} else if (B <= 4.3e-88) {
		tmp = -sqrt((2.0 * (fma(B, B, (C * (A * -4.0))) * (F * (2.0 * A))))) / ((B * B) - (4.0 * (A * C)));
	} else if (B <= 1.744e+61) {
		tmp = -t_1 / t_0;
	} else if (B <= 6.8e+121) {
		tmp = t_2 * -sqrt((F * (-0.5 * ((B * B) / C))));
	} else {
		tmp = t_2 * -sqrt((B * -F));
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_1 = sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A - Float64(hypot(B, Float64(A - C)) - C))))))
	t_2 = Float64(sqrt(2.0) / B)
	tmp = 0.0
	if (B <= -3.9e+110)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= -4.9e+36)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A - Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(A * A) - Float64(A * A)) - Float64(B * B)) / C)) - A))))))) / t_0);
	elseif (B <= -9.6e-39)
		tmp = Float64(t_1 * Float64(-1.0 / t_0));
	elseif (B <= 4.3e-88)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(fma(B, B, Float64(C * Float64(A * -4.0))) * Float64(F * Float64(2.0 * A)))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	elseif (B <= 1.744e+61)
		tmp = Float64(Float64(-t_1) / t_0);
	elseif (B <= 6.8e+121)
		tmp = Float64(t_2 * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B * B) / C))))));
	else
		tmp = Float64(t_2 * Float64(-sqrt(Float64(B * Float64(-F)))));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A - N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[B, -3.9e+110], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -4.9e+36], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A - N[(N[(-0.5 * N[(N[(N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, -9.6e-39], N[(t$95$1 * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 4.3e-88], N[((-N[Sqrt[N[(2.0 * N[(N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.744e+61], N[((-t$95$1) / t$95$0), $MachinePrecision], If[LessEqual[B, 6.8e+121], N[(t$95$2 * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(t$95$2 * (-N[Sqrt[N[(B * (-F)), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -3.9000000000000003e110

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -3.9000000000000003e110 < B < -4.89999999999999981e36

   1. Initial program 16.3%

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

   3. Simplified 16.3%

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

      1. distribute-frac-neg 16.3%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in C around inf 15.7%

      \[\leadsto -\frac{\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 15.9%

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

      2. unpow2 15.9%

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

      3. unpow2 15.9%

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

      4. unpow2 15.9%

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

      5. mul-1-neg 15.9%

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

      6. mul-1-neg 15.9%

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

      7. sqr-neg 15.9%

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

   8. Simplified 15.9%

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

   if -4.89999999999999981e36 < B < -9.60000000000000063e-39

   1. Initial program 43.5%

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

   3. Simplified 43.5%

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

      1. div-inv 43.6%

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

   5. Applied egg-rr 50.7%

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

   if -9.60000000000000063e-39 < B < 4.2999999999999997e-88

   1. Initial program 21.5%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in A around -inf 25.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 25.9%

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

   6. Simplified 25.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 25.9%

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

      2. associate-*l* 25.1%

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

      3. cancel-sign-sub-inv 25.1%

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

      4. metadata-eval 25.1%

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

      5. *-commutative 25.1%

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

      6. *-commutative 25.1%

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

      7. associate-*r* 25.1%

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

      8. fma-udef 25.1%

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

      9. *-commutative 25.1%

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

   8. Applied egg-rr 25.1%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 25.1%

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

   10. Simplified 25.1%

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

   if 4.2999999999999997e-88 < B < 1.7439999999999999e61

   1. Initial program 44.6%

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

   3. Simplified 44.6%

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

      1. distribute-frac-neg 44.6%

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

   5. Applied egg-rr 50.3%

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

   if 1.7439999999999999e61 < B < 6.80000000000000021e121

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

      1. div-inv 15.2%

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

   5. Applied egg-rr 19.7%

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

   6. Taylor expanded in A around 0 17.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 17.4%

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

      2. unpow2 17.4%

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

      3. unpow2 17.4%

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

      4. hypot-def 17.7%

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

   8. Simplified 17.7%

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

   9. Taylor expanded in C around inf 23.2%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot F} \]
   10. Step-by-step derivation

       1. unpow2 23.2%

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

   11. Simplified 23.2%

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

   if 6.80000000000000021e121 < B

   1. Initial program 5.8%

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

   3. Simplified 5.8%

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

      1. div-inv 5.8%

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

   5. Applied egg-rr 5.8%

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

   6. Taylor expanded in A around 0 7.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 7.7%

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

      2. unpow2 7.7%

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

      3. unpow2 7.7%

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

      4. hypot-def 50.1%

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

   8. Simplified 50.1%

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

   9. Taylor expanded in C around 0 41.6%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{-1 \cdot \left(F \cdot B\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 41.6%

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

       2. mul-1-neg 41.6%

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

   11. Simplified 41.6%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-F\right) \cdot B}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.9 \cdot 10^{+110}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -4.9 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -9.6 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{-88}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \]

Alternative 10: 35.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := t_0 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ t_2 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_3 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.5 \cdot 10^{+109}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -6.6 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -9.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(A + \left(B + C\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-32}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B))
        (t_1 (* t_0 (- (sqrt (* F (* -0.5 (/ (* B B) C)))))))
        (t_2 (+ (* B B) (* -4.0 (* A C))))
        (t_3 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -1.5e+109)
     (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
     (if (<= B -6.6e+36)
       (/
        (-
         (sqrt
          (*
           2.0
           (*
            t_2
            (*
             F
             (- A (- (* -0.5 (/ (- (- (* A A) (* A A)) (* B B)) C)) A)))))))
        t_2)
       (if (<= B -9.2e-30)
         (/ (- (sqrt (* 2.0 (* t_2 (* F (+ A (+ B C))))))) t_2)
         (if (<= B -1.75e-32)
           t_1
           (if (<= B 1.15e-32)
             (/
              (- (sqrt (* 2.0 (* (fma B B (* C (* A -4.0))) (* F (* 2.0 A))))))
              t_3)
             (if (<= B 1.744e+61)
               (/ (- (sqrt (* 2.0 (* (- A (hypot A B)) (* F (* B B)))))) t_3)
               (if (<= B 2.4e+122) t_1 (* t_0 (- (sqrt (* B (- F))))))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt(2.0) / B;
	double t_1 = t_0 * -sqrt((F * (-0.5 * ((B * B) / C))));
	double t_2 = (B * B) + (-4.0 * (A * C));
	double t_3 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -1.5e+109) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= -6.6e+36) {
		tmp = -sqrt((2.0 * (t_2 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_2;
	} else if (B <= -9.2e-30) {
		tmp = -sqrt((2.0 * (t_2 * (F * (A + (B + C)))))) / t_2;
	} else if (B <= -1.75e-32) {
		tmp = t_1;
	} else if (B <= 1.15e-32) {
		tmp = -sqrt((2.0 * (fma(B, B, (C * (A * -4.0))) * (F * (2.0 * A))))) / t_3;
	} else if (B <= 1.744e+61) {
		tmp = -sqrt((2.0 * ((A - hypot(A, B)) * (F * (B * B))))) / t_3;
	} else if (B <= 2.4e+122) {
		tmp = t_1;
	} else {
		tmp = t_0 * -sqrt((B * -F));
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(sqrt(2.0) / B)
	t_1 = Float64(t_0 * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B * B) / C))))))
	t_2 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_3 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -1.5e+109)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= -6.6e+36)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_2 * Float64(F * Float64(A - Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(A * A) - Float64(A * A)) - Float64(B * B)) / C)) - A))))))) / t_2);
	elseif (B <= -9.2e-30)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_2 * Float64(F * Float64(A + Float64(B + C))))))) / t_2);
	elseif (B <= -1.75e-32)
		tmp = t_1;
	elseif (B <= 1.15e-32)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(fma(B, B, Float64(C * Float64(A * -4.0))) * Float64(F * Float64(2.0 * A)))))) / t_3);
	elseif (B <= 1.744e+61)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(A - hypot(A, B)) * Float64(F * Float64(B * B)))))) / t_3);
	elseif (B <= 2.4e+122)
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(-sqrt(Float64(B * Float64(-F)))));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.5e+109], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -6.6e+36], N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * N[(F * N[(A - N[(N[(-0.5 * N[(N[(N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, -9.2e-30], N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * N[(F * N[(A + N[(B + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, -1.75e-32], t$95$1, If[LessEqual[B, 1.15e-32], N[((-N[Sqrt[N[(2.0 * N[(N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[B, 1.744e+61], N[((-N[Sqrt[N[(2.0 * N[(N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision] * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[B, 2.4e+122], t$95$1, N[(t$95$0 * (-N[Sqrt[N[(B * (-F)), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -1.50000000000000008e109

