ABCF->ab-angle b

Percentage Accurate: 19.1% → 40.2%

Time: 19.4s

Alternatives: 19

Speedup: 6.1×

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: 19.1% 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: 40.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ t_1 := \left(F \cdot \left({B}^{2} - t\_0\right)\right) \cdot 2\\ t_2 := t\_0 - {B}^{2}\\ t_3 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot t\_1}}{t\_2}\\ t_4 := \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot {\left(\left(F \cdot 2\right) \cdot t\_4\right)}^{0.5}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_4 \cdot t\_1}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \sqrt{\frac{1}{A}}\right) \cdot 0.25}{C} \cdot \sqrt{\mathsf{fma}\left(\left(-4 \cdot C\right) \cdot \left(F \cdot 2\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}\\ \end{array} \end{array} \]

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 (/ (* B B) C) -0.5 A) A)))
   (if (<= t_3 (- INFINITY))
     (/
      (* (sqrt (fma (* C A) -4.0 (* B B))) (pow (* (* F 2.0) t_4) 0.5))
      (- (fma (* -4.0 C) A (* B B))))
     (if (<= t_3 -1e-191)
       (/
        (*
         (sqrt (* (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B)))) F))
         (sqrt (* (fma -4.0 (* C A) (* B B)) 2.0)))
        t_2)
       (if (<= t_3 INFINITY)
         (/ (sqrt (* t_4 t_1)) t_2)
         (*
          (/ (* (* (sqrt 2.0) (sqrt (/ 1.0 A))) 0.25) C)
          (sqrt (fma (* (* -4.0 C) (* F 2.0)) A (* (* (* F 2.0) B) B)))))))))

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(((B * B) / C), -0.5, A) + A;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt(fma((C * A), -4.0, (B * B))) * pow(((F * 2.0) * t_4), 0.5)) / -fma((-4.0 * C), A, (B * B));
	} else if (t_3 <= -1e-191) {
		tmp = (sqrt((((C + A) - sqrt(fma((A - C), (A - C), (B * B)))) * F)) * sqrt((fma(-4.0, (C * A), (B * B)) * 2.0))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_4 * t_1)) / t_2;
	} else {
		tmp = (((sqrt(2.0) * sqrt((1.0 / A))) * 0.25) / C) * sqrt(fma(((-4.0 * C) * (F * 2.0)), A, (((F * 2.0) * B) * B)));
	}
	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 = Float64(fma(Float64(Float64(B * B) / C), -0.5, A) + A)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(fma(Float64(C * A), -4.0, Float64(B * B))) * (Float64(Float64(F * 2.0) * t_4) ^ 0.5)) / Float64(-fma(Float64(-4.0 * C), A, Float64(B * B))));
	elseif (t_3 <= -1e-191)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))) * F)) * sqrt(Float64(fma(-4.0, Float64(C * A), Float64(B * B)) * 2.0))) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(t_4 * t_1)) / t_2);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 / A))) * 0.25) / C) * sqrt(fma(Float64(Float64(-4.0 * C) * Float64(F * 2.0)), A, Float64(Float64(Float64(F * 2.0) * B) * 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[(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[(N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(F * 2.0), $MachinePrecision] * t$95$4), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / (-N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, -1e-191], N[(N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(t$95$4 * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] / C), $MachinePrecision] * N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * A + N[(N[(N[(F * 2.0), $MachinePrecision] * B), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 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 3.2%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 19.4

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

   5. Applied rewrites 19.4%

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

   7. Applied rewrites 19.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

   9. Applied rewrites 32.5%

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

   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))) < -1e-191

   1. Initial program 99.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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 99.2%

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

   if -1e-191 < (/.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 20.0%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 19.3

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

   5. Applied rewrites 19.3%

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

   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)))

   1. Initial program 0.0%

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around inf

      \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{16} \cdot \left(\frac{{B}^{2}}{{A}^{2}} \cdot \sqrt{A - -1 \cdot A}\right) + \frac{1}{16} \cdot \left(\frac{{B}^{2}}{A} \cdot \sqrt{\frac{1}{A - -1 \cdot A}}\right)}{C} - \frac{-1}{4} \cdot \left(\frac{1}{A} \cdot \sqrt{A - -1 \cdot A}\right)}{C}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{16} \cdot \left(\frac{{B}^{2}}{{A}^{2}} \cdot \sqrt{A - -1 \cdot A}\right) + \frac{1}{16} \cdot \left(\frac{{B}^{2}}{A} \cdot \sqrt{\frac{1}{A - -1 \cdot A}}\right)}{C} - \frac{-1}{4} \cdot \left(\frac{1}{A} \cdot \sqrt{A - -1 \cdot A}\right)}{C}} \]

   7. Applied rewrites 2.1%

      \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(-0.0625, \frac{B \cdot B}{A \cdot A} \cdot \sqrt{A - \left(-A\right)}, 0.0625 \cdot \left(\frac{B \cdot B}{A} \cdot \sqrt{\frac{1}{A - \left(-A\right)}}\right)\right)}{C}\right) - -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{A - \left(-A\right)}\right)}{C}} \]

   8. Taylor expanded in C around inf

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

   10. Applied rewrites 3.8%

       \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \sqrt{2}\right)}{C} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*r* N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lift-*.f64 N/A

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 N/A

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

       23. associate-*r* N/A

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

       24. lower-*.f64 N/A

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

   12. Applied rewrites 4.7%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(2 \cdot F\right) \cdot \left(C \cdot -4\right), A, \left(\left(2 \cdot F\right) \cdot B\right) \cdot B\right)}} \cdot \frac{0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \sqrt{2}\right)}{C} \]
3. Recombined 4 regimes into one program.

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot {\left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)}^{0.5}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \sqrt{\frac{1}{A}}\right) \cdot 0.25}{C} \cdot \sqrt{\mathsf{fma}\left(\left(-4 \cdot C\right) \cdot \left(F \cdot 2\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 40.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ t_1 := \frac{B \cdot B}{C}\\ t_2 := t\_0 - {B}^{2}\\ t_3 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_0\right)\right) \cdot 2\right)}}{t\_2}\\ t_4 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_5 := -t\_4\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot {\left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(t\_1, -0.5, A\right) + A\right)\right)}^{0.5}}{t\_5}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, t\_1, A\right) + A\right) \cdot \left(t\_4 \cdot \left(F \cdot 2\right)\right)}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \sqrt{\frac{1}{A}}\right) \cdot 0.25}{C} \cdot \sqrt{\mathsf{fma}\left(\left(-4 \cdot C\right) \cdot \left(F \cdot 2\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}\\ \end{array} \end{array} \]

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 (/ (* B B) C))
        (t_2 (- t_0 (pow B 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))
            (* (* F (- (pow B 2.0) t_0)) 2.0)))
          t_2))
        (t_4 (fma (* -4.0 C) A (* B B)))
        (t_5 (- t_4)))
   (if (<= t_3 (- INFINITY))
     (/
      (*
       (sqrt (fma (* C A) -4.0 (* B B)))
       (pow (* (* F 2.0) (+ (fma t_1 -0.5 A) A)) 0.5))
      t_5)
     (if (<= t_3 -1e-191)
       (/
        (*
         (sqrt (* (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B)))) F))
         (sqrt (* (fma -4.0 (* C A) (* B B)) 2.0)))
        t_2)
       (if (<= t_3 INFINITY)
         (/ (sqrt (* (+ (fma -0.5 t_1 A) A) (* t_4 (* F 2.0)))) t_5)
         (*
          (/ (* (* (sqrt 2.0) (sqrt (/ 1.0 A))) 0.25) C)
          (sqrt (fma (* (* -4.0 C) (* F 2.0)) A (* (* (* F 2.0) B) B)))))))))

