ABCF->ab-angle b

Percentage Accurate: 19.3% → 39.8%

Time: 22.8s

Alternatives: 23

Speedup: 7.1×

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: 19.3% 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: 39.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \frac{B \cdot B}{C}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := t\_2 - {B}^{2}\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{t\_0} \cdot \sqrt{\left(2 \cdot F\right) \cdot \left(A + \mathsf{fma}\left(-0.5, t\_1, A\right)\right)}}}\\ \mathbf{elif}\;t\_4 \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot t\_0\right) \cdot \left(F \cdot \left(A + \left(A + \frac{-0.5 \cdot \left(B \cdot B\right)}{C}\right)\right)\right)}}{-C \cdot \mathsf{fma}\left(-4, A, t\_1\right)}\\ \end{array} \end{array} \]

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 B B (* -4.0 (* A C))))
        (t_1 (/ (* B B) C))
        (t_2 (* (* 4.0 A) C))
        (t_3 (- t_2 (pow B 2.0)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_2) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/
      -1.0
      (/ t_0 (* (sqrt t_0) (sqrt (* (* 2.0 F) (+ A (fma -0.5 t_1 A)))))))
     (if (<= t_4 -4e-214)
       (/
        (*
         (sqrt (* 2.0 (fma B B (* C (* A -4.0)))))
         (sqrt (* F (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B)))))))
        t_3)
       (/
        (sqrt (* (* 2.0 t_0) (* F (+ A (+ A (/ (* -0.5 (* B B)) C))))))
        (- (* C (fma -4.0 A t_1))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (-4.0 * (A * C)));
	double t_1 = (B * B) / C;
	double t_2 = (4.0 * A) * C;
	double t_3 = t_2 - pow(B, 2.0);
	double t_4 = sqrt(((2.0 * ((pow(B, 2.0) - t_2) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = -1.0 / (t_0 / (sqrt(t_0) * sqrt(((2.0 * F) * (A + fma(-0.5, t_1, A))))));
	} else if (t_4 <= -4e-214) {
		tmp = (sqrt((2.0 * fma(B, B, (C * (A * -4.0))))) * sqrt((F * ((A + C) - sqrt(fma((A - C), (A - C), (B * B))))))) / t_3;
	} else {
		tmp = sqrt(((2.0 * t_0) * (F * (A + (A + ((-0.5 * (B * B)) / C)))))) / -(C * fma(-4.0, A, t_1));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(Float64(B * B) / C)
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(t_2 - (B ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(-1.0 / Float64(t_0 / Float64(sqrt(t_0) * sqrt(Float64(Float64(2.0 * F) * Float64(A + fma(-0.5, t_1, A)))))));
	elseif (t_4 <= -4e-214)
		tmp = Float64(Float64(sqrt(Float64(2.0 * fma(B, B, Float64(C * Float64(A * -4.0))))) * sqrt(Float64(F * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))))) / t_3);
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * t_0) * Float64(F * Float64(A + Float64(A + Float64(Float64(-0.5 * Float64(B * B)) / C)))))) / Float64(-Float64(C * fma(-4.0, A, 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[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(-1.0 / N[(t$95$0 / N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A + N[(-0.5 * t$95$1 + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -4e-214], N[(N[(N[Sqrt[N[(2.0 * N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(F * N[(A + N[(A + N[(N[(-0.5 * N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(C * N[(-4.0 * A + t$95$1), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 2.8%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 35.1

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

   5. Applied rewrites 35.1%

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

   7. Applied rewrites 35.1%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. unpow-prod-down N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

   9. Applied rewrites 57.3%

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

   10. Taylor expanded in C around inf

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

       1. lower--.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 63.2

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

   12. Applied rewrites 63.2%

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

   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))) < -3.99999999999999965e-214

   1. Initial program 96.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 97.6%

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

   if -3.99999999999999965e-214 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 8.4%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 23.2

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

   5. Applied rewrites 23.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 24.4%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 26.3

