ABCF->ab-angle b

Percentage Accurate: 18.6% → 38.9%

Time: 17.5s

Alternatives: 17

Speedup: 8.9×

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: 18.6% 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: 38.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A - C, A - C, B \cdot B\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \left(F \cdot \left({B}^{2} - t\_1\right)\right) \cdot 2\\ t_3 := t\_1 - {B}^{2}\\ t_4 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot t\_2}}{t\_3}\\ t_5 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \left(\sqrt{2} \cdot \sqrt{t\_5}\right)}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(C, C, -t\_0\right), \frac{1}{\sqrt{t\_0} + C}, A\right) \cdot t\_2}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_5\right) \cdot \left(\left(\frac{B \cdot B}{C} \cdot -0.5 + A\right) + A\right)}}{-t\_5}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (- A C) (- A C) (* B B)))
        (t_1 (* C (* A 4.0)))
        (t_2 (* (* F (- (pow B 2.0) t_1)) 2.0))
        (t_3 (- t_1 (pow B 2.0)))
        (t_4
         (/
          (sqrt (* (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))) t_2))
          t_3))
        (t_5 (fma (* -4.0 C) A (* B B))))
   (if (<= t_4 (- INFINITY))
     (/ (* (sqrt (* (+ A A) F)) (* (sqrt 2.0) (sqrt t_5))) t_3)
     (if (<= t_4 -5e-195)
       (/
        (sqrt (* (fma (fma C C (- t_0)) (/ 1.0 (+ (sqrt t_0) C)) A) t_2))
        t_3)
       (/
        (sqrt (* (* (* F 2.0) t_5) (+ (+ (* (/ (* B B) C) -0.5) A) A)))
        (- t_5))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((A - C), (A - C), (B * B));
	double t_1 = C * (A * 4.0);
	double t_2 = (F * (pow(B, 2.0) - t_1)) * 2.0;
	double t_3 = t_1 - pow(B, 2.0);
	double t_4 = sqrt((((C + A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))) * t_2)) / t_3;
	double t_5 = fma((-4.0 * C), A, (B * B));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt(((A + A) * F)) * (sqrt(2.0) * sqrt(t_5))) / t_3;
	} else if (t_4 <= -5e-195) {
		tmp = sqrt((fma(fma(C, C, -t_0), (1.0 / (sqrt(t_0) + C)), A) * t_2)) / t_3;
	} else {
		tmp = sqrt((((F * 2.0) * t_5) * (((((B * B) / C) * -0.5) + A) + A))) / -t_5;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(A - C), Float64(A - C), Float64(B * B))
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = Float64(Float64(F * Float64((B ^ 2.0) - t_1)) * 2.0)
	t_3 = Float64(t_1 - (B ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))) * t_2)) / t_3)
	t_5 = fma(Float64(-4.0 * C), A, Float64(B * B))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(A + A) * F)) * Float64(sqrt(2.0) * sqrt(t_5))) / t_3);
	elseif (t_4 <= -5e-195)
		tmp = Float64(sqrt(Float64(fma(fma(C, C, Float64(-t_0)), Float64(1.0 / Float64(sqrt(t_0) + C)), A) * t_2)) / t_3);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_5) * Float64(Float64(Float64(Float64(Float64(B * B) / C) * -0.5) + A) + A))) / Float64(-t_5));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -5e-195], N[(N[Sqrt[N[(N[(N[(C * C + (-t$95$0)), $MachinePrecision] * N[(1.0 / N[(N[Sqrt[t$95$0], $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(N[(N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$5)), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0

   1. Initial program 3.3%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 28.5

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

   5. Applied rewrites 28.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

   7. Applied rewrites 36.1%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 36.2%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000009e-195

   1. Initial program 96.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-- N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 70.0%

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

   if -5.00000000000000009e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 7.3%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 14.7

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

   5. Applied rewrites 14.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 16.4

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

   10. Applied rewrites 16.4%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - \left(-A\right)\right)} \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\right)}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(C, C, -\mathsf{fma}\left(A - C, A - C, B \cdot B\right)\right), \frac{1}{\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} + C}, A\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right) \cdot \left(\left(\frac{B \cdot B}{C} \cdot -0.5 + A\right) + A\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 38.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ t_1 := \left(F \cdot \left({B}^{2} - t\_0\right)\right) \cdot 2\\ t_2 := t\_0 - {B}^{2}\\ t_3 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot t\_1}}{t\_2}\\ t_4 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \left(\sqrt{2} \cdot \sqrt{t\_4}\right)}{t\_2}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(C - A\right) \cdot \left(C + A\right), \frac{1}{C - A}, -\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot t\_1}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_4\right) \cdot \left(\left(\frac{B \cdot B}{C} \cdot -0.5 + A\right) + A\right)}}{-t\_4}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* C (* A 4.0)))
        (t_1 (* (* F (- (pow B 2.0) t_0)) 2.0))
        (t_2 (- t_0 (pow B 2.0)))
        (t_3
         (/
          (sqrt (* (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))) t_1))
          t_2))
        (t_4 (fma (* -4.0 C) A (* B B))))
   (if (<= t_3 (- INFINITY))
     (/ (* (sqrt (* (+ A A) F)) (* (sqrt 2.0) (sqrt t_4))) t_2)
     (if (<= t_3 -5e-195)
       (/
        (sqrt
         (*
          (fma
           (* (- C A) (+ C A))
           (/ 1.0 (- C A))
           (- (sqrt (fma (- A C) (- A C) (* B B)))))
          t_1))
        t_2)
       (/
        (sqrt (* (* (* F 2.0) t_4) (+ (+ (* (/ (* B B) C) -0.5) A) A)))
        (- t_4))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = C * (A * 4.0);
	double t_1 = (F * (pow(B, 2.0) - t_0)) * 2.0;
	double t_2 = t_0 - pow(B, 2.0);
	double t_3 = sqrt((((C + A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))) * t_1)) / t_2;
	double t_4 = fma((-4.0 * C), A, (B * B));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt(((A + A) * F)) * (sqrt(2.0) * sqrt(t_4))) / t_2;
	} else if (t_3 <= -5e-195) {
		tmp = sqrt((fma(((C - A) * (C + A)), (1.0 / (C - A)), -sqrt(fma((A - C), (A - C), (B * B)))) * t_1)) / t_2;
	} else {
		tmp = sqrt((((F * 2.0) * t_4) * (((((B * B) / C) * -0.5) + A) + A))) / -t_4;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(C * Float64(A * 4.0))
	t_1 = Float64(Float64(F * Float64((B ^ 2.0) - t_0)) * 2.0)
	t_2 = Float64(t_0 - (B ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))) * t_1)) / t_2)
	t_4 = fma(Float64(-4.0 * C), A, Float64(B * B))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(A + A) * F)) * Float64(sqrt(2.0) * sqrt(t_4))) / t_2);
	elseif (t_3 <= -5e-195)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(C - A) * Float64(C + A)), Float64(1.0 / Float64(C - A)), Float64(-sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))) * t_1)) / t_2);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_4) * Float64(Float64(Float64(Float64(Float64(B * B) / C) * -0.5) + A) + A))) / Float64(-t_4));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -5e-195], N[(N[Sqrt[N[(N[(N[(N[(C - A), $MachinePrecision] * N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(C - A), $MachinePrecision]), $MachinePrecision] + (-N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(N[(N[(N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0