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 = (B * B) / C;
	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)))) * ((F * (pow(B, 2.0) - t_0)) * 2.0))) / t_2;
	double t_4 = fma((-4.0 * C), A, (B * B));
	double t_5 = -t_4;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt(fma((C * A), -4.0, (B * B))) * pow(((F * 2.0) * (fma(t_1, -0.5, A) + A)), 0.5)) / t_5;
	} else if (t_3 <= -1e-191) {
		tmp = (sqrt((((C + A) - sqrt(fma((A - C), (A - C), (B * B)))) * F)) * sqrt((fma(-4.0, (C * A), (B * B)) * 2.0))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((fma(-0.5, t_1, A) + A) * (t_4 * (F * 2.0)))) / t_5;
	} else {
		tmp = (((sqrt(2.0) * sqrt((1.0 / A))) * 0.25) / C) * sqrt(fma(((-4.0 * C) * (F * 2.0)), A, (((F * 2.0) * B) * B)));
	}
	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(B * B) / C)
	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)))) * Float64(Float64(F * Float64((B ^ 2.0) - t_0)) * 2.0))) / t_2)
	t_4 = fma(Float64(-4.0 * C), A, Float64(B * B))
	t_5 = Float64(-t_4)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(fma(Float64(C * A), -4.0, Float64(B * B))) * (Float64(Float64(F * 2.0) * Float64(fma(t_1, -0.5, A) + A)) ^ 0.5)) / t_5);
	elseif (t_3 <= -1e-191)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))) * F)) * sqrt(Float64(fma(-4.0, Float64(C * A), Float64(B * B)) * 2.0))) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(fma(-0.5, t_1, A) + A) * Float64(t_4 * Float64(F * 2.0)))) / t_5);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 / A))) * 0.25) / C) * sqrt(fma(Float64(Float64(-4.0 * C) * Float64(F * 2.0)), A, Float64(Float64(Float64(F * 2.0) * B) * 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[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] / C), $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] * N[(N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-t$95$4)}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(F * 2.0), $MachinePrecision] * N[(N[(t$95$1 * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$3, -1e-191], N[(N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(N[(-0.5 * t$95$1 + A), $MachinePrecision] + A), $MachinePrecision] * N[(t$95$4 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] / C), $MachinePrecision] * N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * A + N[(N[(N[(F * 2.0), $MachinePrecision] * B), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 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 3.2%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 19.4

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

   5. Applied rewrites 19.4%

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

   7. Applied rewrites 19.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

   9. Applied rewrites 32.5%

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

   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))) < -1e-191

   1. Initial program 99.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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 99.2%

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

   if -1e-191 < (/.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 20.0%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 19.3

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.3%

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

   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)))

   1. Initial program 0.0%

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around inf

      \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{16} \cdot \left(\frac{{B}^{2}}{{A}^{2}} \cdot \sqrt{A - -1 \cdot A}\right) + \frac{1}{16} \cdot \left(\frac{{B}^{2}}{A} \cdot \sqrt{\frac{1}{A - -1 \cdot A}}\right)}{C} - \frac{-1}{4} \cdot \left(\frac{1}{A} \cdot \sqrt{A - -1 \cdot A}\right)}{C}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{16} \cdot \left(\frac{{B}^{2}}{{A}^{2}} \cdot \sqrt{A - -1 \cdot A}\right) + \frac{1}{16} \cdot \left(\frac{{B}^{2}}{A} \cdot \sqrt{\frac{1}{A - -1 \cdot A}}\right)}{C} - \frac{-1}{4} \cdot \left(\frac{1}{A} \cdot \sqrt{A - -1 \cdot A}\right)}{C}} \]

   7. Applied rewrites 2.1%

      \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(-0.0625, \frac{B \cdot B}{A \cdot A} \cdot \sqrt{A - \left(-A\right)}, 0.0625 \cdot \left(\frac{B \cdot B}{A} \cdot \sqrt{\frac{1}{A - \left(-A\right)}}\right)\right)}{C}\right) - -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{A - \left(-A\right)}\right)}{C}} \]

   8. Taylor expanded in C around inf

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

   10. Applied rewrites 3.8%

       \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \sqrt{2}\right)}{C} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*r* N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lift-*.f64 N/A

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 N/A

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

       23. associate-*r* N/A

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

       24. lower-*.f64 N/A

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

   12. Applied rewrites 4.7%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(2 \cdot F\right) \cdot \left(C \cdot -4\right), A, \left(\left(2 \cdot F\right) \cdot B\right) \cdot B\right)}} \cdot \frac{0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \sqrt{2}\right)}{C} \]
3. Recombined 4 regimes into one program.

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot {\left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)}^{0.5}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \sqrt{\frac{1}{A}}\right) \cdot 0.25}{C} \cdot \sqrt{\mathsf{fma}\left(\left(-4 \cdot C\right) \cdot \left(F \cdot 2\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 40.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B}^{2}}\\ t_3 := \frac{B \cdot B}{C}\\ t_4 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_5 := -t\_4\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot {\left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(t\_3, -0.5, A\right) + A\right)\right)}^{0.5}}{t\_5}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-t\_0}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, t\_3, A\right) + A\right) \cdot \left(t\_4 \cdot \left(F \cdot 2\right)\right)}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \sqrt{\frac{1}{A}}\right) \cdot 0.25}{C} \cdot \sqrt{\mathsf{fma}\left(\left(-4 \cdot C\right) \cdot \left(F \cdot 2\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}\\ \end{array} \end{array} \]

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 (/ (* B B) C))
        (t_4 (fma (* -4.0 C) A (* B B)))
        (t_5 (- t_4)))
   (if (<= t_2 (- INFINITY))
     (/
      (*
       (sqrt (fma (* C A) -4.0 (* B B)))
       (pow (* (* F 2.0) (+ (fma t_3 -0.5 A) A)) 0.5))
      t_5)
     (if (<= t_2 -1e-191)
       (/
        (sqrt
         (*
          (* t_0 (* F 2.0))
          (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B))))))
        (- t_0))
       (if (<= t_2 INFINITY)
         (/ (sqrt (* (+ (fma -0.5 t_3 A) A) (* t_4 (* F 2.0)))) t_5)
         (*
          (/ (* (* (sqrt 2.0) (sqrt (/ 1.0 A))) 0.25) C)
          (sqrt (fma (* (* -4.0 C) (* F 2.0)) A (* (* (* F 2.0) B) 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 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 = (B * B) / C;
	double t_4 = fma((-4.0 * C), A, (B * B));
	double t_5 = -t_4;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (sqrt(fma((C * A), -4.0, (B * B))) * pow(((F * 2.0) * (fma(t_3, -0.5, A) + A)), 0.5)) / t_5;
	} else if (t_2 <= -1e-191) {
		tmp = sqrt(((t_0 * (F * 2.0)) * ((C + A) - sqrt(fma((A - C), (A - C), (B * B)))))) / -t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((fma(-0.5, t_3, A) + A) * (t_4 * (F * 2.0)))) / t_5;
	} else {
		tmp = (((sqrt(2.0) * sqrt((1.0 / A))) * 0.25) / C) * sqrt(fma(((-4.0 * C) * (F * 2.0)), A, (((F * 2.0) * B) * B)));
	}
	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 = 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(Float64(B * B) / C)
	t_4 = fma(Float64(-4.0 * C), A, Float64(B * B))
	t_5 = Float64(-t_4)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(fma(Float64(C * A), -4.0, Float64(B * B))) * (Float64(Float64(F * 2.0) * Float64(fma(t_3, -0.5, A) + A)) ^ 0.5)) / t_5);
	elseif (t_2 <= -1e-191)
		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));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(fma(-0.5, t_3, A) + A) * Float64(t_4 * Float64(F * 2.0)))) / t_5);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 / A))) * 0.25) / C) * sqrt(fma(Float64(Float64(-4.0 * C) * Float64(F * 2.0)), A, Float64(Float64(Float64(F * 2.0) * B) * 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[(-4.0 * N[(C * A), $MachinePrecision] + 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 = N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-t$95$4)}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[Sqrt[N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(F * 2.0), $MachinePrecision] * N[(N[(t$95$3 * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$2, -1e-191], 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], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(N[(-0.5 * t$95$3 + A), $MachinePrecision] + A), $MachinePrecision] * N[(t$95$4 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] / C), $MachinePrecision] * N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * A + N[(N[(N[(F * 2.0), $MachinePrecision] * B), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 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 3.2%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 19.4