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -1.50000000000000008e109 < B < -6.5999999999999997e36

   1. Initial program 16.3%

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

   3. Simplified 16.3%

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

      1. distribute-frac-neg 16.3%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in C around inf 15.7%

      \[\leadsto -\frac{\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 15.9%

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

      2. unpow2 15.9%

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

      3. unpow2 15.9%

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

      4. unpow2 15.9%

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

      5. mul-1-neg 15.9%

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

      6. mul-1-neg 15.9%

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

      7. sqr-neg 15.9%

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

   8. Simplified 15.9%

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

   if -6.5999999999999997e36 < B < -9.19999999999999937e-30

   1. Initial program 46.0%

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

   3. Simplified 46.0%

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

      1. distribute-frac-neg 46.0%

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

   5. Applied egg-rr 55.1%

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

   6. Taylor expanded in B around -inf 47.8%

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

   if -9.19999999999999937e-30 < B < -1.7499999999999999e-32 or 1.7439999999999999e61 < B < 2.4000000000000002e122

   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. Step-by-step derivation

   3. Simplified 17.4%

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

      1. div-inv 17.4%

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

   5. Applied egg-rr 21.3%

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

   6. Taylor expanded in A around 0 15.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 15.4%

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

      2. unpow2 15.4%

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

      3. unpow2 15.4%

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

      4. hypot-def 15.8%

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

   8. Simplified 15.8%

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

   9. Taylor expanded in C around inf 24.7%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot F} \]
   10. Step-by-step derivation

       1. unpow2 24.7%

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

   11. Simplified 24.7%

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

   if -1.7499999999999999e-32 < B < 1.15e-32

   1. Initial program 21.6%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in A around -inf 25.8%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 25.8%

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

   6. Simplified 25.8%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 25.8%

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

      2. associate-*l* 25.0%

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

      3. cancel-sign-sub-inv 25.0%

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

      4. metadata-eval 25.0%

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

      5. *-commutative 25.0%

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

      6. *-commutative 25.0%

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

      7. associate-*r* 25.0%

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

      8. fma-udef 25.0%

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

      9. *-commutative 25.0%

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

   8. Applied egg-rr 25.0%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 25.0%

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

   10. Simplified 25.0%

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

   if 1.15e-32 < B < 1.7439999999999999e61

   1. Initial program 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in C around 0 45.3%

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

      1. +-commutative 45.3%

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

      2. unpow2 45.3%

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

      3. unpow2 45.3%

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

      4. hypot-def 51.5%

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

   6. Simplified 51.5%

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

   7. Taylor expanded in B around inf 52.6%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\color{blue}{{B}^{2}} \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   8. Step-by-step derivation

      1. unpow2 52.6%

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

   9. Simplified 52.6%

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

   if 2.4000000000000002e122 < B

   1. Initial program 5.8%

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

   3. Simplified 5.8%

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

      1. div-inv 5.8%

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

   5. Applied egg-rr 5.8%

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

   6. Taylor expanded in A around 0 7.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 7.7%

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

      2. unpow2 7.7%

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

      3. unpow2 7.7%

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

      4. hypot-def 50.1%

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

   8. Simplified 50.1%

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

   9. Taylor expanded in C around 0 41.6%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{-1 \cdot \left(F \cdot B\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 41.6%

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

       2. mul-1-neg 41.6%

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

   11. Simplified 41.6%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-F\right) \cdot B}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.5 \cdot 10^{+109}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -6.6 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -9.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \left(B + C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-32}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \]