   1. Initial program 3.3%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 28.5

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

   5. Applied rewrites 28.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

   7. Applied rewrites 36.1%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 36.2%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000009e-195

   1. Initial program 96.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. difference-of-squares N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-neg.f64 96.5

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

      19. lift-+.f64 N/A

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

      20. lift-pow.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-fma.f64 96.5

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

   4. Applied rewrites 96.5%

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

   if -5.00000000000000009e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 7.3%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 14.7

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

   5. Applied rewrites 14.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 16.4

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

   10. Applied rewrites 16.4%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - \left(-A\right)\right)} \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\right)}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(C - A\right) \cdot \left(C + A\right), \frac{1}{C - A}, -\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right) \cdot \left(\left(\frac{B \cdot B}{C} \cdot -0.5 + A\right) + A\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 38.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := t\_2 - {B}^{2}\\ t_4 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{+280}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \left(\sqrt{2} \cdot \sqrt{t\_1}\right)}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_1\right) \cdot \left(\left(\frac{B \cdot B}{C} \cdot -0.5 + A\right) + A\right)}}{-t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B B)))
        (t_1 (fma (* -4.0 C) A (* B B)))
        (t_2 (* C (* A 4.0)))
        (t_3 (- t_2 (pow B 2.0)))
        (t_4
         (/
          (sqrt
           (*
            (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))
            (* (* F (- (pow B 2.0) t_2)) 2.0)))
          t_3)))
   (if (<= t_4 -1e+280)
     (/ (* (sqrt (* (+ A A) F)) (* (sqrt 2.0) (sqrt t_1))) t_3)
     (if (<= t_4 -5e-195)
       (/
        (sqrt
         (*
          (* t_0 (* F 2.0))
          (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B))))))
        (- t_0))
       (/
        (sqrt (* (* (* F 2.0) t_1) (+ (+ (* (/ (* B B) C) -0.5) A) A)))
        (- t_1))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B * B));
	double t_1 = fma((-4.0 * C), A, (B * B));
	double t_2 = C * (A * 4.0);
	double t_3 = t_2 - pow(B, 2.0);
	double t_4 = sqrt((((C + A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))) * ((F * (pow(B, 2.0) - t_2)) * 2.0))) / t_3;
	double tmp;
	if (t_4 <= -1e+280) {
		tmp = (sqrt(((A + A) * F)) * (sqrt(2.0) * sqrt(t_1))) / t_3;
	} else if (t_4 <= -5e-195) {
		tmp = sqrt(((t_0 * (F * 2.0)) * ((C + A) - sqrt(fma((A - C), (A - C), (B * B)))))) / -t_0;
	} else {
		tmp = sqrt((((F * 2.0) * t_1) * (((((B * B) / C) * -0.5) + A) + A))) / -t_1;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B * B))
	t_1 = fma(Float64(-4.0 * C), A, Float64(B * B))
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(t_2 - (B ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))) * Float64(Float64(F * Float64((B ^ 2.0) - t_2)) * 2.0))) / t_3)
	tmp = 0.0
	if (t_4 <= -1e+280)
		tmp = Float64(Float64(sqrt(Float64(Float64(A + A) * F)) * Float64(sqrt(2.0) * sqrt(t_1))) / t_3);
	elseif (t_4 <= -5e-195)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(F * 2.0)) * Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_1) * Float64(Float64(Float64(Float64(Float64(B * B) / C) * -0.5) + A) + A))) / Float64(-t_1));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, -1e+280], N[(N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -5e-195], N[(N[Sqrt[N[(N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[(N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e280

   1. Initial program 4.7%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 29.4

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

   5. Applied rewrites 29.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

   7. Applied rewrites 36.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 37.0%

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

   if -1e280 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000009e-195

   1. Initial program 96.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 96.9%

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

   if -5.00000000000000009e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 7.3%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 14.7