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

   5. Applied rewrites 19.4%

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

   7. Applied rewrites 19.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

   9. Applied rewrites 32.5%

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

   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))) < -1e-191

   1. Initial program 99.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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 99.0%

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

   if -1e-191 < (/.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 20.0%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 19.3

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.3%

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

   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)))

   1. Initial program 0.0%

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around inf

      \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{16} \cdot \left(\frac{{B}^{2}}{{A}^{2}} \cdot \sqrt{A - -1 \cdot A}\right) + \frac{1}{16} \cdot \left(\frac{{B}^{2}}{A} \cdot \sqrt{\frac{1}{A - -1 \cdot A}}\right)}{C} - \frac{-1}{4} \cdot \left(\frac{1}{A} \cdot \sqrt{A - -1 \cdot A}\right)}{C}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{16} \cdot \left(\frac{{B}^{2}}{{A}^{2}} \cdot \sqrt{A - -1 \cdot A}\right) + \frac{1}{16} \cdot \left(\frac{{B}^{2}}{A} \cdot \sqrt{\frac{1}{A - -1 \cdot A}}\right)}{C} - \frac{-1}{4} \cdot \left(\frac{1}{A} \cdot \sqrt{A - -1 \cdot A}\right)}{C}} \]

   7. Applied rewrites 2.1%

      \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(-0.0625, \frac{B \cdot B}{A \cdot A} \cdot \sqrt{A - \left(-A\right)}, 0.0625 \cdot \left(\frac{B \cdot B}{A} \cdot \sqrt{\frac{1}{A - \left(-A\right)}}\right)\right)}{C}\right) - -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{A - \left(-A\right)}\right)}{C}} \]

   8. Taylor expanded in C around inf

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

   10. Applied rewrites 3.8%

       \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \sqrt{2}\right)}{C} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*r* N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lift-*.f64 N/A

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 N/A

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

       23. associate-*r* N/A

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

       24. lower-*.f64 N/A

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

   12. Applied rewrites 4.7%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(2 \cdot F\right) \cdot \left(C \cdot -4\right), A, \left(\left(2 \cdot F\right) \cdot B\right) \cdot B\right)}} \cdot \frac{0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \sqrt{2}\right)}{C} \]
3. Recombined 4 regimes into one program.

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot {\left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)}^{0.5}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \sqrt{\frac{1}{A}}\right) \cdot 0.25}{C} \cdot \sqrt{\mathsf{fma}\left(\left(-4 \cdot C\right) \cdot \left(F \cdot 2\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 40.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B}^{2}}\\ t_3 := \frac{B \cdot B}{C}\\ t_4 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_5 := -t\_4\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot {\left(\left(\mathsf{fma}\left(t\_3, -0.5, A\right) + A\right) \cdot F\right)}^{0.5}}{t\_5}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-t\_0}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, t\_3, A\right) + A\right) \cdot \left(t\_4 \cdot \left(F \cdot 2\right)\right)}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \sqrt{\frac{1}{A}}\right) \cdot 0.25}{C} \cdot \sqrt{\mathsf{fma}\left(\left(-4 \cdot C\right) \cdot \left(F \cdot 2\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}\\ \end{array} \end{array} \]

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 (/ (* B B) C))
        (t_4 (fma (* -4.0 C) A (* B B)))
        (t_5 (- t_4)))
   (if (<= t_2 (- INFINITY))
     (/
      (*
       (sqrt (* (fma (* C A) -4.0 (* B B)) 2.0))
       (pow (* (+ (fma t_3 -0.5 A) A) F) 0.5))
      t_5)
     (if (<= t_2 -1e-191)
       (/
        (sqrt
         (*
          (* t_0 (* F 2.0))
          (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B))))))
        (- t_0))
       (if (<= t_2 INFINITY)
         (/ (sqrt (* (+ (fma -0.5 t_3 A) A) (* t_4 (* F 2.0)))) t_5)
         (*
          (/ (* (* (sqrt 2.0) (sqrt (/ 1.0 A))) 0.25) C)
          (sqrt (fma (* (* -4.0 C) (* F 2.0)) A (* (* (* F 2.0) B) 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 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 = (B * B) / C;
	double t_4 = fma((-4.0 * C), A, (B * B));
	double t_5 = -t_4;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (sqrt((fma((C * A), -4.0, (B * B)) * 2.0)) * pow(((fma(t_3, -0.5, A) + A) * F), 0.5)) / t_5;
	} else if (t_2 <= -1e-191) {
		tmp = sqrt(((t_0 * (F * 2.0)) * ((C + A) - sqrt(fma((A - C), (A - C), (B * B)))))) / -t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((fma(-0.5, t_3, A) + A) * (t_4 * (F * 2.0)))) / t_5;
	} else {
		tmp = (((sqrt(2.0) * sqrt((1.0 / A))) * 0.25) / C) * sqrt(fma(((-4.0 * C) * (F * 2.0)), A, (((F * 2.0) * B) * B)));
	}
	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 = 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(Float64(B * B) / C)
	t_4 = fma(Float64(-4.0 * C), A, Float64(B * B))
	t_5 = Float64(-t_4)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B * B)) * 2.0)) * (Float64(Float64(fma(t_3, -0.5, A) + A) * F) ^ 0.5)) / t_5);
	elseif (t_2 <= -1e-191)
		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));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(fma(-0.5, t_3, A) + A) * Float64(t_4 * Float64(F * 2.0)))) / t_5);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 / A))) * 0.25) / C) * sqrt(fma(Float64(Float64(-4.0 * C) * Float64(F * 2.0)), A, Float64(Float64(Float64(F * 2.0) * B) * 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[(-4.0 * N[(C * A), $MachinePrecision] + 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 = N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-t$95$4)}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(N[(t$95$3 * -0.5 + A), $MachinePrecision] + A), $MachinePrecision] * F), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$2, -1e-191], 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], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(N[(-0.5 * t$95$3 + A), $MachinePrecision] + A), $MachinePrecision] * N[(t$95$4 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] / C), $MachinePrecision] * N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * A + N[(N[(N[(F * 2.0), $MachinePrecision] * B), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 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 3.2%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 19.4

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

   5. Applied rewrites 19.4%

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

   7. Applied rewrites 19.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

   9. Applied rewrites 32.5%

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

   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))) < -1e-191

   1. Initial program 99.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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 99.0%

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

   if -1e-191 < (/.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 20.0%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 19.3

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.3%

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

   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)))

   1. Initial program 0.0%

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around inf

      \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{16} \cdot \left(\frac{{B}^{2}}{{A}^{2}} \cdot \sqrt{A - -1 \cdot A}\right) + \frac{1}{16} \cdot \left(\frac{{B}^{2}}{A} \cdot \sqrt{\frac{1}{A - -1 \cdot A}}\right)}{C} - \frac{-1}{4} \cdot \left(\frac{1}{A} \cdot \sqrt{A - -1 \cdot A}\right)}{C}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{16} \cdot \left(\frac{{B}^{2}}{{A}^{2}} \cdot \sqrt{A - -1 \cdot A}\right) + \frac{1}{16} \cdot \left(\frac{{B}^{2}}{A} \cdot \sqrt{\frac{1}{A - -1 \cdot A}}\right)}{C} - \frac{-1}{4} \cdot \left(\frac{1}{A} \cdot \sqrt{A - -1 \cdot A}\right)}{C}} \]

   7. Applied rewrites 2.1%

      \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(-0.0625, \frac{B \cdot B}{A \cdot A} \cdot \sqrt{A - \left(-A\right)}, 0.0625 \cdot \left(\frac{B \cdot B}{A} \cdot \sqrt{\frac{1}{A - \left(-A\right)}}\right)\right)}{C}\right) - -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{A - \left(-A\right)}\right)}{C}} \]

   8. Taylor expanded in C around inf

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

   10. Applied rewrites 3.8%

       \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \sqrt{2}\right)}{C} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*r* N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lift-*.f64 N/A

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 N/A

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

       23. associate-*r* N/A

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

       24. lower-*.f64 N/A

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

   12. Applied rewrites 4.7%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(2 \cdot F\right) \cdot \left(C \cdot -4\right), A, \left(\left(2 \cdot F\right) \cdot B\right) \cdot B\right)}} \cdot \frac{0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \sqrt{2}\right)}{C} \]
3. Recombined 4 regimes into one program.