Alternative 11: 35.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{t_0}\\ t_2 := \frac{\sqrt{2}}{B}\\ t_3 := t_2 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ t_4 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -2.9 \cdot 10^{+110}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.3 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -3.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A + \left(B + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-32}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{-33}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_4\right) \cdot \left(2 \cdot A\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.85 \cdot 10^{+122}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C))))
        (t_1
         (/
          (-
           (sqrt
            (*
             2.0
             (*
              t_0
              (*
               F
               (- A (- (* -0.5 (/ (- (- (* A A) (* A A)) (* B B)) C)) A)))))))
          t_0))
        (t_2 (/ (sqrt 2.0) B))
        (t_3 (* t_2 (- (sqrt (* F (* -0.5 (/ (* B B) C)))))))
        (t_4 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -2.9e+110)
     (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
     (if (<= B -1.3e+37)
       t_1
       (if (<= B -3.8e-28)
         (/ (- (sqrt (* 2.0 (* t_0 (* F (+ A (+ B C))))))) t_0)
         (if (<= B -1.75e-32)
           t_3
           (if (<= B 3.9e-33)
             (- (/ (sqrt (* 2.0 (* (* F t_4) (* 2.0 A)))) t_4))
             (if (<= B 2.4e+29)
               (/ (- (sqrt (* 2.0 (* (- A (hypot A B)) (* F (* B B)))))) t_4)
               (if (<= B 6.8e+29)
                 t_1
                 (if (<= B 2.85e+122)
                   t_3
                   (* t_2 (- (sqrt (* B (- F)))))))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = -sqrt((2.0 * (t_0 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_0;
	double t_2 = sqrt(2.0) / B;
	double t_3 = t_2 * -sqrt((F * (-0.5 * ((B * B) / C))));
	double t_4 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -2.9e+110) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= -1.3e+37) {
		tmp = t_1;
	} else if (B <= -3.8e-28) {
		tmp = -sqrt((2.0 * (t_0 * (F * (A + (B + C)))))) / t_0;
	} else if (B <= -1.75e-32) {
		tmp = t_3;
	} else if (B <= 3.9e-33) {
		tmp = -(sqrt((2.0 * ((F * t_4) * (2.0 * A)))) / t_4);
	} else if (B <= 2.4e+29) {
		tmp = -sqrt((2.0 * ((A - hypot(A, B)) * (F * (B * B))))) / t_4;
	} else if (B <= 6.8e+29) {
		tmp = t_1;
	} else if (B <= 2.85e+122) {
		tmp = t_3;
	} else {
		tmp = t_2 * -sqrt((B * -F));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = -Math.sqrt((2.0 * (t_0 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_0;
	double t_2 = Math.sqrt(2.0) / B;
	double t_3 = t_2 * -Math.sqrt((F * (-0.5 * ((B * B) / C))));
	double t_4 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -2.9e+110) {
		tmp = 2.0 * (Math.sqrt((A * F)) * (1.0 / B));
	} else if (B <= -1.3e+37) {
		tmp = t_1;
	} else if (B <= -3.8e-28) {
		tmp = -Math.sqrt((2.0 * (t_0 * (F * (A + (B + C)))))) / t_0;
	} else if (B <= -1.75e-32) {
		tmp = t_3;
	} else if (B <= 3.9e-33) {
		tmp = -(Math.sqrt((2.0 * ((F * t_4) * (2.0 * A)))) / t_4);
	} else if (B <= 2.4e+29) {
		tmp = -Math.sqrt((2.0 * ((A - Math.hypot(A, B)) * (F * (B * B))))) / t_4;
	} else if (B <= 6.8e+29) {
		tmp = t_1;
	} else if (B <= 2.85e+122) {
		tmp = t_3;
	} else {
		tmp = t_2 * -Math.sqrt((B * -F));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + (-4.0 * (A * C))
	t_1 = -math.sqrt((2.0 * (t_0 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_0
	t_2 = math.sqrt(2.0) / B
	t_3 = t_2 * -math.sqrt((F * (-0.5 * ((B * B) / C))))
	t_4 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= -2.9e+110:
		tmp = 2.0 * (math.sqrt((A * F)) * (1.0 / B))
	elif B <= -1.3e+37:
		tmp = t_1
	elif B <= -3.8e-28:
		tmp = -math.sqrt((2.0 * (t_0 * (F * (A + (B + C)))))) / t_0
	elif B <= -1.75e-32:
		tmp = t_3
	elif B <= 3.9e-33:
		tmp = -(math.sqrt((2.0 * ((F * t_4) * (2.0 * A)))) / t_4)
	elif B <= 2.4e+29:
		tmp = -math.sqrt((2.0 * ((A - math.hypot(A, B)) * (F * (B * B))))) / t_4
	elif B <= 6.8e+29:
		tmp = t_1
	elif B <= 2.85e+122:
		tmp = t_3
	else:
		tmp = t_2 * -math.sqrt((B * -F))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A - Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(A * A) - Float64(A * A)) - Float64(B * B)) / C)) - A))))))) / t_0)
	t_2 = Float64(sqrt(2.0) / B)
	t_3 = Float64(t_2 * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B * B) / C))))))
	t_4 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -2.9e+110)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= -1.3e+37)
		tmp = t_1;
	elseif (B <= -3.8e-28)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A + Float64(B + C))))))) / t_0);
	elseif (B <= -1.75e-32)
		tmp = t_3;
	elseif (B <= 3.9e-33)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_4) * Float64(2.0 * A)))) / t_4));
	elseif (B <= 2.4e+29)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(A - hypot(A, B)) * Float64(F * Float64(B * B)))))) / t_4);
	elseif (B <= 6.8e+29)
		tmp = t_1;
	elseif (B <= 2.85e+122)
		tmp = t_3;
	else
		tmp = Float64(t_2 * Float64(-sqrt(Float64(B * Float64(-F)))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + (-4.0 * (A * C));
	t_1 = -sqrt((2.0 * (t_0 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_0;
	t_2 = sqrt(2.0) / B;
	t_3 = t_2 * -sqrt((F * (-0.5 * ((B * B) / C))));
	t_4 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= -2.9e+110)
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	elseif (B <= -1.3e+37)
		tmp = t_1;
	elseif (B <= -3.8e-28)
		tmp = -sqrt((2.0 * (t_0 * (F * (A + (B + C)))))) / t_0;
	elseif (B <= -1.75e-32)
		tmp = t_3;
	elseif (B <= 3.9e-33)
		tmp = -(sqrt((2.0 * ((F * t_4) * (2.0 * A)))) / t_4);
	elseif (B <= 2.4e+29)
		tmp = -sqrt((2.0 * ((A - hypot(A, B)) * (F * (B * B))))) / t_4;
	elseif (B <= 6.8e+29)
		tmp = t_1;
	elseif (B <= 2.85e+122)
		tmp = t_3;
	else
		tmp = t_2 * -sqrt((B * -F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A - N[(N[(-0.5 * N[(N[(N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$4 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -2.9e+110], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.3e+37], t$95$1, If[LessEqual[B, -3.8e-28], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A + N[(B + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, -1.75e-32], t$95$3, If[LessEqual[B, 3.9e-33], (-N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$4), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), If[LessEqual[B, 2.4e+29], N[((-N[Sqrt[N[(2.0 * N[(N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision] * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision], If[LessEqual[B, 6.8e+29], t$95$1, If[LessEqual[B, 2.85e+122], t$95$3, N[(t$95$2 * (-N[Sqrt[N[(B * (-F)), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -2.9e110

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -2.9e110 < B < -1.3e37 or 2.4000000000000001e29 < B < 6.79999999999999963e29

   1. Initial program 15.8%

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

   3. Simplified 15.8%

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

      1. distribute-frac-neg 15.8%

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

   5. Applied egg-rr 17.0%

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

   6. Taylor expanded in C around inf 19.1%

      \[\leadsto -\frac{\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 19.3%

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

      2. unpow2 19.3%

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

      3. unpow2 19.3%

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

      4. unpow2 19.3%

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

      5. mul-1-neg 19.3%

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

      6. mul-1-neg 19.3%

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

      7. sqr-neg 19.3%

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

   8. Simplified 19.3%

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

   if -1.3e37 < B < -3.80000000000000009e-28

   1. Initial program 46.0%

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

   3. Simplified 46.0%

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

      1. distribute-frac-neg 46.0%

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

   5. Applied egg-rr 55.1%

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

   6. Taylor expanded in B around -inf 47.8%

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

   if -3.80000000000000009e-28 < B < -1.7499999999999999e-32 or 6.79999999999999963e29 < B < 2.85000000000000003e122

   1. Initial program 23.0%

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

   3. Simplified 23.0%

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

      1. div-inv 22.9%

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

   5. Applied egg-rr 26.2%

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

   6. Taylor expanded in A around 0 21.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 21.6%

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

      2. unpow2 21.6%

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

      3. unpow2 21.6%

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

      4. hypot-def 21.9%

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

   8. Simplified 21.9%

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

   9. Taylor expanded in C around inf 25.5%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot F} \]
   10. Step-by-step derivation

       1. unpow2 25.5%

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

   11. Simplified 25.5%

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

   if -1.7499999999999999e-32 < B < 3.89999999999999974e-33

   1. Initial program 21.6%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in A around -inf 25.8%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 25.8%

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

   6. Simplified 25.8%

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

   if 3.89999999999999974e-33 < B < 2.4000000000000001e29

   1. Initial program 60.2%

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

   3. Simplified 60.2%

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

   4. Taylor expanded in C around 0 60.2%

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

      1. +-commutative 60.2%

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

      2. unpow2 60.2%

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

      3. unpow2 60.2%

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

      4. hypot-def 71.2%

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

   6. Simplified 71.2%

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

   7. Taylor expanded in B around inf 71.2%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\color{blue}{{B}^{2}} \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   8. Step-by-step derivation

      1. unpow2 71.2%

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

   9. Simplified 71.2%

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

   if 2.85000000000000003e122 < B

   1. Initial program 5.8%

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

   3. Simplified 5.8%

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

      1. div-inv 5.8%

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

   5. Applied egg-rr 5.8%

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

   6. Taylor expanded in A around 0 7.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 7.7%

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

      2. unpow2 7.7%

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

      3. unpow2 7.7%

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

      4. hypot-def 50.1%

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

   8. Simplified 50.1%

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

   9. Taylor expanded in C around 0 41.6%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{-1 \cdot \left(F \cdot B\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 41.6%

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

       2. mul-1-neg 41.6%

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

   11. Simplified 41.6%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-F\right) \cdot B}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.9 \cdot 10^{+110}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -3.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \left(B + C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{-33}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.85 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \]