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

   5. Applied rewrites 14.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 16.4

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

   10. Applied rewrites 16.4%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - \left(-A\right)\right)} \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{+280}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\right)}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right) \cdot \left(\left(\frac{B \cdot B}{C} \cdot -0.5 + A\right) + A\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 38.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_2 - {B}^{2}}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+280}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \left(A + A\right)} \cdot \sqrt{t\_1}}{\left(C \cdot 4 - \frac{B \cdot B}{A}\right) \cdot A}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_1\right) \cdot \left(\left(\frac{B \cdot B}{C} \cdot -0.5 + A\right) + A\right)}}{-t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B B)))
        (t_1 (fma (* -4.0 C) A (* B B)))
        (t_2 (* C (* A 4.0)))
        (t_3
         (/
          (sqrt
           (*
            (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))
            (* (* F (- (pow B 2.0) t_2)) 2.0)))
          (- t_2 (pow B 2.0)))))
   (if (<= t_3 -1e+280)
     (/
      (* (sqrt (* (* F 2.0) (+ A A))) (sqrt t_1))
      (* (- (* C 4.0) (/ (* B B) A)) A))
     (if (<= t_3 -5e-195)
       (/
        (sqrt
         (*
          (* t_0 (* F 2.0))
          (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B))))))
        (- t_0))
       (/
        (sqrt (* (* (* F 2.0) t_1) (+ (+ (* (/ (* B B) C) -0.5) A) A)))
        (- t_1))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B * B));
	double t_1 = fma((-4.0 * C), A, (B * B));
	double t_2 = C * (A * 4.0);
	double t_3 = sqrt((((C + A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))) * ((F * (pow(B, 2.0) - t_2)) * 2.0))) / (t_2 - pow(B, 2.0));
	double tmp;
	if (t_3 <= -1e+280) {
		tmp = (sqrt(((F * 2.0) * (A + A))) * sqrt(t_1)) / (((C * 4.0) - ((B * B) / A)) * A);
	} else if (t_3 <= -5e-195) {
		tmp = sqrt(((t_0 * (F * 2.0)) * ((C + A) - sqrt(fma((A - C), (A - C), (B * B)))))) / -t_0;
	} else {
		tmp = sqrt((((F * 2.0) * t_1) * (((((B * B) / C) * -0.5) + A) + A))) / -t_1;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B * B))
	t_1 = fma(Float64(-4.0 * C), A, Float64(B * B))
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))) * Float64(Float64(F * Float64((B ^ 2.0) - t_2)) * 2.0))) / Float64(t_2 - (B ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -1e+280)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * 2.0) * Float64(A + A))) * sqrt(t_1)) / Float64(Float64(Float64(C * 4.0) - Float64(Float64(B * B) / A)) * A));
	elseif (t_3 <= -5e-195)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(F * 2.0)) * Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_1) * Float64(Float64(Float64(Float64(Float64(B * B) / C) * -0.5) + A) + A))) / Float64(-t_1));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+280], N[(N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(C * 4.0), $MachinePrecision] - N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * A), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-195], N[(N[Sqrt[N[(N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[(N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e280

   1. Initial program 4.7%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 29.4

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

   5. Applied rewrites 29.4%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 29.4%

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

   8. Taylor expanded in A around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 29.3

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

   10. Applied rewrites 29.3%

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

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. unpow-prod-down N/A

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

   12. Applied rewrites 36.8%

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

   if -1e280 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000009e-195

   1. Initial program 96.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 96.9%

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

   if -5.00000000000000009e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 7.3%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 14.7

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

   5. Applied rewrites 14.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 16.4

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

   10. Applied rewrites 16.4%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - \left(-A\right)\right)} \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{+280}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \left(A + A\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}{\left(C \cdot 4 - \frac{B \cdot B}{A}\right) \cdot A}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right) \cdot \left(\left(\frac{B \cdot B}{C} \cdot -0.5 + A\right) + A\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B}^{2}}\\ t_3 := -t\_0\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+236}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \left(A + A\right)} \cdot \sqrt{t\_0}}{\left(C \cdot 4 - \frac{B \cdot B}{A}\right) \cdot A}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(B, B, \left(C - A\right) \cdot \left(C - A\right)\right)}\right) \cdot F\right) \cdot 2\right) \cdot t\_0}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_0\right) \cdot \left(\left(\frac{B \cdot B}{C} \cdot -0.5 + A\right) + A\right)}}{t\_3}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B B)))
        (t_1 (* C (* A 4.0)))
        (t_2
         (/
          (sqrt
           (*
            (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))
            (* (* F (- (pow B 2.0) t_1)) 2.0)))
          (- t_1 (pow B 2.0))))
        (t_3 (- t_0)))
   (if (<= t_2 -2e+236)
     (/
      (* (sqrt (* (* F 2.0) (+ A A))) (sqrt t_0))
      (* (- (* C 4.0) (/ (* B B) A)) A))
     (if (<= t_2 -5e-195)
       (/
        (sqrt
         (*
          (* (* (- (+ C A) (sqrt (fma B B (* (- C A) (- C A))))) F) 2.0)
          t_0))
        t_3)
       (/
        (sqrt (* (* (* F 2.0) t_0) (+ (+ (* (/ (* B B) C) -0.5) A) A)))
        t_3)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B * B));
	double t_1 = C * (A * 4.0);
	double t_2 = sqrt((((C + A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))) * ((F * (pow(B, 2.0) - t_1)) * 2.0))) / (t_1 - pow(B, 2.0));
	double t_3 = -t_0;
	double tmp;
	if (t_2 <= -2e+236) {
		tmp = (sqrt(((F * 2.0) * (A + A))) * sqrt(t_0)) / (((C * 4.0) - ((B * B) / A)) * A);
	} else if (t_2 <= -5e-195) {
		tmp = sqrt((((((C + A) - sqrt(fma(B, B, ((C - A) * (C - A))))) * F) * 2.0) * t_0)) / t_3;
	} else {
		tmp = sqrt((((F * 2.0) * t_0) * (((((B * B) / C) * -0.5) + A) + A))) / t_3;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B * B))
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))) * Float64(Float64(F * Float64((B ^ 2.0) - t_1)) * 2.0))) / Float64(t_1 - (B ^ 2.0)))
	t_3 = Float64(-t_0)
	tmp = 0.0
	if (t_2 <= -2e+236)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * 2.0) * Float64(A + A))) * sqrt(t_0)) / Float64(Float64(Float64(C * 4.0) - Float64(Float64(B * B) / A)) * A));
	elseif (t_2 <= -5e-195)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(C + A) - sqrt(fma(B, B, Float64(Float64(C - A) * Float64(C - A))))) * F) * 2.0) * t_0)) / t_3);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_0) * Float64(Float64(Float64(Float64(Float64(B * B) / C) * -0.5) + A) + A))) / t_3);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-t$95$0)}, If[LessEqual[t$95$2, -2e+236], N[(N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(C * 4.0), $MachinePrecision] - N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * A), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-195], N[(N[Sqrt[N[(N[(N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(B * B + N[(N[(C - A), $MachinePrecision] * N[(C - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.00000000000000011e236