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot {\left(\left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right) \cdot F\right)}^{0.5}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \sqrt{\frac{1}{A}}\right) \cdot 0.25}{C} \cdot \sqrt{\mathsf{fma}\left(\left(-4 \cdot C\right) \cdot \left(F \cdot 2\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B}^{2}}\\ t_3 := \frac{B \cdot B}{C}\\ t_4 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_5 := -t\_4\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(t\_3, -0.5, A\right) + A\right)\right)}}{t\_5}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-t\_0}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, t\_3, A\right) + A\right) \cdot \left(t\_4 \cdot \left(F \cdot 2\right)\right)}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \sqrt{\frac{1}{A}}\right) \cdot 0.25}{C} \cdot \sqrt{\mathsf{fma}\left(\left(-4 \cdot C\right) \cdot \left(F \cdot 2\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}\\ \end{array} \end{array} \]

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 (/ (* B B) C))
        (t_4 (fma (* -4.0 C) A (* B B)))
        (t_5 (- t_4)))
   (if (<= t_2 (- INFINITY))
     (/
      (sqrt
       (* (fma (* C A) -4.0 (* B B)) (* (* F 2.0) (+ (fma t_3 -0.5 A) A))))
      t_5)
     (if (<= t_2 -1e-191)
       (/
        (sqrt
         (*
          (* t_0 (* F 2.0))
          (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B))))))
        (- t_0))
       (if (<= t_2 INFINITY)
         (/ (sqrt (* (+ (fma -0.5 t_3 A) A) (* t_4 (* F 2.0)))) t_5)
         (*
          (/ (* (* (sqrt 2.0) (sqrt (/ 1.0 A))) 0.25) C)
          (sqrt (fma (* (* -4.0 C) (* F 2.0)) A (* (* (* F 2.0) B) 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 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 = (B * B) / C;
	double t_4 = fma((-4.0 * C), A, (B * B));
	double t_5 = -t_4;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = sqrt((fma((C * A), -4.0, (B * B)) * ((F * 2.0) * (fma(t_3, -0.5, A) + A)))) / t_5;
	} else if (t_2 <= -1e-191) {
		tmp = sqrt(((t_0 * (F * 2.0)) * ((C + A) - sqrt(fma((A - C), (A - C), (B * B)))))) / -t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((fma(-0.5, t_3, A) + A) * (t_4 * (F * 2.0)))) / t_5;
	} else {
		tmp = (((sqrt(2.0) * sqrt((1.0 / A))) * 0.25) / C) * sqrt(fma(((-4.0 * C) * (F * 2.0)), A, (((F * 2.0) * B) * B)));
	}
	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 = 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(Float64(B * B) / C)
	t_4 = fma(Float64(-4.0 * C), A, Float64(B * B))
	t_5 = Float64(-t_4)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B * B)) * Float64(Float64(F * 2.0) * Float64(fma(t_3, -0.5, A) + A)))) / t_5);
	elseif (t_2 <= -1e-191)
		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));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(fma(-0.5, t_3, A) + A) * Float64(t_4 * Float64(F * 2.0)))) / t_5);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 / A))) * 0.25) / C) * sqrt(fma(Float64(Float64(-4.0 * C) * Float64(F * 2.0)), A, Float64(Float64(Float64(F * 2.0) * B) * 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[(-4.0 * N[(C * A), $MachinePrecision] + 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 = N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-t$95$4)}, If[LessEqual[t$95$2, (-Infinity)], N[(N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * N[(N[(t$95$3 * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$2, -1e-191], 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], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(N[(-0.5 * t$95$3 + A), $MachinePrecision] + A), $MachinePrecision] * N[(t$95$4 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] / C), $MachinePrecision] * N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * A + N[(N[(N[(F * 2.0), $MachinePrecision] * B), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 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 3.2%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 19.4

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

   5. Applied rewrites 19.4%

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

   7. Applied rewrites 19.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

   9. Applied rewrites 21.5%

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

   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))) < -1e-191

   1. Initial program 99.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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 99.0%

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

   if -1e-191 < (/.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 20.0%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 19.3

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.3%

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

   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)))

   1. Initial program 0.0%

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around inf

      \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{16} \cdot \left(\frac{{B}^{2}}{{A}^{2}} \cdot \sqrt{A - -1 \cdot A}\right) + \frac{1}{16} \cdot \left(\frac{{B}^{2}}{A} \cdot \sqrt{\frac{1}{A - -1 \cdot A}}\right)}{C} - \frac{-1}{4} \cdot \left(\frac{1}{A} \cdot \sqrt{A - -1 \cdot A}\right)}{C}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{16} \cdot \left(\frac{{B}^{2}}{{A}^{2}} \cdot \sqrt{A - -1 \cdot A}\right) + \frac{1}{16} \cdot \left(\frac{{B}^{2}}{A} \cdot \sqrt{\frac{1}{A - -1 \cdot A}}\right)}{C} - \frac{-1}{4} \cdot \left(\frac{1}{A} \cdot \sqrt{A - -1 \cdot A}\right)}{C}} \]

   7. Applied rewrites 2.1%

      \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(-0.0625, \frac{B \cdot B}{A \cdot A} \cdot \sqrt{A - \left(-A\right)}, 0.0625 \cdot \left(\frac{B \cdot B}{A} \cdot \sqrt{\frac{1}{A - \left(-A\right)}}\right)\right)}{C}\right) - -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{A - \left(-A\right)}\right)}{C}} \]

   8. Taylor expanded in C around inf

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

   10. Applied rewrites 3.8%

       \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \sqrt{2}\right)}{C} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*r* N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lift-*.f64 N/A

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 N/A

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

       23. associate-*r* N/A

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

       24. lower-*.f64 N/A

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

   12. Applied rewrites 4.7%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(2 \cdot F\right) \cdot \left(C \cdot -4\right), A, \left(\left(2 \cdot F\right) \cdot B\right) \cdot B\right)}} \cdot \frac{0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \sqrt{2}\right)}{C} \]
3. Recombined 4 regimes into one program.