Alternative 12: 36.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := 4 \cdot \left(A \cdot C\right)\\ t_2 := B \cdot B - t_1\\ t_3 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -8 \cdot 10^{+108}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.05 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq -6.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(A, B\right) - A\right) \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{+121}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B))
        (t_1 (* 4.0 (* A C)))
        (t_2 (- (* B B) t_1))
        (t_3 (+ (* B B) (* -4.0 (* A C)))))
   (if (<= B -8e+108)
     (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
     (if (<= B -2.05e+37)
       (/
        (-
         (sqrt
          (*
           2.0
           (*
            t_3
            (*
             F
             (- A (- (* -0.5 (/ (- (- (* A A) (* A A)) (* B B)) C)) A)))))))
        t_3)
       (if (<= B -6.6e-29)
         (/ (- (sqrt (* 2.0 (* (- (hypot A B) A) (* F (- t_1 (* B B))))))) t_2)
         (if (<= B 7.6e-35)
           (/
            (- (sqrt (* 2.0 (* (fma B B (* C (* A -4.0))) (* F (* 2.0 A))))))
            t_2)
           (if (<= B 1.744e+61)
             (/ (- (sqrt (* 2.0 (* (- A (hypot A B)) (* F (* B B)))))) t_2)
             (if (<= B 7.8e+121)
               (* t_0 (- (sqrt (* F (* -0.5 (/ (* B B) C))))))
               (* t_0 (- (sqrt (* B (- F)))))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt(2.0) / B;
	double t_1 = 4.0 * (A * C);
	double t_2 = (B * B) - t_1;
	double t_3 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (B <= -8e+108) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= -2.05e+37) {
		tmp = -sqrt((2.0 * (t_3 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_3;
	} else if (B <= -6.6e-29) {
		tmp = -sqrt((2.0 * ((hypot(A, B) - A) * (F * (t_1 - (B * B)))))) / t_2;
	} else if (B <= 7.6e-35) {
		tmp = -sqrt((2.0 * (fma(B, B, (C * (A * -4.0))) * (F * (2.0 * A))))) / t_2;
	} else if (B <= 1.744e+61) {
		tmp = -sqrt((2.0 * ((A - hypot(A, B)) * (F * (B * B))))) / t_2;
	} else if (B <= 7.8e+121) {
		tmp = t_0 * -sqrt((F * (-0.5 * ((B * B) / C))));
	} else {
		tmp = t_0 * -sqrt((B * -F));
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(sqrt(2.0) / B)
	t_1 = Float64(4.0 * Float64(A * C))
	t_2 = Float64(Float64(B * B) - t_1)
	t_3 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -8e+108)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= -2.05e+37)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_3 * Float64(F * Float64(A - Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(A * A) - Float64(A * A)) - Float64(B * B)) / C)) - A))))))) / t_3);
	elseif (B <= -6.6e-29)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(hypot(A, B) - A) * Float64(F * Float64(t_1 - Float64(B * B))))))) / t_2);
	elseif (B <= 7.6e-35)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(fma(B, B, Float64(C * Float64(A * -4.0))) * Float64(F * Float64(2.0 * A)))))) / t_2);
	elseif (B <= 1.744e+61)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(A - hypot(A, B)) * Float64(F * Float64(B * B)))))) / t_2);
	elseif (B <= 7.8e+121)
		tmp = Float64(t_0 * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B * B) / C))))));
	else
		tmp = Float64(t_0 * Float64(-sqrt(Float64(B * Float64(-F)))));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(B * B), $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -8e+108], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -2.05e+37], N[((-N[Sqrt[N[(2.0 * N[(t$95$3 * N[(F * N[(A - N[(N[(-0.5 * N[(N[(N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[B, -6.6e-29], N[((-N[Sqrt[N[(2.0 * N[(N[(N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision] - A), $MachinePrecision] * N[(F * N[(t$95$1 - N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 7.6e-35], N[((-N[Sqrt[N[(2.0 * N[(N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 1.744e+61], N[((-N[Sqrt[N[(2.0 * N[(N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision] * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 7.8e+121], N[(t$95$0 * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(t$95$0 * (-N[Sqrt[N[(B * (-F)), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -8.0000000000000003e108

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -8.0000000000000003e108 < B < -2.0499999999999999e37

   1. Initial program 16.3%

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

   3. Simplified 16.3%

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

      1. distribute-frac-neg 16.3%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in C around inf 15.7%

      \[\leadsto -\frac{\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 15.9%

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

      2. unpow2 15.9%

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

      3. unpow2 15.9%

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

      4. unpow2 15.9%

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

      5. mul-1-neg 15.9%

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

      6. mul-1-neg 15.9%

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

      7. sqr-neg 15.9%

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

   8. Simplified 15.9%

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

   if -2.0499999999999999e37 < B < -6.60000000000000055e-29

   1. Initial program 46.0%

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

   3. Simplified 46.0%

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

   4. Taylor expanded in C around 0 46.2%

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

      1. +-commutative 46.2%

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

      2. unpow2 46.2%

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

      3. unpow2 46.2%

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

      4. hypot-def 55.0%

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

   6. Simplified 55.0%

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

   if -6.60000000000000055e-29 < B < 7.6000000000000002e-35

   1. Initial program 21.9%

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

   3. Simplified 21.7%

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

   4. Taylor expanded in A around -inf 25.2%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 25.2%

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

   6. Simplified 25.2%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 25.2%

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

      2. associate-*l* 24.4%

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

      3. cancel-sign-sub-inv 24.4%

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

      4. metadata-eval 24.4%

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

      5. *-commutative 24.4%

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

      6. *-commutative 24.4%

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

      7. associate-*r* 24.4%

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

      8. fma-udef 24.4%

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

      9. *-commutative 24.4%

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

   8. Applied egg-rr 24.4%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 24.4%

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

   10. Simplified 24.4%

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

   if 7.6000000000000002e-35 < B < 1.7439999999999999e61

   1. Initial program 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in C around 0 45.3%

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

      1. +-commutative 45.3%

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

      2. unpow2 45.3%

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

      3. unpow2 45.3%

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

      4. hypot-def 51.5%

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

   6. Simplified 51.5%

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

   7. Taylor expanded in B around inf 52.6%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\color{blue}{{B}^{2}} \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   8. Step-by-step derivation

      1. unpow2 52.6%

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

   9. Simplified 52.6%

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

   if 1.7439999999999999e61 < B < 7.79999999999999967e121

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

      1. div-inv 15.2%

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

   5. Applied egg-rr 19.7%

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

   6. Taylor expanded in A around 0 17.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 17.4%

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

      2. unpow2 17.4%

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

      3. unpow2 17.4%

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

      4. hypot-def 17.7%

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

   8. Simplified 17.7%

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

   9. Taylor expanded in C around inf 23.2%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot F} \]
   10. Step-by-step derivation

       1. unpow2 23.2%

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

   11. Simplified 23.2%

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

   if 7.79999999999999967e121 < B

   1. Initial program 5.8%

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

   3. Simplified 5.8%

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

      1. div-inv 5.8%

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

   5. Applied egg-rr 5.8%

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

   6. Taylor expanded in A around 0 7.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 7.7%

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

      2. unpow2 7.7%

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

      3. unpow2 7.7%

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

      4. hypot-def 50.1%

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

   8. Simplified 50.1%

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

   9. Taylor expanded in C around 0 41.6%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{-1 \cdot \left(F \cdot B\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 41.6%

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

       2. mul-1-neg 41.6%

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

   11. Simplified 41.6%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-F\right) \cdot B}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -8 \cdot 10^{+108}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.05 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -6.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(A, B\right) - A\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.744 \cdot 10^{+61}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \]