   1. Initial program 6.5%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 28.9

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

   5. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 28.9%

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

   8. Taylor expanded in A around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 28.8

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

   10. Applied rewrites 28.8%

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

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. unpow-prod-down N/A

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

   12. Applied rewrites 36.2%

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

   if -2.00000000000000011e236 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000009e-195

   1. Initial program 96.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 96.9%

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

   6. Applied rewrites 96.6%

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

   8. Applied rewrites 97.0%

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

   if -5.00000000000000009e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 7.3%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 14.7

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

   5. Applied rewrites 14.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 16.4

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

   10. Applied rewrites 16.4%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - \left(-A\right)\right)} \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{+236}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \left(A + A\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}{\left(C \cdot 4 - \frac{B \cdot B}{A}\right) \cdot A}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(B, B, \left(C - A\right) \cdot \left(C - A\right)\right)}\right) \cdot F\right) \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right) \cdot \left(\left(\frac{B \cdot B}{C} \cdot -0.5 + A\right) + A\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B}^{2}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \left(A + A\right)} \cdot \sqrt{t\_0}}{\left(C \cdot 4 - \frac{B \cdot B}{A}\right) \cdot A}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_0\right) \cdot \left(\left(\frac{B \cdot B}{C} \cdot -0.5 + A\right) + A\right)}}{-t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B B)))
        (t_1 (* C (* A 4.0)))
        (t_2
         (/
          (sqrt
           (*
            (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))
            (* (* F (- (pow B 2.0) t_1)) 2.0)))
          (- t_1 (pow B 2.0)))))
   (if (<= t_2 -1e+135)
     (/
      (* (sqrt (* (* F 2.0) (+ A A))) (sqrt t_0))
      (* (- (* C 4.0) (/ (* B B) A)) A))
     (if (<= t_2 -1e-173)
       (*
        (sqrt
         (*
          (/
           (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B))))
           (fma (* -4.0 A) C (* B B)))
          F))
        (- (sqrt 2.0)))
       (/
        (sqrt (* (* (* F 2.0) t_0) (+ (+ (* (/ (* B B) C) -0.5) A) A)))
        (- t_0))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B * B));
	double t_1 = C * (A * 4.0);
	double t_2 = sqrt((((C + A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))) * ((F * (pow(B, 2.0) - t_1)) * 2.0))) / (t_1 - pow(B, 2.0));
	double tmp;
	if (t_2 <= -1e+135) {
		tmp = (sqrt(((F * 2.0) * (A + A))) * sqrt(t_0)) / (((C * 4.0) - ((B * B) / A)) * A);
	} else if (t_2 <= -1e-173) {
		tmp = sqrt(((((C + A) - sqrt(fma((A - C), (A - C), (B * B)))) / fma((-4.0 * A), C, (B * B))) * F)) * -sqrt(2.0);
	} else {
		tmp = sqrt((((F * 2.0) * t_0) * (((((B * B) / C) * -0.5) + A) + A))) / -t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B * B))
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))) * Float64(Float64(F * Float64((B ^ 2.0) - t_1)) * 2.0))) / Float64(t_1 - (B ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -1e+135)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * 2.0) * Float64(A + A))) * sqrt(t_0)) / Float64(Float64(Float64(C * 4.0) - Float64(Float64(B * B) / A)) * A));
	elseif (t_2 <= -1e-173)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))) / fma(Float64(-4.0 * A), C, Float64(B * B))) * F)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_0) * Float64(Float64(Float64(Float64(Float64(B * B) / C) * -0.5) + A) + A))) / Float64(-t_0));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+135], N[(N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(C * 4.0), $MachinePrecision] - N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * A), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-173], N[(N[Sqrt[N[(N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999962e134

   1. Initial program 14.3%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 31.6

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

   5. Applied rewrites 31.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 31.6%

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

   8. Taylor expanded in A around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 31.4

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

   10. Applied rewrites 31.4%

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

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. unpow-prod-down N/A

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

   12. Applied rewrites 38.3%

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

   if -9.99999999999999962e134 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-173

   1. Initial program 98.4%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 95.1%

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

   if -1e-173 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 8.0%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 15.1

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

   5. Applied rewrites 15.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 15.1%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 16.8