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \sqrt{\frac{1}{A}}\right) \cdot 0.25}{C} \cdot \sqrt{\mathsf{fma}\left(\left(-4 \cdot C\right) \cdot \left(F \cdot 2\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ t_1 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_0\right)\right) \cdot 2\right)}}{t\_0 - {B}^{2}}\\ t_2 := \frac{B \cdot B}{C}\\ t_3 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_4 := -t\_3\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(t\_2, -0.5, A\right) + A\right)\right)}}{t\_4}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(\left(t\_3 \cdot F\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)\right) \cdot 2}}{t\_4}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, t\_2, A\right) + A\right) \cdot \left(t\_3 \cdot \left(F \cdot 2\right)\right)}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \sqrt{\frac{1}{A}}\right) \cdot 0.25}{C} \cdot \sqrt{\mathsf{fma}\left(\left(-4 \cdot C\right) \cdot \left(F \cdot 2\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}\\ \end{array} \end{array} \]

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))))
        (t_2 (/ (* B B) C))
        (t_3 (fma (* -4.0 C) A (* B B)))
        (t_4 (- t_3)))
   (if (<= t_1 (- INFINITY))
     (/
      (sqrt
       (* (fma (* C A) -4.0 (* B B)) (* (* F 2.0) (+ (fma t_2 -0.5 A) A))))
      t_4)
     (if (<= t_1 -1e-191)
       (/
        (sqrt
         (*
          (* (* t_3 F) (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B)))))
          2.0))
        t_4)
       (if (<= t_1 INFINITY)
         (/ (sqrt (* (+ (fma -0.5 t_2 A) A) (* t_3 (* F 2.0)))) t_4)
         (*
          (/ (* (* (sqrt 2.0) (sqrt (/ 1.0 A))) 0.25) C)
          (sqrt (fma (* (* -4.0 C) (* F 2.0)) A (* (* (* F 2.0) B) B)))))))))

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 t_2 = (B * B) / C;
	double t_3 = fma((-4.0 * C), A, (B * B));
	double t_4 = -t_3;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = sqrt((fma((C * A), -4.0, (B * B)) * ((F * 2.0) * (fma(t_2, -0.5, A) + A)))) / t_4;
	} else if (t_1 <= -1e-191) {
		tmp = sqrt((((t_3 * F) * ((C + A) - sqrt(fma((A - C), (A - C), (B * B))))) * 2.0)) / t_4;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(((fma(-0.5, t_2, A) + A) * (t_3 * (F * 2.0)))) / t_4;
	} else {
		tmp = (((sqrt(2.0) * sqrt((1.0 / A))) * 0.25) / C) * sqrt(fma(((-4.0 * C) * (F * 2.0)), A, (((F * 2.0) * B) * B)));
	}
	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)))
	t_2 = Float64(Float64(B * B) / C)
	t_3 = fma(Float64(-4.0 * C), A, Float64(B * B))
	t_4 = Float64(-t_3)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B * B)) * Float64(Float64(F * 2.0) * Float64(fma(t_2, -0.5, A) + A)))) / t_4);
	elseif (t_1 <= -1e-191)
		tmp = Float64(sqrt(Float64(Float64(Float64(t_3 * F) * Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))) * 2.0)) / t_4);
	elseif (t_1 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(fma(-0.5, t_2, A) + A) * Float64(t_3 * Float64(F * 2.0)))) / t_4);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 / A))) * 0.25) / C) * sqrt(fma(Float64(Float64(-4.0 * C) * Float64(F * 2.0)), A, Float64(Float64(Float64(F * 2.0) * B) * 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[(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]}, Block[{t$95$2 = N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-t$95$3)}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * N[(N[(t$95$2 * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$1, -1e-191], N[(N[Sqrt[N[(N[(N[(t$95$3 * F), $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] * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(N[(N[(-0.5 * t$95$2 + A), $MachinePrecision] + A), $MachinePrecision] * N[(t$95$3 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] / C), $MachinePrecision] * N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * A + N[(N[(N[(F * 2.0), $MachinePrecision] * B), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 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 3.2%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 19.4

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

   5. Applied rewrites 19.4%

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

   7. Applied rewrites 19.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

   9. Applied rewrites 21.5%

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

   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))) < -1e-191

   1. Initial program 99.0%

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 98.9%

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

   if -1e-191 < (/.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 20.0%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 19.3

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.3%

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

   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)))

   1. Initial program 0.0%

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around inf

      \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{16} \cdot \left(\frac{{B}^{2}}{{A}^{2}} \cdot \sqrt{A - -1 \cdot A}\right) + \frac{1}{16} \cdot \left(\frac{{B}^{2}}{A} \cdot \sqrt{\frac{1}{A - -1 \cdot A}}\right)}{C} - \frac{-1}{4} \cdot \left(\frac{1}{A} \cdot \sqrt{A - -1 \cdot A}\right)}{C}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{16} \cdot \left(\frac{{B}^{2}}{{A}^{2}} \cdot \sqrt{A - -1 \cdot A}\right) + \frac{1}{16} \cdot \left(\frac{{B}^{2}}{A} \cdot \sqrt{\frac{1}{A - -1 \cdot A}}\right)}{C} - \frac{-1}{4} \cdot \left(\frac{1}{A} \cdot \sqrt{A - -1 \cdot A}\right)}{C}} \]

   7. Applied rewrites 2.1%

      \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(-0.0625, \frac{B \cdot B}{A \cdot A} \cdot \sqrt{A - \left(-A\right)}, 0.0625 \cdot \left(\frac{B \cdot B}{A} \cdot \sqrt{\frac{1}{A - \left(-A\right)}}\right)\right)}{C}\right) - -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{A - \left(-A\right)}\right)}{C}} \]

   8. Taylor expanded in C around inf

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

   10. Applied rewrites 3.8%

       \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \sqrt{2}\right)}{C} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*r* N/A

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

       15. lower-fma.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lift-*.f64 N/A

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 N/A

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

       23. associate-*r* N/A

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

       24. lower-*.f64 N/A

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

   12. Applied rewrites 4.7%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(2 \cdot F\right) \cdot \left(C \cdot -4\right), A, \left(\left(2 \cdot F\right) \cdot B\right) \cdot B\right)}} \cdot \frac{0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \sqrt{2}\right)}{C} \]
3. Recombined 4 regimes into one program.

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)\right) \cdot 2}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \sqrt{\frac{1}{A}}\right) \cdot 0.25}{C} \cdot \sqrt{\mathsf{fma}\left(\left(-4 \cdot C\right) \cdot \left(F \cdot 2\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 23.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := -\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B}^{2}}\\ t_3 := \frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{t\_0}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+153}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right) \cdot -2}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 (/ (sqrt (* (* (* (* (+ A A) F) C) A) -8.0)) t_0)))
   (if (<= t_2 -2e+153)
     t_3
     (if (<= t_2 -1e-164) (/ (sqrt (* (* (* (* B B) B) F) -2.0)) t_0) 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 = sqrt((((((A + A) * F) * C) * A) * -8.0)) / t_0;
	double tmp;
	if (t_2 <= -2e+153) {
		tmp = t_3;
	} else if (t_2 <= -1e-164) {
		tmp = sqrt(((((B * B) * B) * F) * -2.0)) / t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(-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(sqrt(Float64(Float64(Float64(Float64(Float64(A + A) * F) * C) * A) * -8.0)) / t_0)
	tmp = 0.0
	if (t_2 <= -2e+153)
		tmp = t_3;
	elseif (t_2 <= -1e-164)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(B * B) * B) * F) * -2.0)) / t_0);
	else
		tmp = 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 = 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]}, If[LessEqual[t$95$2, -2e+153], t$95$3, If[LessEqual[t$95$2, -1e-164], N[(N[Sqrt[N[(N[(N[(N[(B * B), $MachinePrecision] * B), $MachinePrecision] * F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 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))) < -2e153 or -9.99999999999999962e-165 < (/.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 9.5%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 10.5

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

   5. Applied rewrites 10.5%

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 9.4

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

   10. Applied rewrites 9.4%

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

   if -2e153 < (/.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))) < -9.99999999999999962e-165