Alternative 13: 35.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := t_1 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ t_3 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -2.7 \cdot 10^{+109}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A + \left(B + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{+31}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \left(2 \cdot A\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C))))
        (t_1 (/ (sqrt 2.0) B))
        (t_2 (* t_1 (- (sqrt (* F (* -0.5 (/ (* B B) C)))))))
        (t_3 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -2.7e+109)
     (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
     (if (<= B -1.15e+37)
       (/
        (-
         (sqrt
          (*
           2.0
           (*
            t_0
            (*
             F
             (- A (- (* -0.5 (/ (- (- (* A A) (* A A)) (* B B)) C)) A)))))))
        t_0)
       (if (<= B -7.2e-29)
         (/ (- (sqrt (* 2.0 (* t_0 (* F (+ A (+ B C))))))) t_0)
         (if (<= B -1.75e-32)
           t_2
           (if (<= B 1.2e+31)
             (- (/ (sqrt (* 2.0 (* (* F t_3) (* 2.0 A)))) t_3))
             (if (<= B 1.9e+122) t_2 (* t_1 (- (sqrt (* B (- F)))))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = sqrt(2.0) / B;
	double t_2 = t_1 * -sqrt((F * (-0.5 * ((B * B) / C))));
	double t_3 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -2.7e+109) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= -1.15e+37) {
		tmp = -sqrt((2.0 * (t_0 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_0;
	} else if (B <= -7.2e-29) {
		tmp = -sqrt((2.0 * (t_0 * (F * (A + (B + C)))))) / t_0;
	} else if (B <= -1.75e-32) {
		tmp = t_2;
	} else if (B <= 1.2e+31) {
		tmp = -(sqrt((2.0 * ((F * t_3) * (2.0 * A)))) / t_3);
	} else if (B <= 1.9e+122) {
		tmp = t_2;
	} else {
		tmp = t_1 * -sqrt((B * -F));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this 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) :: tmp
    t_0 = (b * b) + ((-4.0d0) * (a * c))
    t_1 = sqrt(2.0d0) / b
    t_2 = t_1 * -sqrt((f * ((-0.5d0) * ((b * b) / c))))
    t_3 = (b * b) - (4.0d0 * (a * c))
    if (b <= (-2.7d+109)) then
        tmp = 2.0d0 * (sqrt((a * f)) * (1.0d0 / b))
    else if (b <= (-1.15d+37)) then
        tmp = -sqrt((2.0d0 * (t_0 * (f * (a - (((-0.5d0) * ((((a * a) - (a * a)) - (b * b)) / c)) - a)))))) / t_0
    else if (b <= (-7.2d-29)) then
        tmp = -sqrt((2.0d0 * (t_0 * (f * (a + (b + c)))))) / t_0
    else if (b <= (-1.75d-32)) then
        tmp = t_2
    else if (b <= 1.2d+31) then
        tmp = -(sqrt((2.0d0 * ((f * t_3) * (2.0d0 * a)))) / t_3)
    else if (b <= 1.9d+122) then
        tmp = t_2
    else
        tmp = t_1 * -sqrt((b * -f))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = Math.sqrt(2.0) / B;
	double t_2 = t_1 * -Math.sqrt((F * (-0.5 * ((B * B) / C))));
	double t_3 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -2.7e+109) {
		tmp = 2.0 * (Math.sqrt((A * F)) * (1.0 / B));
	} else if (B <= -1.15e+37) {
		tmp = -Math.sqrt((2.0 * (t_0 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_0;
	} else if (B <= -7.2e-29) {
		tmp = -Math.sqrt((2.0 * (t_0 * (F * (A + (B + C)))))) / t_0;
	} else if (B <= -1.75e-32) {
		tmp = t_2;
	} else if (B <= 1.2e+31) {
		tmp = -(Math.sqrt((2.0 * ((F * t_3) * (2.0 * A)))) / t_3);
	} else if (B <= 1.9e+122) {
		tmp = t_2;
	} else {
		tmp = t_1 * -Math.sqrt((B * -F));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + (-4.0 * (A * C))
	t_1 = math.sqrt(2.0) / B
	t_2 = t_1 * -math.sqrt((F * (-0.5 * ((B * B) / C))))
	t_3 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= -2.7e+109:
		tmp = 2.0 * (math.sqrt((A * F)) * (1.0 / B))
	elif B <= -1.15e+37:
		tmp = -math.sqrt((2.0 * (t_0 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_0
	elif B <= -7.2e-29:
		tmp = -math.sqrt((2.0 * (t_0 * (F * (A + (B + C)))))) / t_0
	elif B <= -1.75e-32:
		tmp = t_2
	elif B <= 1.2e+31:
		tmp = -(math.sqrt((2.0 * ((F * t_3) * (2.0 * A)))) / t_3)
	elif B <= 1.9e+122:
		tmp = t_2
	else:
		tmp = t_1 * -math.sqrt((B * -F))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(sqrt(2.0) / B)
	t_2 = Float64(t_1 * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B * B) / C))))))
	t_3 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -2.7e+109)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= -1.15e+37)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A - Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(A * A) - Float64(A * A)) - Float64(B * B)) / C)) - A))))))) / t_0);
	elseif (B <= -7.2e-29)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(A + Float64(B + C))))))) / t_0);
	elseif (B <= -1.75e-32)
		tmp = t_2;
	elseif (B <= 1.2e+31)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_3) * Float64(2.0 * A)))) / t_3));
	elseif (B <= 1.9e+122)
		tmp = t_2;
	else
		tmp = Float64(t_1 * Float64(-sqrt(Float64(B * Float64(-F)))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + (-4.0 * (A * C));
	t_1 = sqrt(2.0) / B;
	t_2 = t_1 * -sqrt((F * (-0.5 * ((B * B) / C))));
	t_3 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= -2.7e+109)
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	elseif (B <= -1.15e+37)
		tmp = -sqrt((2.0 * (t_0 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_0;
	elseif (B <= -7.2e-29)
		tmp = -sqrt((2.0 * (t_0 * (F * (A + (B + C)))))) / t_0;
	elseif (B <= -1.75e-32)
		tmp = t_2;
	elseif (B <= 1.2e+31)
		tmp = -(sqrt((2.0 * ((F * t_3) * (2.0 * A)))) / t_3);
	elseif (B <= 1.9e+122)
		tmp = t_2;
	else
		tmp = t_1 * -sqrt((B * -F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$3 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -2.7e+109], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.15e+37], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A - N[(N[(-0.5 * N[(N[(N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, -7.2e-29], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(A + N[(B + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, -1.75e-32], t$95$2, If[LessEqual[B, 1.2e+31], (-N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$3), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]), If[LessEqual[B, 1.9e+122], t$95$2, N[(t$95$1 * (-N[Sqrt[N[(B * (-F)), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if B < -2.70000000000000001e109

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -2.70000000000000001e109 < B < -1.15000000000000001e37

   1. Initial program 16.3%

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

   3. Simplified 16.3%

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

      1. distribute-frac-neg 16.3%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in C around inf 15.7%

      \[\leadsto -\frac{\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 15.9%

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

      2. unpow2 15.9%

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

      3. unpow2 15.9%

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

      4. unpow2 15.9%

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

      5. mul-1-neg 15.9%

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

      6. mul-1-neg 15.9%

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

      7. sqr-neg 15.9%

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

   8. Simplified 15.9%

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

   if -1.15000000000000001e37 < B < -7.19999999999999948e-29

   1. Initial program 46.0%

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

   3. Simplified 46.0%

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

      1. distribute-frac-neg 46.0%

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

   5. Applied egg-rr 55.1%

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

   6. Taylor expanded in B around -inf 47.8%

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

   if -7.19999999999999948e-29 < B < -1.7499999999999999e-32 or 1.19999999999999991e31 < B < 1.8999999999999999e122

   1. Initial program 23.0%

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

   3. Simplified 23.0%

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

      1. div-inv 22.9%

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

   5. Applied egg-rr 26.2%

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

   6. Taylor expanded in A around 0 21.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 21.6%

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

      2. unpow2 21.6%

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

      3. unpow2 21.6%

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

      4. hypot-def 21.9%

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

   8. Simplified 21.9%

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

   9. Taylor expanded in C around inf 25.5%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot F} \]
   10. Step-by-step derivation