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

   10. Applied rewrites 16.8%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - \left(-A\right)\right)} \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \left(A + A\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}{\left(C \cdot 4 - \frac{B \cdot B}{A}\right) \cdot A}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right) \cdot \left(\left(\frac{B \cdot B}{C} \cdot -0.5 + A\right) + A\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ t_1 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_0\right)\right) \cdot 2\right)}}{t\_0 - {B}^{2}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \left(A + A\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}{\left(C \cdot 4 - \frac{B \cdot B}{A}\right) \cdot A}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{\left(C \cdot A\right) \cdot 4}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* C (* A 4.0)))
        (t_1
         (/
          (sqrt
           (*
            (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))
            (* (* F (- (pow B 2.0) t_0)) 2.0)))
          (- t_0 (pow B 2.0)))))
   (if (<= t_1 -1e+135)
     (/
      (* (sqrt (* (* F 2.0) (+ A A))) (sqrt (fma (* -4.0 C) A (* B B))))
      (* (- (* C 4.0) (/ (* B B) A)) A))
     (if (<= t_1 -1e-140)
       (*
        (sqrt
         (*
          (/
           (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B))))
           (fma (* -4.0 A) C (* B B)))
          F))
        (- (sqrt 2.0)))
       (/ (sqrt (* (* (* (* (+ A A) F) C) A) -8.0)) (* (* C A) 4.0))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = C * (A * 4.0);
	double t_1 = sqrt((((C + A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))) * ((F * (pow(B, 2.0) - t_0)) * 2.0))) / (t_0 - pow(B, 2.0));
	double tmp;
	if (t_1 <= -1e+135) {
		tmp = (sqrt(((F * 2.0) * (A + A))) * sqrt(fma((-4.0 * C), A, (B * B)))) / (((C * 4.0) - ((B * B) / A)) * A);
	} else if (t_1 <= -1e-140) {
		tmp = sqrt(((((C + A) - sqrt(fma((A - C), (A - C), (B * B)))) / fma((-4.0 * A), C, (B * B))) * F)) * -sqrt(2.0);
	} else {
		tmp = sqrt((((((A + A) * F) * C) * A) * -8.0)) / ((C * A) * 4.0);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(C * Float64(A * 4.0))
	t_1 = Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))) * Float64(Float64(F * Float64((B ^ 2.0) - t_0)) * 2.0))) / Float64(t_0 - (B ^ 2.0)))
	tmp = 0.0
	if (t_1 <= -1e+135)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * 2.0) * Float64(A + A))) * sqrt(fma(Float64(-4.0 * C), A, Float64(B * B)))) / Float64(Float64(Float64(C * 4.0) - Float64(Float64(B * B) / A)) * A));
	elseif (t_1 <= -1e-140)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))) / fma(Float64(-4.0 * A), C, Float64(B * B))) * F)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(A + A) * F) * C) * A) * -8.0)) / Float64(Float64(C * A) * 4.0));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+135], N[(N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(C * 4.0), $MachinePrecision] - N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * A), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-140], N[(N[Sqrt[N[(N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision] * C), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision]], $MachinePrecision] / N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999962e134

   1. Initial program 14.3%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 31.6

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

   5. Applied rewrites 31.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 31.6%

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

   8. Taylor expanded in A around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 31.4

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

   10. Applied rewrites 31.4%

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

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. unpow-prod-down N/A

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

   12. Applied rewrites 38.3%

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

   if -9.99999999999999962e134 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999998e-141

   1. Initial program 98.3%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 97.9%

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

   if -9.9999999999999998e-141 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 8.6%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 15.7

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

   5. Applied rewrites 15.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 15.7%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 15.3

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

   10. Applied rewrites 15.3%

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

   11. Taylor expanded in C around inf

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]
   12. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. rem-square-sqrt N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower--.f64 N/A

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

       10. mul-1-neg N/A

           \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{\left(\mathsf{neg}\left(A\right)\right)}\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)} \]

       11. lower-neg.f64 17.4

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

   13. Applied rewrites 17.4%

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \left(A + A\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}{\left(C \cdot 4 - \frac{B \cdot B}{A}\right) \cdot A}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{\left(C \cdot A\right) \cdot 4}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_1 := \left(A + A\right) \cdot F\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_2 - {B}^{2}}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\left(-\sqrt{t\_0 \cdot 2}\right) \cdot \sqrt{t\_1}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(t\_1 \cdot C\right) \cdot A\right) \cdot -8}}{\left(C \cdot A\right) \cdot 4}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B B)))
        (t_1 (* (+ A A) F))
        (t_2 (* C (* A 4.0)))
        (t_3
         (/
          (sqrt
           (*
            (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))
            (* (* F (- (pow B 2.0) t_2)) 2.0)))
          (- t_2 (pow B 2.0)))))
   (if (<= t_3 -1e+135)
     (/ (* (- (sqrt (* t_0 2.0))) (sqrt t_1)) t_0)
     (if (<= t_3 -1e-173)
       (*
        (sqrt
         (*
          (/
           (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B))))
           (fma (* -4.0 A) C (* B B)))
          F))
        (- (sqrt 2.0)))
       (/ (sqrt (* (* (* t_1 C) A) -8.0)) (* (* C A) 4.0))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B * B));
	double t_1 = (A + A) * F;
	double t_2 = C * (A * 4.0);
	double t_3 = sqrt((((C + A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))) * ((F * (pow(B, 2.0) - t_2)) * 2.0))) / (t_2 - pow(B, 2.0));
	double tmp;
	if (t_3 <= -1e+135) {
		tmp = (-sqrt((t_0 * 2.0)) * sqrt(t_1)) / t_0;
	} else if (t_3 <= -1e-173) {
		tmp = sqrt(((((C + A) - sqrt(fma((A - C), (A - C), (B * B)))) / fma((-4.0 * A), C, (B * B))) * F)) * -sqrt(2.0);
	} else {
		tmp = sqrt((((t_1 * C) * A) * -8.0)) / ((C * A) * 4.0);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B * B))
	t_1 = Float64(Float64(A + A) * F)
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))) * Float64(Float64(F * Float64((B ^ 2.0) - t_2)) * 2.0))) / Float64(t_2 - (B ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -1e+135)
		tmp = Float64(Float64(Float64(-sqrt(Float64(t_0 * 2.0))) * sqrt(t_1)) / t_0);
	elseif (t_3 <= -1e-173)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))) / fma(Float64(-4.0 * A), C, Float64(B * B))) * F)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(t_1 * C) * A) * -8.0)) / Float64(Float64(C * A) * 4.0));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+135], N[(N[((-N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$3, -1e-173], N[(N[Sqrt[N[(N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(t$95$1 * C), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision]], $MachinePrecision] / N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999962e134