   1. Initial program 98.7%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 7.6

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

   5. Applied rewrites 7.6%

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

   7. Applied rewrites 7.6%

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

   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 16.9

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

   10. Applied rewrites 16.9%

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

4. Final simplification 10.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right) \cdot -2}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 17.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := -\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B}^{2}}\\ t_3 := \frac{\sqrt{\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16}}{t\_0}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+153}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right) \cdot -2}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 (/ (sqrt (* (* (* (* A A) C) F) -16.0)) t_0)))
   (if (<= t_2 -2e+153)
     t_3
     (if (<= t_2 -1e-164) (/ (sqrt (* (* (* (* B B) B) F) -2.0)) t_0) 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 = sqrt(((((A * A) * C) * F) * -16.0)) / t_0;
	double tmp;
	if (t_2 <= -2e+153) {
		tmp = t_3;
	} else if (t_2 <= -1e-164) {
		tmp = sqrt(((((B * B) * B) * F) * -2.0)) / t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(-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(sqrt(Float64(Float64(Float64(Float64(A * A) * C) * F) * -16.0)) / t_0)
	tmp = 0.0
	if (t_2 <= -2e+153)
		tmp = t_3;
	elseif (t_2 <= -1e-164)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(B * B) * B) * F) * -2.0)) / t_0);
	else
		tmp = 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 = N[(N[Sqrt[N[(N[(N[(N[(A * A), $MachinePrecision] * C), $MachinePrecision] * F), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+153], t$95$3, If[LessEqual[t$95$2, -1e-164], N[(N[Sqrt[N[(N[(N[(N[(B * B), $MachinePrecision] * B), $MachinePrecision] * F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 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))) < -2e153 or -9.99999999999999962e-165 < (/.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 9.5%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 10.5

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

   5. Applied rewrites 10.5%

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\color{blue}{\left({A}^{2} \cdot C\right)} \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

      6. lower-*.f64 7.5

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

   10. Applied rewrites 7.5%

       \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

   if -2e153 < (/.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))) < -9.99999999999999962e-165

   1. Initial program 98.7%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 7.6

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

   5. Applied rewrites 7.6%

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

   7. Applied rewrites 7.6%

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

   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 16.9

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

   10. Applied rewrites 16.9%

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

4. Final simplification 8.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(B \cdot B\right) \cdot B\right) \cdot F\right) \cdot -2}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(t\_0 \cdot \left(F \cdot 2\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

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-41)
     (/ (sqrt (* (+ A A) (* t_0 (* F 2.0)))) (- 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-41) {
		tmp = sqrt(((A + A) * (t_0 * (F * 2.0)))) / -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-41)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(t_0 * Float64(F * 2.0)))) / 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-41], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $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)) < 4.9999999999999996e-41

   1. Initial program 24.0%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 16.9%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 18.2

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

   10. Applied rewrites 18.2%

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

   if 4.9999999999999996e-41 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 13.9%

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

   3. Taylor expanded in C around 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 8.8

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

   5. Applied rewrites 8.8%

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

4. Final simplification 13.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, -4 \cdot C, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;C \leq -1.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(t\_1 \cdot \left(F \cdot 2\right)\right)}}{-t\_1}\\ \mathbf{elif}\;C \leq 2.8 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5 \cdot B, \frac{B}{C}, A + A\right) \cdot \left(\left(t\_0 \cdot 2\right) \cdot F\right)}}{-t\_0}\\ \end{array} \end{array} \]

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 A (* -4.0 C) (* B B))) (t_1 (fma (* -4.0 C) A (* B B))))
   (if (<= C -1.2e-250)
     (/ (sqrt (* (+ A A) (* t_1 (* F 2.0)))) (- t_1))
     (if (<= C 2.8e-74)
       (*
        (sqrt
         (*
          (/
           (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B))))
           (fma (* -4.0 A) C (* B B)))
          F))
        (- (sqrt 2.0)))
       (/
        (sqrt (* (fma (* -0.5 B) (/ B C) (+ A A)) (* (* t_0 2.0) F)))
        (- t_0))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(A, (-4.0 * C), (B * B));
	double t_1 = fma((-4.0 * C), A, (B * B));
	double tmp;
	if (C <= -1.2e-250) {
		tmp = sqrt(((A + A) * (t_1 * (F * 2.0)))) / -t_1;
	} else if (C <= 2.8e-74) {
		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((fma((-0.5 * B), (B / C), (A + A)) * ((t_0 * 2.0) * F))) / -t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(A, Float64(-4.0 * C), Float64(B * B))
	t_1 = fma(Float64(-4.0 * C), A, Float64(B * B))
	tmp = 0.0
	if (C <= -1.2e-250)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(t_1 * Float64(F * 2.0)))) / Float64(-t_1));
	elseif (C <= 2.8e-74)
		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(fma(Float64(-0.5 * B), Float64(B / C), Float64(A + A)) * Float64(Float64(t_0 * 2.0) * F))) / 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[(A * N[(-4.0 * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -1.2e-250], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(t$95$1 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[C, 2.8e-74], 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[(-0.5 * B), $MachinePrecision] * N[(B / C), $MachinePrecision] + N[(A + A), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * 2.0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.1999999999999999e-250

   1. Initial program 21.2%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 2.6

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

   5. Applied rewrites 2.6%

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

   7. Applied rewrites 2.6%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 3.8

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

   10. Applied rewrites 3.8%

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

   if -1.1999999999999999e-250 < C < 2.79999999999999988e-74

   1. Initial program 34.1%

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

   3. Taylor expanded in 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 34.4%

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

   if 2.79999999999999988e-74 < C

   1. Initial program 3.5%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 22.7

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

   5. Applied rewrites 22.7%

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

   7. Applied rewrites 22.7%

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

   9. Applied rewrites 22.7%

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

4. Final simplification 16.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;C \leq 2.8 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5 \cdot B, \frac{B}{C}, A + A\right) \cdot \left(\left(\mathsf{fma}\left(A, -4 \cdot C, B \cdot B\right) \cdot 2\right) \cdot F\right)}}{-\mathsf{fma}\left(A, -4 \cdot C, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 31.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_1 := -t\_0\\ \mathbf{if}\;C \leq -1.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(t\_0 \cdot \left(F \cdot 2\right)\right)}}{t\_1}\\ \mathbf{elif}\;C \leq 3 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{t\_1}\\ \end{array} \end{array} \]

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 (- t_0)))
   (if (<= C -1.2e-250)
     (/ (sqrt (* (+ A A) (* t_0 (* F 2.0)))) t_1)
     (if (<= C 3e+49)
       (*
        (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)) 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 = -t_0;
	double tmp;
	if (C <= -1.2e-250) {
		tmp = sqrt(((A + A) * (t_0 * (F * 2.0)))) / t_1;
	} else if (C <= 3e+49) {
		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)) / t_1;
	}
	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(-t_0)
	tmp = 0.0
	if (C <= -1.2e-250)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(t_0 * Float64(F * 2.0)))) / t_1);
	elseif (C <= 3e+49)
		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)) / 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[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, If[LessEqual[C, -1.2e-250], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[C, 3e+49], 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] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.1999999999999999e-250

   1. Initial program 21.2%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 2.6

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

   5. Applied rewrites 2.6%

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

   7. Applied rewrites 2.6%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 3.8

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

   10. Applied rewrites 3.8%

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

   if -1.1999999999999999e-250 < C < 3.0000000000000002e49

   1. Initial program 27.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 28.1%

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

   if 3.0000000000000002e49 < C

   1. Initial program 1.4%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 24.8

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

   5. Applied rewrites 24.8%

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

   7. Applied rewrites 24.8%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 19.7

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

   10. Applied rewrites 19.7%

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

4. Final simplification 14.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;C \leq 3 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_1 := -t\_0\\ \mathbf{if}\;C \leq -2.8 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(t\_0 \cdot \left(F \cdot 2\right)\right)}}{t\_1}\\ \mathbf{elif}\;C \leq 9.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{1}{\frac{-B}{\sqrt{2}} \cdot \sqrt{\frac{1}{\left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right) \cdot F}}}\\ \mathbf{elif}\;C \leq 1.1 \cdot 10^{+49}:\\ \;\;\;\;-\sqrt{\left(\frac{F}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{t\_1}\\ \end{array} \end{array} \]