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

   10. Applied rewrites 26.3%

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

   11. Taylor expanded in C around inf

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

       1. lower-*.f64 N/A

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

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

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

       3. metadata-eval N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 26.3

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

   13. Applied rewrites 26.3%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(F \cdot \left(A + \left(A + \frac{-0.5 \cdot \left(B \cdot B\right)}{C}\right)\right)\right)}}{-C \cdot \mathsf{fma}\left(-4, A, \frac{B \cdot B}{C}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 40.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \frac{B \cdot B}{C}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{t\_0} \cdot \sqrt{\left(2 \cdot F\right) \cdot \left(A + \mathsf{fma}\left(-0.5, t\_1, A\right)\right)}}}\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot t\_0\right) \cdot \left(F \cdot \left(A + \left(A + \frac{-0.5 \cdot \left(B \cdot B\right)}{C}\right)\right)\right)}}{-C \cdot \mathsf{fma}\left(-4, A, t\_1\right)}\\ \end{array} \end{array} \]

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 B B (* -4.0 (* A C))))
        (t_1 (/ (* B B) C))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_2) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B 2.0)))))
   (if (<= t_3 (- INFINITY))
     (/
      -1.0
      (/ t_0 (* (sqrt t_0) (sqrt (* (* 2.0 F) (+ A (fma -0.5 t_1 A)))))))
     (if (<= t_3 -4e-214)
       (/
        (sqrt
         (*
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B))))
          (* (* 2.0 F) (fma B B (* C (* A -4.0))))))
        (fma B (- B) (* A (* 4.0 C))))
       (/
        (sqrt (* (* 2.0 t_0) (* F (+ A (+ A (/ (* -0.5 (* B B)) C))))))
        (- (* C (fma -4.0 A t_1))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (-4.0 * (A * C)));
	double t_1 = (B * B) / C;
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B, 2.0) - t_2) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B, 2.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -1.0 / (t_0 / (sqrt(t_0) * sqrt(((2.0 * F) * (A + fma(-0.5, t_1, A))))));
	} else if (t_3 <= -4e-214) {
		tmp = sqrt((((A + C) - sqrt(fma((A - C), (A - C), (B * B)))) * ((2.0 * F) * fma(B, B, (C * (A * -4.0)))))) / fma(B, -B, (A * (4.0 * C)));
	} else {
		tmp = sqrt(((2.0 * t_0) * (F * (A + (A + ((-0.5 * (B * B)) / C)))))) / -(C * fma(-4.0, A, t_1));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(Float64(B * B) / C)
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B ^ 2.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-1.0 / Float64(t_0 / Float64(sqrt(t_0) * sqrt(Float64(Float64(2.0 * F) * Float64(A + fma(-0.5, t_1, A)))))));
	elseif (t_3 <= -4e-214)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))) * Float64(Float64(2.0 * F) * fma(B, B, Float64(C * Float64(A * -4.0)))))) / fma(B, Float64(-B), Float64(A * Float64(4.0 * C))));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * t_0) * Float64(F * Float64(A + Float64(A + Float64(Float64(-0.5 * Float64(B * B)) / C)))))) / Float64(-Float64(C * fma(-4.0, A, 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[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(-1.0 / N[(t$95$0 / N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A + N[(-0.5 * t$95$1 + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -4e-214], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B * (-B) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(F * N[(A + N[(A + N[(N[(-0.5 * N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(C * N[(-4.0 * A + t$95$1), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 30.1

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

   5. Applied rewrites 30.1%

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

   7. Applied rewrites 30.0%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. unpow-prod-down N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

   9. Applied rewrites 46.6%

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

   10. Taylor expanded in C around inf

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

       1. lower--.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 47.7

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

   12. Applied rewrites 47.7%

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

   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))) < -3.99999999999999965e-214

   1. Initial program 97.6%

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

   4. Applied rewrites 97.6%

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

   if -3.99999999999999965e-214 < (/.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 6.6%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 21.8

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

   5. Applied rewrites 21.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 22.3%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 25.0

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

   10. Applied rewrites 25.0%

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

   11. Taylor expanded in C around inf

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

       1. lower-*.f64 N/A

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

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

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

       3. metadata-eval N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 25.0

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

   13. Applied rewrites 25.0%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(F \cdot \left(A + \left(A + \frac{-0.5 \cdot \left(B \cdot B\right)}{C}\right)\right)\right)}}{-C \cdot \mathsf{fma}\left(-4, A, \frac{B \cdot B}{C}\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))