   1. Initial program 14.3%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 31.6

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

   5. Applied rewrites 31.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

   7. Applied rewrites 38.3%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 38.3

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

   9. Applied rewrites 38.3%

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

   if -9.99999999999999962e134 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-173

   1. Initial program 98.4%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 95.1%

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

   if -1e-173 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 8.0%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 15.1

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

   5. Applied rewrites 15.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 15.1%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 15.4

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

   10. Applied rewrites 15.4%

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

   11. Taylor expanded in C around inf

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]
   12. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. rem-square-sqrt N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower--.f64 N/A

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

       10. mul-1-neg N/A

           \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{\left(\mathsf{neg}\left(A\right)\right)}\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)} \]

       11. lower-neg.f64 17.5

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

   13. Applied rewrites 17.5%

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\left(-\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2}\right) \cdot \sqrt{\left(A + A\right) \cdot F}}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{\left(C \cdot A\right) \cdot 4}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 28.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_0\right) \cdot \left(A + A\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B B))))
   (if (<= (pow B 2.0) 5e+53)
     (/ (sqrt (* (* (* F 2.0) t_0) (+ A A))) (- t_0))
     (* (sqrt (* (- A (sqrt (fma B B (* A A)))) F)) (/ (- (sqrt 2.0)) B)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B * B));
	double tmp;
	if (pow(B, 2.0) <= 5e+53) {
		tmp = sqrt((((F * 2.0) * t_0) * (A + A))) / -t_0;
	} else {
		tmp = sqrt(((A - sqrt(fma(B, B, (A * A)))) * F)) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B * B))
	tmp = 0.0
	if ((B ^ 2.0) <= 5e+53)
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_0) * Float64(A + A))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(A - sqrt(fma(B, B, Float64(A * A)))) * F)) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e+53], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000004e53

   1. Initial program 26.6%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 33.2

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

   5. Applied rewrites 33.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 33.2%

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

   if 5.0000000000000004e53 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.1%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 7.8

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

   5. Applied rewrites 7.8%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right) \cdot \left(A + A\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F\right) \cdot 2\right) \cdot \left(A + A\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 6.2e+20)
   (/
    (sqrt (* (* (* (fma (* -4.0 C) A (* B B)) F) 2.0) (+ A A)))
    (* C (* A 4.0)))
   (* (sqrt (* (- A (sqrt (fma B B (* A A)))) F)) (/ (- (sqrt 2.0)) B))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 6.2e+20) {
		tmp = sqrt((((fma((-4.0 * C), A, (B * B)) * F) * 2.0) * (A + A))) / (C * (A * 4.0));
	} else {
		tmp = sqrt(((A - sqrt(fma(B, B, (A * A)))) * F)) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 6.2e+20)
		tmp = Float64(sqrt(Float64(Float64(Float64(fma(Float64(-4.0 * C), A, Float64(B * B)) * F) * 2.0) * Float64(A + A))) / Float64(C * Float64(A * 4.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(A - sqrt(fma(B, B, Float64(A * A)))) * F)) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 6.2e+20], N[(N[Sqrt[N[(N[(N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.2e20

   1. Initial program 25.2%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 25.2

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

   5. Applied rewrites 25.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 25.2%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 23.1

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

   10. Applied rewrites 23.1%

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

   12. Applied rewrites 23.1%

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

   if 6.2e20 < B

   1. Initial program 7.6%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 13.4

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

   5. Applied rewrites 13.4%

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

4. Final simplification 20.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F\right) \cdot 2\right) \cdot \left(A + A\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 24.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;B \leq 6.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{\left(\left(t\_0 \cdot F\right) \cdot 2\right) \cdot \left(A + A\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 1.26 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right) \cdot -2}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{\left(C \cdot A\right) \cdot 4}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B B))))
   (if (<= B 6.2e+20)
     (/ (sqrt (* (* (* t_0 F) 2.0) (+ A A))) (* C (* A 4.0)))
     (if (<= B 1.26e+91)
       (/ (sqrt (* (* (* (* B B) B) F) -2.0)) (- t_0))
       (/ (sqrt (* (* (* (* (+ A A) F) C) A) -8.0)) (* (* C A) 4.0))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B * B));
	double tmp;
	if (B <= 6.2e+20) {
		tmp = sqrt((((t_0 * F) * 2.0) * (A + A))) / (C * (A * 4.0));
	} else if (B <= 1.26e+91) {
		tmp = sqrt(((((B * B) * B) * F) * -2.0)) / -t_0;
	} else {
		tmp = sqrt((((((A + A) * F) * C) * A) * -8.0)) / ((C * A) * 4.0);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B * B))
	tmp = 0.0
	if (B <= 6.2e+20)
		tmp = Float64(sqrt(Float64(Float64(Float64(t_0 * F) * 2.0) * Float64(A + A))) / Float64(C * Float64(A * 4.0)));
	elseif (B <= 1.26e+91)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(B * B) * B) * F) * -2.0)) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(A + A) * F) * C) * A) * -8.0)) / Float64(Float64(C * A) * 4.0));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 6.2e+20], N[(N[Sqrt[N[(N[(N[(t$95$0 * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.26e+91], N[(N[Sqrt[N[(N[(N[(N[(B * B), $MachinePrecision] * B), $MachinePrecision] * F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision] * C), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision]], $MachinePrecision] / N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.2e20

   1. Initial program 25.2%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 25.2

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

   5. Applied rewrites 25.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 25.2%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 23.1