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 (- t_0)))
   (if (<= C -2.8e-225)
     (/ (sqrt (* (+ A A) (* t_0 (* F 2.0)))) t_1)
     (if (<= C 9.5e-186)
       (/
        1.0
        (*
         (/ (- B) (sqrt 2.0))
         (sqrt (/ 1.0 (* (- A (sqrt (fma A A (* B B)))) F)))))
       (if (<= C 1.1e+49)
         (-
          (sqrt
           (*
            (*
             (/ F (fma (* C A) -4.0 (* B B)))
             (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B)))))
            2.0)))
         (/ (sqrt (* (* (* (* (+ A A) F) C) A) -8.0)) 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 = -t_0;
	double tmp;
	if (C <= -2.8e-225) {
		tmp = sqrt(((A + A) * (t_0 * (F * 2.0)))) / t_1;
	} else if (C <= 9.5e-186) {
		tmp = 1.0 / ((-B / sqrt(2.0)) * sqrt((1.0 / ((A - sqrt(fma(A, A, (B * B)))) * F))));
	} else if (C <= 1.1e+49) {
		tmp = -sqrt((((F / fma((C * A), -4.0, (B * B))) * ((C + A) - sqrt(fma((A - C), (A - C), (B * B))))) * 2.0));
	} else {
		tmp = sqrt((((((A + A) * F) * C) * A) * -8.0)) / t_1;
	}
	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(-t_0)
	tmp = 0.0
	if (C <= -2.8e-225)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(t_0 * Float64(F * 2.0)))) / t_1);
	elseif (C <= 9.5e-186)
		tmp = Float64(1.0 / Float64(Float64(Float64(-B) / sqrt(2.0)) * sqrt(Float64(1.0 / Float64(Float64(A - sqrt(fma(A, A, Float64(B * B)))) * F)))));
	elseif (C <= 1.1e+49)
		tmp = Float64(-sqrt(Float64(Float64(Float64(F / fma(Float64(C * A), -4.0, Float64(B * B))) * Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))) * 2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(A + A) * F) * C) * A) * -8.0)) / 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[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, If[LessEqual[C, -2.8e-225], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[C, 9.5e-186], N[(1.0 / N[(N[((-B) / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(A - N[Sqrt[N[(A * A + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 1.1e+49], (-N[Sqrt[N[(N[(N[(F / N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $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] * 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] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if C < -2.8e-225

   1. Initial program 20.9%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 2.7

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 4.1

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

   10. Applied rewrites 4.1%

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

   if -2.8e-225 < C < 9.4999999999999998e-186

   1. Initial program 32.0%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 1.7

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

   5. Applied rewrites 1.7%

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 19.9

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

   10. Applied rewrites 19.9%

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

   if 9.4999999999999998e-186 < C < 1.1e49

   1. Initial program 22.7%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 17.4

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

   5. Applied rewrites 17.4%

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

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   7. 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. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   8. Applied rewrites 23.7%

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

   10. Applied rewrites 25.8%

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

   if 1.1e49 < C

   1. Initial program 1.4%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 24.8

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

   5. Applied rewrites 24.8%

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

   7. Applied rewrites 24.8%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 19.7

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

   10. Applied rewrites 19.7%

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

4. Final simplification 14.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -2.8 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;C \leq 9.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{1}{\frac{-B}{\sqrt{2}} \cdot \sqrt{\frac{1}{\left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right) \cdot F}}}\\ \mathbf{elif}\;C \leq 1.1 \cdot 10^{+49}:\\ \;\;\;\;-\sqrt{\left(\frac{F}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_1 := -t\_0\\ \mathbf{if}\;C \leq -3.8 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(t\_0 \cdot \left(F \cdot 2\right)\right)}}{t\_1}\\ \mathbf{elif}\;C \leq 3.2 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;C \leq 1.1 \cdot 10^{+49}:\\ \;\;\;\;-\sqrt{\left(\frac{F}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{t\_1}\\ \end{array} \end{array} \]

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 (- t_0)))
   (if (<= C -3.8e-223)
     (/ (sqrt (* (+ A A) (* t_0 (* F 2.0)))) t_1)
     (if (<= C 3.2e-184)
       (* (sqrt (* (- A (sqrt (fma B B (* A A)))) F)) (/ (- (sqrt 2.0)) B))
       (if (<= C 1.1e+49)
         (-
          (sqrt
           (*
            (*
             (/ F (fma (* C A) -4.0 (* B B)))
             (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B)))))
            2.0)))
         (/ (sqrt (* (* (* (* (+ A A) F) C) A) -8.0)) 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 = -t_0;
	double tmp;
	if (C <= -3.8e-223) {
		tmp = sqrt(((A + A) * (t_0 * (F * 2.0)))) / t_1;
	} else if (C <= 3.2e-184) {
		tmp = sqrt(((A - sqrt(fma(B, B, (A * A)))) * F)) * (-sqrt(2.0) / B);
	} else if (C <= 1.1e+49) {
		tmp = -sqrt((((F / fma((C * A), -4.0, (B * B))) * ((C + A) - sqrt(fma((A - C), (A - C), (B * B))))) * 2.0));
	} else {
		tmp = sqrt((((((A + A) * F) * C) * A) * -8.0)) / t_1;
	}
	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(-t_0)
	tmp = 0.0
	if (C <= -3.8e-223)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(t_0 * Float64(F * 2.0)))) / t_1);
	elseif (C <= 3.2e-184)
		tmp = Float64(sqrt(Float64(Float64(A - sqrt(fma(B, B, Float64(A * A)))) * F)) * Float64(Float64(-sqrt(2.0)) / B));
	elseif (C <= 1.1e+49)
		tmp = Float64(-sqrt(Float64(Float64(Float64(F / fma(Float64(C * A), -4.0, Float64(B * B))) * Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))) * 2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(A + A) * F) * C) * A) * -8.0)) / 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[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, If[LessEqual[C, -3.8e-223], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[C, 3.2e-184], 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], If[LessEqual[C, 1.1e+49], (-N[Sqrt[N[(N[(N[(F / N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $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] * 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] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if C < -3.80000000000000012e-223

   1. Initial program 21.3%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 2.8

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

   5. Applied rewrites 2.8%

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

   7. Applied rewrites 2.8%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 4.1

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

   10. Applied rewrites 4.1%

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

   if -3.80000000000000012e-223 < C < 3.2e-184

   1. Initial program 30.7%

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

   3. Taylor expanded in C around 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 19.5

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

   5. Applied rewrites 19.5%

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

   if 3.2e-184 < C < 1.1e49

   1. Initial program 22.7%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 17.4

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

   5. Applied rewrites 17.4%

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

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   7. 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. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   8. Applied rewrites 23.7%

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

   10. Applied rewrites 25.8%

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

   if 1.1e49 < C

   1. Initial program 1.4%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 24.8

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

   5. Applied rewrites 24.8%

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

   7. Applied rewrites 24.8%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 19.7

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

   10. Applied rewrites 19.7%

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

4. Final simplification 14.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -3.8 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;C \leq 3.2 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;C \leq 1.1 \cdot 10^{+49}:\\ \;\;\;\;-\sqrt{\left(\frac{F}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;A \leq -3.1 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(t\_0 \cdot \left(F \cdot 2\right)\right)}}{-t\_0}\\ \mathbf{elif}\;A \leq 4.1 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{2}{A}} \cdot 0.25}{C} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}\\ \end{array} \end{array} \]

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 (<= A -3.1e-111)
     (/ (sqrt (* (+ A A) (* t_0 (* F 2.0)))) (- t_0))
     (if (<= A 4.1e-231)
       (* (sqrt (* (- A (sqrt (fma B B (* A A)))) F)) (/ (- (sqrt 2.0)) B))
       (*
        (/ (* (sqrt (/ 2.0 A)) 0.25) C)
        (sqrt (* (fma -4.0 (* C A) (* B B)) (* F 2.0))))))))