       1. unpow2 25.5%

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

   11. Simplified 25.5%

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

   if -1.7499999999999999e-32 < B < 1.19999999999999991e31

   1. Initial program 24.8%

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

   3. Simplified 24.5%

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

   4. Taylor expanded in A around -inf 25.5%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 25.5%

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

   6. Simplified 25.5%

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

   if 1.8999999999999999e122 < B

   1. Initial program 5.8%

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

   3. Simplified 5.8%

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

      1. div-inv 5.8%

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

   5. Applied egg-rr 5.8%

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

   6. Taylor expanded in A around 0 7.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 7.7%

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

      2. unpow2 7.7%

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

      3. unpow2 7.7%

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

      4. hypot-def 50.1%

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

   8. Simplified 50.1%

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

   9. Taylor expanded in C around 0 41.6%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{-1 \cdot \left(F \cdot B\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 41.6%

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

       2. mul-1-neg 41.6%

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

   11. Simplified 41.6%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{\left(-F\right) \cdot B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.7 \cdot 10^{+109}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \left(B + C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{+31}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \]

Alternative 14: 35.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -2.2 \cdot 10^{+109}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-29}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (+ (* B B) (* -4.0 (* A C)))))
   (if (<= B -2.2e+109)
     (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
     (if (<= B -2.9e+35)
       (/
        (-
         (sqrt
          (*
           2.0
           (*
            t_1
            (*
             F
             (- A (- (* -0.5 (/ (- (- (* A A) (* A A)) (* B B)) C)) A)))))))
        t_1)
       (if (<= B 5.8e-29)
         (- (/ (sqrt (* 2.0 (* (* F t_0) (* 2.0 A)))) t_0))
         (* (/ (sqrt 2.0) B) (- (sqrt (* B (- F))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (B <= -2.2e+109) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= -2.9e+35) {
		tmp = -sqrt((2.0 * (t_1 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_1;
	} else if (B <= 5.8e-29) {
		tmp = -(sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0);
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((B * -F));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this 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) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    t_1 = (b * b) + ((-4.0d0) * (a * c))
    if (b <= (-2.2d+109)) then
        tmp = 2.0d0 * (sqrt((a * f)) * (1.0d0 / b))
    else if (b <= (-2.9d+35)) then
        tmp = -sqrt((2.0d0 * (t_1 * (f * (a - (((-0.5d0) * ((((a * a) - (a * a)) - (b * b)) / c)) - a)))))) / t_1
    else if (b <= 5.8d-29) then
        tmp = -(sqrt((2.0d0 * ((f * t_0) * (2.0d0 * a)))) / t_0)
    else
        tmp = (sqrt(2.0d0) / b) * -sqrt((b * -f))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (B <= -2.2e+109) {
		tmp = 2.0 * (Math.sqrt((A * F)) * (1.0 / B));
	} else if (B <= -2.9e+35) {
		tmp = -Math.sqrt((2.0 * (t_1 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_1;
	} else if (B <= 5.8e-29) {
		tmp = -(Math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0);
	} else {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((B * -F));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = (B * B) + (-4.0 * (A * C))
	tmp = 0
	if B <= -2.2e+109:
		tmp = 2.0 * (math.sqrt((A * F)) * (1.0 / B))
	elif B <= -2.9e+35:
		tmp = -math.sqrt((2.0 * (t_1 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_1
	elif B <= 5.8e-29:
		tmp = -(math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0)
	else:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((B * -F))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -2.2e+109)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= -2.9e+35)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(F * Float64(A - Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(A * A) - Float64(A * A)) - Float64(B * B)) / C)) - A))))))) / t_1);
	elseif (B <= 5.8e-29)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(2.0 * A)))) / t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(B * Float64(-F)))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = (B * B) + (-4.0 * (A * C));
	tmp = 0.0;
	if (B <= -2.2e+109)
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	elseif (B <= -2.9e+35)
		tmp = -sqrt((2.0 * (t_1 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_1;
	elseif (B <= 5.8e-29)
		tmp = -(sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0);
	else
		tmp = (sqrt(2.0) / B) * -sqrt((B * -F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -2.2e+109], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -2.9e+35], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(F * N[(A - N[(N[(-0.5 * N[(N[(N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 5.8e-29], (-N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(B * (-F)), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -2.1999999999999999e109

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -2.1999999999999999e109 < B < -2.89999999999999995e35

   1. Initial program 16.3%

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

   3. Simplified 16.3%

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

      1. distribute-frac-neg 16.3%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in C around inf 15.7%

      \[\leadsto -\frac{\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 15.9%

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

      2. unpow2 15.9%

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

      3. unpow2 15.9%

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

      4. unpow2 15.9%

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

      5. mul-1-neg 15.9%

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

      6. mul-1-neg 15.9%

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

      7. sqr-neg 15.9%

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

   8. Simplified 15.9%

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

   if -2.89999999999999995e35 < B < 5.80000000000000048e-29

   1. Initial program 24.8%

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

   3. Simplified 24.6%

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

   4. Taylor expanded in A around -inf 24.4%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 24.4%

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

   6. Simplified 24.4%

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

   if 5.80000000000000048e-29 < B

   1. Initial program 17.9%

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

   3. Simplified 17.9%

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

      1. div-inv 17.9%

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

   5. Applied egg-rr 20.9%

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

   6. Taylor expanded in A around 0 19.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 19.8%

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

      2. unpow2 19.8%

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

      3. unpow2 19.8%

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

      4. hypot-def 40.2%

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

   8. Simplified 40.2%

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

   9. Taylor expanded in C around 0 35.4%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{-1 \cdot \left(F \cdot B\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 35.4%

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

       2. mul-1-neg 35.4%

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

   11. Simplified 35.4%

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

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.2 \cdot 10^{+109}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-29}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \]

Alternative 15: 29.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -4.8 \cdot 10^{-50}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (+ (* B B) (* -4.0 (* A C)))))
   (if (<= A -4.8e-50)
     (- (/ (sqrt (* 2.0 (* (* F t_0) (* 2.0 A)))) t_0))
     (/
      (-
       (sqrt
        (*
         2.0
         (*
          t_1
          (* F (- A (- (* -0.5 (/ (- (- (* A A) (* A A)) (* B B)) C)) A)))))))
      t_1))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (A <= -4.8e-50) {
		tmp = -(sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0);
	} else {
		tmp = -sqrt((2.0 * (t_1 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_1;
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this 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) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    t_1 = (b * b) + ((-4.0d0) * (a * c))
    if (a <= (-4.8d-50)) then
        tmp = -(sqrt((2.0d0 * ((f * t_0) * (2.0d0 * a)))) / t_0)
    else
        tmp = -sqrt((2.0d0 * (t_1 * (f * (a - (((-0.5d0) * ((((a * a) - (a * a)) - (b * b)) / c)) - a)))))) / t_1
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = (B * B) + (-4.0 * (A * C));
	double tmp;
	if (A <= -4.8e-50) {
		tmp = -(Math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0);
	} else {
		tmp = -Math.sqrt((2.0 * (t_1 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_1;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = (B * B) + (-4.0 * (A * C))
	tmp = 0
	if A <= -4.8e-50:
		tmp = -(math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0)
	else:
		tmp = -math.sqrt((2.0 * (t_1 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_1
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (A <= -4.8e-50)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(2.0 * A)))) / t_0));
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(F * Float64(A - Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(A * A) - Float64(A * A)) - Float64(B * B)) / C)) - A))))))) / t_1);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = (B * B) + (-4.0 * (A * C));
	tmp = 0.0;
	if (A <= -4.8e-50)
		tmp = -(sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0);
	else
		tmp = -sqrt((2.0 * (t_1 * (F * (A - ((-0.5 * ((((A * A) - (A * A)) - (B * B)) / C)) - A)))))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -4.8e-50], (-N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(F * N[(A - N[(N[(-0.5 * N[(N[(N[(N[(A * A), $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if A < -4.80000000000000004e-50