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

   10. Applied rewrites 23.1%

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

   12. Applied rewrites 23.1%

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

   if 6.2e20 < B < 1.26e91

   1. Initial program 29.9%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 15.6

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

   5. Applied rewrites 15.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 15.6%

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

   8. Taylor expanded in B around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 20.7

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

   10. Applied rewrites 20.7%

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

   if 1.26e91 < B

   1. Initial program 0.6%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 0.8

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

   5. Applied rewrites 0.8%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 1.2

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

   10. Applied rewrites 1.2%

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

   11. Taylor expanded in C around inf

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]
   12. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. rem-square-sqrt N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower--.f64 N/A

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

       10. mul-1-neg N/A

           \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{\left(\mathsf{neg}\left(A\right)\right)}\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)} \]

       11. lower-neg.f64 6.1

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

   13. Applied rewrites 6.1%

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 20.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F\right) \cdot 2\right) \cdot \left(A + A\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 1.26 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right) \cdot -2}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{\left(C \cdot A\right) \cdot 4}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 24.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \left(C \cdot A\right) \cdot 4\\ t_1 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;B \leq 6.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_1\right) \cdot \left(A + A\right)}}{t\_0}\\ \mathbf{elif}\;B \leq 1.26 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right) \cdot -2}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (* C A) 4.0)) (t_1 (fma (* -4.0 C) A (* B B))))
   (if (<= B 6.2e+20)
     (/ (sqrt (* (* (* F 2.0) t_1) (+ A A))) t_0)
     (if (<= B 1.26e+91)
       (/ (sqrt (* (* (* (* B B) B) F) -2.0)) (- t_1))
       (/ (sqrt (* (* (* (* (+ A A) F) C) A) -8.0)) t_0)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = (C * A) * 4.0;
	double t_1 = fma((-4.0 * C), A, (B * B));
	double tmp;
	if (B <= 6.2e+20) {
		tmp = sqrt((((F * 2.0) * t_1) * (A + A))) / t_0;
	} else if (B <= 1.26e+91) {
		tmp = sqrt(((((B * B) * B) * F) * -2.0)) / -t_1;
	} else {
		tmp = sqrt((((((A + A) * F) * C) * A) * -8.0)) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(C * A) * 4.0)
	t_1 = fma(Float64(-4.0 * C), A, Float64(B * B))
	tmp = 0.0
	if (B <= 6.2e+20)
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_1) * Float64(A + A))) / t_0);
	elseif (B <= 1.26e+91)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(B * B) * B) * F) * -2.0)) / Float64(-t_1));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(A + A) * F) * C) * A) * -8.0)) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 6.2e+20], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 1.26e+91], N[(N[Sqrt[N[(N[(N[(N[(B * B), $MachinePrecision] * B), $MachinePrecision] * F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision] * C), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.2e20

   1. Initial program 25.2%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 25.2

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

   5. Applied rewrites 25.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 25.2%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 23.1

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

   10. Applied rewrites 23.1%

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

   if 6.2e20 < B < 1.26e91

   1. Initial program 29.9%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 15.6

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

   5. Applied rewrites 15.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 15.6%

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

   8. Taylor expanded in B around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 20.7

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

   10. Applied rewrites 20.7%

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

   if 1.26e91 < B

   1. Initial program 0.6%

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

   3. Taylor expanded in C around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 0.8

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

   5. Applied rewrites 0.8%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 1.2

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

   10. Applied rewrites 1.2%

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

   11. Taylor expanded in C around inf

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]
   12. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. rem-square-sqrt N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower--.f64 N/A

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

       10. mul-1-neg N/A

           \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{\left(\mathsf{neg}\left(A\right)\right)}\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)} \]

       11. lower-neg.f64 6.1

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

   13. Applied rewrites 6.1%

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 19.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right) \cdot \left(A + A\right)}}{\left(C \cdot A\right) \cdot 4}\\ \mathbf{elif}\;B \leq 1.26 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right) \cdot -2}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{\left(C \cdot A\right) \cdot 4}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 24.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{\left(C \cdot A\right) \cdot 4} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (/ (sqrt (* (* (* (* (+ A A) F) C) A) -8.0)) (* (* C A) 4.0)))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return sqrt((((((A + A) * F) * C) * A) * -8.0)) / ((C * A) * 4.0);
}

Fortran

NOTE: A, B, C, and F 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 = sqrt((((((a + a) * f) * c) * a) * (-8.0d0))) / ((c * a) * 4.0d0)
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return Math.sqrt((((((A + A) * F) * C) * A) * -8.0)) / ((C * A) * 4.0);
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return math.sqrt((((((A + A) * F) * C) * A) * -8.0)) / ((C * A) * 4.0)

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(sqrt(Float64(Float64(Float64(Float64(Float64(A + A) * F) * C) * A) * -8.0)) / Float64(Float64(C * A) * 4.0))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = sqrt((((((A + A) * F) * C) * A) * -8.0)) / ((C * A) * 4.0);
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(N[Sqrt[N[(N[(N[(N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision] * C), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision]], $MachinePrecision] / N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.1%

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

3. Taylor expanded in C around inf

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

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower-+.f64 20.4

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

5. Applied rewrites 20.4%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 20.4%

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

8. Taylor expanded in C around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 18.1

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

10. Applied rewrites 18.1%

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

11. Taylor expanded in C around inf

    \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]
12. Step-by-step derivation

    1. rem-square-sqrt N/A

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

    2. unpow2 N/A

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

    3. lower-*.f64 N/A

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

    4. unpow2 N/A

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

    5. rem-square-sqrt N/A

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

    6. lower-*.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. lower--.f64 N/A

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

    10. mul-1-neg N/A

        \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{\left(\mathsf{neg}\left(A\right)\right)}\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)} \]

    11. lower-neg.f64 19.1

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

13. Applied rewrites 19.1%

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

14. Final simplification 19.1%

    \[\leadsto \frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{\left(C \cdot A\right) \cdot 4} \]
15. Add Preprocessing