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 (A <= -3.1e-111) {
		tmp = sqrt(((A + A) * (t_0 * (F * 2.0)))) / -t_0;
	} else if (A <= 4.1e-231) {
		tmp = sqrt(((A - sqrt(fma(B, B, (A * A)))) * F)) * (-sqrt(2.0) / B);
	} else {
		tmp = ((sqrt((2.0 / A)) * 0.25) / C) * sqrt((fma(-4.0, (C * A), (B * B)) * (F * 2.0)));
	}
	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 (A <= -3.1e-111)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(t_0 * Float64(F * 2.0)))) / Float64(-t_0));
	elseif (A <= 4.1e-231)
		tmp = Float64(sqrt(Float64(Float64(A - sqrt(fma(B, B, Float64(A * A)))) * F)) * Float64(Float64(-sqrt(2.0)) / B));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(2.0 / A)) * 0.25) / C) * sqrt(Float64(fma(-4.0, Float64(C * A), Float64(B * B)) * Float64(F * 2.0))));
	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[A, -3.1e-111], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[A, 4.1e-231], 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], N[(N[(N[(N[Sqrt[N[(2.0 / A), $MachinePrecision]], $MachinePrecision] * 0.25), $MachinePrecision] / C), $MachinePrecision] * N[Sqrt[N[(N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if A < -3.10000000000000014e-111

   1. Initial program 15.5%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 16.3

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

   5. Applied rewrites 16.3%

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

   7. Applied rewrites 16.3%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 21.4

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

   10. Applied rewrites 21.4%

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

   if -3.10000000000000014e-111 < A < 4.1000000000000002e-231

   1. Initial program 34.3%

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

   3. Taylor expanded in C around 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 12.7

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

   5. Applied rewrites 12.7%

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

   if 4.1000000000000002e-231 < A

   1. Initial program 12.2%

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in C around inf

      \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{16} \cdot \left(\frac{{B}^{2}}{{A}^{2}} \cdot \sqrt{A - -1 \cdot A}\right) + \frac{1}{16} \cdot \left(\frac{{B}^{2}}{A} \cdot \sqrt{\frac{1}{A - -1 \cdot A}}\right)}{C} - \frac{-1}{4} \cdot \left(\frac{1}{A} \cdot \sqrt{A - -1 \cdot A}\right)}{C}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{16} \cdot \left(\frac{{B}^{2}}{{A}^{2}} \cdot \sqrt{A - -1 \cdot A}\right) + \frac{1}{16} \cdot \left(\frac{{B}^{2}}{A} \cdot \sqrt{\frac{1}{A - -1 \cdot A}}\right)}{C} - \frac{-1}{4} \cdot \left(\frac{1}{A} \cdot \sqrt{A - -1 \cdot A}\right)}{C}} \]

   7. Applied rewrites 4.0%

      \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{\left(-\frac{\mathsf{fma}\left(-0.0625, \frac{B \cdot B}{A \cdot A} \cdot \sqrt{A - \left(-A\right)}, 0.0625 \cdot \left(\frac{B \cdot B}{A} \cdot \sqrt{\frac{1}{A - \left(-A\right)}}\right)\right)}{C}\right) - -0.25 \cdot \left(\frac{1}{A} \cdot \sqrt{A - \left(-A\right)}\right)}{C}} \]

   8. Taylor expanded in C around inf

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

   10. Applied rewrites 7.3%

       \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \sqrt{2}\right)}{C} \]
   11. Step-by-step derivation

   12. Applied rewrites 7.3%

       \[\leadsto \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{\sqrt{\frac{2}{A}} \cdot 0.25}{C}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 12.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.1 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;A \leq 4.1 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{2}{A}} \cdot 0.25}{C} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 22.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{if}\;C \leq -2.6 \cdot 10^{-184}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;C \leq 9 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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
         (/
          (sqrt (* (* (* (* (+ A A) F) C) A) -8.0))
          (- (fma (* -4.0 C) A (* B B))))))
   (if (<= C -2.6e-184)
     t_0
     (if (<= C 9e-141)
       (* (sqrt (* (- A (sqrt (fma B B (* A A)))) F)) (/ (- (sqrt 2.0)) B))
       t_0))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt((((((A + A) * F) * C) * A) * -8.0)) / -fma((-4.0 * C), A, (B * B));
	double tmp;
	if (C <= -2.6e-184) {
		tmp = t_0;
	} else if (C <= 9e-141) {
		tmp = sqrt(((A - sqrt(fma(B, B, (A * A)))) * F)) * (-sqrt(2.0) / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(A + A) * F) * C) * A) * -8.0)) / Float64(-fma(Float64(-4.0 * C), A, Float64(B * B))))
	tmp = 0.0
	if (C <= -2.6e-184)
		tmp = t_0;
	elseif (C <= 9e-141)
		tmp = Float64(sqrt(Float64(Float64(A - sqrt(fma(B, B, Float64(A * A)))) * F)) * Float64(Float64(-sqrt(2.0)) / B));
	else
		tmp = 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[Sqrt[N[(N[(N[(N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision] * C), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[C, -2.6e-184], t$95$0, If[LessEqual[C, 9e-141], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if C < -2.59999999999999978e-184 or 9.0000000000000001e-141 < C

   1. Initial program 15.4%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 12.7

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

   5. Applied rewrites 12.7%

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

   7. Applied rewrites 12.7%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 11.2

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

   10. Applied rewrites 11.2%

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

   if -2.59999999999999978e-184 < C < 9.0000000000000001e-141

   1. Initial program 28.2%

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

   3. Taylor expanded in C around 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 19.1

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

   5. Applied rewrites 19.1%

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

4. Final simplification 13.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -2.6 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;C \leq 9 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{\left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 16.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \frac{\sqrt{\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \end{array} \]

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)) (- (fma (* -4.0 C) A (* B B)))))

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)) / -fma((-4.0 * C), A, (B * B));
}

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(-fma(Float64(-4.0 * C), A, Float64(B * B))))
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[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 18.5%

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

3. Taylor expanded in C around inf

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 10.2

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

5. Applied rewrites 10.2%

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

7. Applied rewrites 10.2%

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

8. Taylor expanded in A around -inf

   \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{-16 \cdot \left(\color{blue}{\left({A}^{2} \cdot C\right)} \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

   5. unpow2 N/A

      \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

   6. lower-*.f64 7.6

      \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

10. Applied rewrites 7.6%

    \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

11. Final simplification 7.6%

    \[\leadsto \frac{\sqrt{\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
12. Add Preprocessing

Alternative 17: 2.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot A\right) \cdot F\right) \cdot -16}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \end{array} \]

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) A) F) -16.0)) (- (fma (* -4.0 C) A (* B B)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return sqrt(((((C * C) * A) * F) * -16.0)) / -fma((-4.0 * C), A, (B * B));
}

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) * A) * F) * -16.0)) / Float64(-fma(Float64(-4.0 * C), A, Float64(B * B))))
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] * A), $MachinePrecision] * F), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 18.5%

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

3. Taylor expanded in C around inf

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 10.2

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

5. Applied rewrites 10.2%

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

7. Applied rewrites 10.2%

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

8. Taylor expanded in C around -inf

   \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot {C}^{2}\right)} \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

   5. unpow2 N/A

      \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \]

   6. lower-*.f64 11.3

      \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

10. Applied rewrites 11.3%

    \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

11. Final simplification 11.3%

    \[\leadsto \frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot A\right) \cdot F\right) \cdot -16}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
12. Add Preprocessing

Alternative 18: 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 18.5%

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

3. Taylor expanded in 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 1.9

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

5. Applied rewrites 1.9%

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

7. Applied rewrites 1.9%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
8. Step-by-step derivation

9. Applied rewrites 1.9%

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

11. Applied rewrites 2.0%

    \[\leadsto \sqrt{\frac{2}{\frac{B}{F}}} \]
12. Add Preprocessing

Alternative 19: 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 18.5%

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

3. Taylor expanded in 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 1.9

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

5. Applied rewrites 1.9%

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

7. Applied rewrites 1.9%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
8. Step-by-step derivation

9. Applied rewrites 1.9%

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

11. Applied rewrites 1.9%

    \[\leadsto \sqrt{F \cdot \frac{2}{B}} \]

12. Final simplification 1.9%

    \[\leadsto \sqrt{\frac{2}{B} \cdot F} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(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))))