   1. Initial program 17.9%

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

   3. Simplified 17.9%

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

   4. Taylor expanded in A around -inf 28.0%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 28.0%

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

   6. Simplified 28.0%

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

   if -4.80000000000000004e-50 < A

   1. Initial program 20.3%

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

   3. Simplified 20.2%

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

      1. distribute-frac-neg 20.2%

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

   5. Applied egg-rr 24.2%

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

   6. Taylor expanded in C around inf 8.7%

      \[\leadsto -\frac{\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 8.8%

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

      2. unpow2 8.8%

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

      3. unpow2 8.8%

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

      4. unpow2 8.8%

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

      5. mul-1-neg 8.8%

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

      6. mul-1-neg 8.8%

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

      7. sqr-neg 8.8%

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

   8. Simplified 8.8%

      \[\leadsto -\frac{\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{B \cdot B + \left(A \cdot A - A \cdot A\right)}{C}\right)}\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4.8 \cdot 10^{-50}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(A - \left(-0.5 \cdot \frac{\left(A \cdot A - A \cdot A\right) - B \cdot B}{C} - A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 16: 25.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq -6 \cdot 10^{+108}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{+121}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B -6e+108)
   (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
   (if (<= B 6.6e+121)
     (-
      (/
       (sqrt (* 2.0 (* (* 2.0 A) (* F (* -4.0 (* A C))))))
       (- (* B B) (* 4.0 (* A C)))))
     (* -2.0 (/ (pow (* A F) 0.5) B)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -6e+108) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= 6.6e+121) {
		tmp = -(sqrt((2.0 * ((2.0 * A) * (F * (-4.0 * (A * C)))))) / ((B * B) - (4.0 * (A * C))));
	} else {
		tmp = -2.0 * (pow((A * F), 0.5) / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this 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) :: tmp
    if (b <= (-6d+108)) then
        tmp = 2.0d0 * (sqrt((a * f)) * (1.0d0 / b))
    else if (b <= 6.6d+121) then
        tmp = -(sqrt((2.0d0 * ((2.0d0 * a) * (f * ((-4.0d0) * (a * c)))))) / ((b * b) - (4.0d0 * (a * c))))
    else
        tmp = (-2.0d0) * (((a * f) ** 0.5d0) / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -6e+108) {
		tmp = 2.0 * (Math.sqrt((A * F)) * (1.0 / B));
	} else if (B <= 6.6e+121) {
		tmp = -(Math.sqrt((2.0 * ((2.0 * A) * (F * (-4.0 * (A * C)))))) / ((B * B) - (4.0 * (A * C))));
	} else {
		tmp = -2.0 * (Math.pow((A * F), 0.5) / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= -6e+108:
		tmp = 2.0 * (math.sqrt((A * F)) * (1.0 / B))
	elif B <= 6.6e+121:
		tmp = -(math.sqrt((2.0 * ((2.0 * A) * (F * (-4.0 * (A * C)))))) / ((B * B) - (4.0 * (A * C))))
	else:
		tmp = -2.0 * (math.pow((A * F), 0.5) / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= -6e+108)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= 6.6e+121)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(2.0 * A) * Float64(F * Float64(-4.0 * Float64(A * C)))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))));
	else
		tmp = Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= -6e+108)
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	elseif (B <= 6.6e+121)
		tmp = -(sqrt((2.0 * ((2.0 * A) * (F * (-4.0 * (A * C)))))) / ((B * B) - (4.0 * (A * C))));
	else
		tmp = -2.0 * (((A * F) ^ 0.5) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, -6e+108], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 6.6e+121], (-N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * A), $MachinePrecision] * N[(F * N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < -5.99999999999999968e108

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -5.99999999999999968e108 < B < 6.59999999999999958e121

   1. Initial program 24.8%

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

   3. Simplified 24.6%

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

   4. Taylor expanded in A around -inf 18.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 18.1%

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

   6. Simplified 18.1%

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

   7. Taylor expanded in B around 0 16.8%

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

   if 6.59999999999999958e121 < B

   1. Initial program 5.8%

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

   3. Simplified 5.8%

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

   4. Taylor expanded in A around -inf 0.5%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.5%

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

   6. Simplified 0.5%

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

   7. Taylor expanded in B around inf 5.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   8. Step-by-step derivation

      1. un-div-inv 5.5%

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

   9. Applied egg-rr 5.5%

      \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]
   10. Step-by-step derivation

       1. *-commutative 5.5%

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

   11. Simplified 5.5%

       \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{F \cdot A}}{B}} \]
   12. Step-by-step derivation

       1. pow1/2 5.7%

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

       2. *-commutative 5.7%

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

   13. Applied egg-rr 5.7%

       \[\leadsto -2 \cdot \frac{\color{blue}{{\left(A \cdot F\right)}^{0.5}}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 14.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -6 \cdot 10^{+108}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{+121}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(2 \cdot A\right) \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \]

Alternative 17: 26.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ -\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(2 \cdot A\right)\right)}}{t_0} \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (- (/ (sqrt (* 2.0 (* (* F t_0) (* 2.0 A)))) t_0))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	return -(sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0);
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this 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
    t_0 = (b * b) - (4.0d0 * (a * c))
    code = -(sqrt((2.0d0 * ((f * t_0) * (2.0d0 * a)))) / t_0)
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	return -(Math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0);
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	return -(math.sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0)

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	return Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(2.0 * A)))) / t_0))
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = -(sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0);
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, (-N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision])]

Derivation

1. Initial program 19.6%

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

3. Simplified 19.5%

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

4. Taylor expanded in A around -inf 13.3%

   \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
5. Step-by-step derivation

   1. *-commutative 13.3%

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

6. Simplified 13.3%

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

7. Final simplification 13.3%

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

Alternative 18: 18.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq -6 \cdot 10^{+108}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B -6e+108)
   (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
   (if (<= B 9.2e-34)
     (/
      (- (sqrt (* 2.0 (* -8.0 (* (* A A) (* C F))))))
      (+ (* B B) (* -4.0 (* A C))))
     (* -2.0 (/ (pow (* A F) 0.5) B)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -6e+108) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else if (B <= 9.2e-34) {
		tmp = -sqrt((2.0 * (-8.0 * ((A * A) * (C * F))))) / ((B * B) + (-4.0 * (A * C)));
	} else {
		tmp = -2.0 * (pow((A * F), 0.5) / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this 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) :: tmp
    if (b <= (-6d+108)) then
        tmp = 2.0d0 * (sqrt((a * f)) * (1.0d0 / b))
    else if (b <= 9.2d-34) then
        tmp = -sqrt((2.0d0 * ((-8.0d0) * ((a * a) * (c * f))))) / ((b * b) + ((-4.0d0) * (a * c)))
    else
        tmp = (-2.0d0) * (((a * f) ** 0.5d0) / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -6e+108) {
		tmp = 2.0 * (Math.sqrt((A * F)) * (1.0 / B));
	} else if (B <= 9.2e-34) {
		tmp = -Math.sqrt((2.0 * (-8.0 * ((A * A) * (C * F))))) / ((B * B) + (-4.0 * (A * C)));
	} else {
		tmp = -2.0 * (Math.pow((A * F), 0.5) / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= -6e+108:
		tmp = 2.0 * (math.sqrt((A * F)) * (1.0 / B))
	elif B <= 9.2e-34:
		tmp = -math.sqrt((2.0 * (-8.0 * ((A * A) * (C * F))))) / ((B * B) + (-4.0 * (A * C)))
	else:
		tmp = -2.0 * (math.pow((A * F), 0.5) / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= -6e+108)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	elseif (B <= 9.2e-34)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-8.0 * Float64(Float64(A * A) * Float64(C * F)))))) / Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))));
	else
		tmp = Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= -6e+108)
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	elseif (B <= 9.2e-34)
		tmp = -sqrt((2.0 * (-8.0 * ((A * A) * (C * F))))) / ((B * B) + (-4.0 * (A * C)));
	else
		tmp = -2.0 * (((A * F) ^ 0.5) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, -6e+108], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 9.2e-34], N[((-N[Sqrt[N[(2.0 * N[(-8.0 * N[(N[(A * A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < -5.99999999999999968e108