Alternative 14: 18.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \frac{\sqrt{\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16}}{\left(C \cdot A\right) \cdot 4} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (/ (sqrt (* (* (* (* A A) C) F) -16.0)) (* (* C A) 4.0)))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return sqrt(((((A * A) * C) * F) * -16.0)) / ((C * A) * 4.0);
}

Fortran

NOTE: A, B, C, and F 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 = sqrt(((((a * a) * c) * f) * (-16.0d0))) / ((c * a) * 4.0d0)
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return Math.sqrt(((((A * A) * C) * F) * -16.0)) / ((C * A) * 4.0);
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return math.sqrt(((((A * A) * C) * F) * -16.0)) / ((C * A) * 4.0)

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(sqrt(Float64(Float64(Float64(Float64(A * A) * C) * F) * -16.0)) / Float64(Float64(C * A) * 4.0))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = sqrt(((((A * A) * C) * F) * -16.0)) / ((C * A) * 4.0);
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(N[Sqrt[N[(N[(N[(N[(A * A), $MachinePrecision] * C), $MachinePrecision] * F), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] / N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.1%

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

3. Taylor expanded in C around inf

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

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower-+.f64 20.4

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

5. Applied rewrites 20.4%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 20.4%

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

8. Taylor expanded in C around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 18.1

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

10. Applied rewrites 18.1%

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

11. Taylor expanded in A around -inf

    \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]

    2. associate-*r* N/A

       \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}}}{4 \cdot \left(A \cdot C\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}}}{4 \cdot \left(A \cdot C\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt{-16 \cdot \left(\color{blue}{\left({A}^{2} \cdot C\right)} \cdot F\right)}}{4 \cdot \left(A \cdot C\right)} \]

    5. unpow2 N/A

       \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)}}{4 \cdot \left(A \cdot C\right)} \]

    6. lower-*.f64 14.8

       \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)}}{4 \cdot \left(A \cdot C\right)} \]

13. Applied rewrites 14.8%

    \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{4 \cdot \left(A \cdot C\right)} \]

14. Final simplification 14.8%

    \[\leadsto \frac{\sqrt{\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16}}{\left(C \cdot A\right) \cdot 4} \]
15. Add Preprocessing

Alternative 15: 1.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16}}{\left(C \cdot A\right) \cdot 4} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (/ (sqrt (* (* (* (* C C) F) A) -16.0)) (* (* C A) 4.0)))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return sqrt(((((C * C) * F) * A) * -16.0)) / ((C * A) * 4.0);
}

Fortran

NOTE: A, B, C, and F 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 = sqrt(((((c * c) * f) * a) * (-16.0d0))) / ((c * a) * 4.0d0)
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return Math.sqrt(((((C * C) * F) * A) * -16.0)) / ((C * A) * 4.0);
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return math.sqrt(((((C * C) * F) * A) * -16.0)) / ((C * A) * 4.0)

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(sqrt(Float64(Float64(Float64(Float64(C * C) * F) * A) * -16.0)) / Float64(Float64(C * A) * 4.0))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = sqrt(((((C * C) * F) * A) * -16.0)) / ((C * A) * 4.0);
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision] * A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] / N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.1%

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

3. Taylor expanded in C around inf

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

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower-+.f64 20.4

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

5. Applied rewrites 20.4%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 20.4%

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

8. Taylor expanded in C around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 18.1

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

10. Applied rewrites 18.1%

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

11. Taylor expanded in C around -inf

    \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \color{blue}{\left({C}^{2} \cdot F\right)}\right)}}{4 \cdot \left(A \cdot C\right)} \]

    4. unpow2 N/A

       \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)} \]

    5. lower-*.f64 8.7

       \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)} \]

13. Applied rewrites 8.7%

    \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{4 \cdot \left(A \cdot C\right)} \]

14. Final simplification 8.7%

    \[\leadsto \frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16}}{\left(C \cdot A\right) \cdot 4} \]
15. Add Preprocessing

Alternative 16: 2.0% accurate, 14.9× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \sqrt{\frac{2}{\frac{B}{F}}} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (sqrt (/ 2.0 (/ B F))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return sqrt((2.0 / (B / F)));
}

Fortran

NOTE: A, B, C, and F 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 = sqrt((2.0d0 / (b / f)))
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return Math.sqrt((2.0 / (B / F)));
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return math.sqrt((2.0 / (B / F)))

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return sqrt(Float64(2.0 / Float64(B / F)))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = sqrt((2.0 / (B / F)));
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[Sqrt[N[(2.0 / N[(B / F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 21.1%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 1.8

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

5. Applied rewrites 1.8%

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

7. Applied rewrites 1.8%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

9. Applied rewrites 1.8%

   \[\leadsto \sqrt{\frac{2}{\frac{B}{F}}} \]
10. Add Preprocessing

Alternative 17: 1.9% accurate, 18.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \sqrt{\frac{2}{B} \cdot F} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (sqrt (* (/ 2.0 B) F)))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return sqrt(((2.0 / B) * F));
}

Fortran

NOTE: A, B, C, and F 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 = sqrt(((2.0d0 / b) * f))
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return Math.sqrt(((2.0 / B) * F));
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return math.sqrt(((2.0 / B) * F))

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return sqrt(Float64(Float64(2.0 / B) * F))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = sqrt(((2.0 / B) * F));
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[Sqrt[N[(N[(2.0 / B), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 21.1%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 1.8

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

5. Applied rewrites 1.8%

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

7. Applied rewrites 1.8%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

9. Applied rewrites 1.8%

   \[\leadsto \sqrt{F \cdot \frac{2}{B}} \]

10. Final simplification 1.8%

    \[\leadsto \sqrt{\frac{2}{B} \cdot F} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024236 
(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))))