   1. Initial program 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in A around -inf 3.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in B around -inf 11.7%

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

   if -5.99999999999999968e108 < B < 9.20000000000000045e-34

   1. Initial program 23.0%

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

   3. Simplified 22.8%

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

      1. distribute-frac-neg 22.8%

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

   5. Applied egg-rr 31.3%

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

   6. Taylor expanded in A around -inf 12.3%

      \[\leadsto -\frac{\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. unpow2 12.3%

         \[\leadsto -\frac{\sqrt{2 \cdot \left(-8 \cdot \left(\color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]

   8. Simplified 12.3%

      \[\leadsto -\frac{\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]

   if 9.20000000000000045e-34 < B

   1. Initial program 19.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

   3. Simplified 19.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 5.4%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 5.4%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Simplified 5.4%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Taylor expanded in B around inf 6.1%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   8. Step-by-step derivation

      1. un-div-inv 6.1%

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]

   9. Applied egg-rr 6.1%

      \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]
   10. Step-by-step derivation

       1. *-commutative 6.1%

          \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot A}}}{B} \]

   11. Simplified 6.1%

       \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{F \cdot A}}{B}} \]
   12. Step-by-step derivation

       1. pow1/2 6.4%

          \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot A\right)}^{0.5}}}{B} \]

       2. *-commutative 6.4%

          \[\leadsto -2 \cdot \frac{{\color{blue}{\left(A \cdot F\right)}}^{0.5}}{B} \]

   13. Applied egg-rr 6.4%

       \[\leadsto -2 \cdot \frac{\color{blue}{{\left(A \cdot F\right)}^{0.5}}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -6 \cdot 10^{+108}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \]

Alternative 19: 9.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq -3.1 \cdot 10^{-273}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B -3.1e-273)
   (* 2.0 (* (sqrt (* A F)) (/ 1.0 B)))
   (* -2.0 (/ (pow (* A F) 0.5) B))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -3.1e-273) {
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	} else {
		tmp = -2.0 * (pow((A * F), 0.5) / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this 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) :: tmp
    if (b <= (-3.1d-273)) then
        tmp = 2.0d0 * (sqrt((a * f)) * (1.0d0 / b))
    else
        tmp = (-2.0d0) * (((a * f) ** 0.5d0) / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -3.1e-273) {
		tmp = 2.0 * (Math.sqrt((A * F)) * (1.0 / B));
	} else {
		tmp = -2.0 * (Math.pow((A * F), 0.5) / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= -3.1e-273:
		tmp = 2.0 * (math.sqrt((A * F)) * (1.0 / B))
	else:
		tmp = -2.0 * (math.pow((A * F), 0.5) / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= -3.1e-273)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	else
		tmp = Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= -3.1e-273)
		tmp = 2.0 * (sqrt((A * F)) * (1.0 / B));
	else
		tmp = -2.0 * (((A * F) ^ 0.5) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, -3.1e-273], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -3.09999999999999988e-273

   1. Initial program 22.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

   3. Simplified 21.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 14.0%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 14.0%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Simplified 14.0%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Taylor expanded in B around -inf 6.3%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]

   if -3.09999999999999988e-273 < B

   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. Step-by-step derivation

   3. Simplified 17.2%

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 12.7%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. *-commutative 12.7%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Simplified 12.7%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Taylor expanded in B around inf 4.2%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   8. Step-by-step derivation

      1. un-div-inv 4.3%

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]

   9. Applied egg-rr 4.3%

      \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]
   10. Step-by-step derivation

       1. *-commutative 4.3%

          \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot A}}}{B} \]

   11. Simplified 4.3%

       \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{F \cdot A}}{B}} \]
   12. Step-by-step derivation

       1. pow1/2 4.5%

          \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot A\right)}^{0.5}}}{B} \]

       2. *-commutative 4.5%

          \[\leadsto -2 \cdot \frac{{\color{blue}{\left(A \cdot F\right)}}^{0.5}}{B} \]

   13. Applied egg-rr 4.5%

       \[\leadsto -2 \cdot \frac{\color{blue}{{\left(A \cdot F\right)}^{0.5}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.1 \cdot 10^{-273}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B}\\ \end{array} \]

Alternative 20: 5.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ -2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (* -2.0 (/ (pow (* A F) 0.5) B)))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	return -2.0 * (pow((A * F), 0.5) / B);
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this 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
    code = (-2.0d0) * (((a * f) ** 0.5d0) / b)
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	return -2.0 * (Math.pow((A * F), 0.5) / B);
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	return -2.0 * (math.pow((A * F), 0.5) / B)

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	return Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) / B))
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp = code(A, B, C, F)
	tmp = -2.0 * (((A * F) ^ 0.5) / B);
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.6%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Step-by-step derivation

3. Simplified 19.5%

   \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

4. Taylor expanded in A around -inf 13.3%

   \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
5. Step-by-step derivation

   1. *-commutative 13.3%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

6. Simplified 13.3%

   \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

7. Taylor expanded in B around inf 2.9%

   \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
8. Step-by-step derivation

   1. un-div-inv 2.9%

      \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]

9. Applied egg-rr 2.9%

   \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]
10. Step-by-step derivation

    1. *-commutative 2.9%

       \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot A}}}{B} \]

11. Simplified 2.9%

    \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{F \cdot A}}{B}} \]
12. Step-by-step derivation

    1. pow1/2 3.0%

       \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot A\right)}^{0.5}}}{B} \]

    2. *-commutative 3.0%

       \[\leadsto -2 \cdot \frac{{\color{blue}{\left(A \cdot F\right)}}^{0.5}}{B} \]

13. Applied egg-rr 3.0%

    \[\leadsto -2 \cdot \frac{\color{blue}{{\left(A \cdot F\right)}^{0.5}}}{B} \]

14. Final simplification 3.0%

    \[\leadsto -2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B} \]

Alternative 21: 5.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ -2 \cdot \frac{\sqrt{A \cdot F}}{B} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (* -2.0 (/ (sqrt (* A F)) B)))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	return -2.0 * (sqrt((A * F)) / B);
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this 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
    code = (-2.0d0) * (sqrt((a * f)) / b)
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	return -2.0 * (Math.sqrt((A * F)) / B);
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	return -2.0 * (math.sqrt((A * F)) / B)

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	return Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B))
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp = code(A, B, C, F)
	tmp = -2.0 * (sqrt((A * F)) / B);
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.6%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Step-by-step derivation

3. Simplified 19.5%

   \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

4. Taylor expanded in A around -inf 13.3%

   \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
5. Step-by-step derivation

   1. *-commutative 13.3%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

6. Simplified 13.3%

   \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\left(A \cdot 2\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

7. Taylor expanded in B around inf 2.9%

   \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
8. Step-by-step derivation

   1. un-div-inv 2.9%

      \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]

9. Applied egg-rr 2.9%

   \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]
10. Step-by-step derivation

    1. *-commutative 2.9%

       \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot A}}}{B} \]

11. Simplified 2.9%

    \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{F \cdot A}}{B}} \]

12. Final simplification 2.9%

    \[\leadsto -2 \cdot \frac{\sqrt{A \cdot F}}{B} \]

Reproduce

herbie shell --seed 2023174 
(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))))