ABCF->ab-angle b

Percentage Accurate: 19.3% → 41.2%

Time: 20.1s

Alternatives: 18

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: 19.3% 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: 41.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := 2 \cdot \left(\left({B}^{2} - t\_1\right) \cdot F\right)\\ t_3 := t\_1 - {B}^{2}\\ t_4 := \frac{\sqrt{t\_2 \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, B \cdot B, -8 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + A\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{A \cdot -16} \cdot \sqrt{A \cdot \left(C \cdot F\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B B)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (* 2.0 (* (- (pow B 2.0) t_1) F)))
        (t_3 (- t_1 (pow B 2.0)))
        (t_4
         (/
          (sqrt (* t_2 (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ (* (sqrt (fma 2.0 (* B B) (* -8.0 (* A C)))) (sqrt (* F (+ A A)))) t_3)
     (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)
       (if (<= t_4 2e+63)
         (/ (sqrt (* t_2 (+ A (fma (/ (* B B) C) -0.5 A)))) t_3)
         (if (<= t_4 INFINITY)
           (/ (* (sqrt (* A -16.0)) (sqrt (* A (* C F)))) (- t_0))
           (/ (* (sqrt (* -16.0 (* C F))) (- A)) t_0)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B * B));
	double t_1 = (4.0 * A) * C;
	double t_2 = 2.0 * ((pow(B, 2.0) - t_1) * F);
	double t_3 = t_1 - pow(B, 2.0);
	double t_4 = sqrt((t_2 * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt(fma(2.0, (B * B), (-8.0 * (A * C)))) * sqrt((F * (A + A)))) / t_3;
	} 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 if (t_4 <= 2e+63) {
		tmp = sqrt((t_2 * (A + fma(((B * B) / C), -0.5, A)))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((A * -16.0)) * sqrt((A * (C * F)))) / -t_0;
	} else {
		tmp = (sqrt((-16.0 * (C * F))) * -A) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B * B))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(2.0 * Float64(Float64((B ^ 2.0) - t_1) * F))
	t_3 = Float64(t_1 - (B ^ 2.0))
	t_4 = Float64(sqrt(Float64(t_2 * 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(Float64(sqrt(fma(2.0, Float64(B * B), Float64(-8.0 * Float64(A * C)))) * sqrt(Float64(F * Float64(A + A)))) / t_3);
	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);
	elseif (t_4 <= 2e+63)
		tmp = Float64(sqrt(Float64(t_2 * Float64(A + fma(Float64(Float64(B * B) / C), -0.5, A)))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(A * -16.0)) * sqrt(Float64(A * Float64(C * F)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(-16.0 * Float64(C * F))) * Float64(-A)) / 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[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$2 * 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[(N[(N[Sqrt[N[(2.0 * N[(B * B), $MachinePrecision] + N[(-8.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $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], If[LessEqual[t$95$4, 2e+63], N[(N[Sqrt[N[(t$95$2 * N[(A + N[(N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(A * -16.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-A)), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 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 14.9

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

   5. Applied rewrites 14.9%

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

   7. Applied rewrites 25.8%

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

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

   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))) < 2.00000000000000012e63

   1. Initial program 16.4%

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 29.7

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

   5. Applied rewrites 29.7%

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

   if 2.00000000000000012e63 < (/.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 30.0%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 30.5

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

   5. Applied rewrites 30.5%

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

   7. Applied rewrites 30.5%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 16.8

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

   10. Applied rewrites 16.8%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. sqrt-prod N/A

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

       5. pow1/2 N/A

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

       6. metadata-eval N/A

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

       7. lower-*.f64 N/A

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

       8. metadata-eval N/A

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

       9. pow1/2 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. lower-*.f64 39.0

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

   12. Applied rewrites 39.0%

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

   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

   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 3.4

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

   5. Applied rewrites 3.4%

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 4.7

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

   10. Applied rewrites 4.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. sqrt-prod N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. sqrt-pow1 N/A

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

       11. metadata-eval N/A

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

       12. unpow1 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-sqrt.f64 N/A

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

       15. lower-*.f64 5.5

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

   12. Applied rewrites 5.5%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(C \cdot F\right) \cdot -16} \cdot A}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 34.5%

   \[\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{\sqrt{\mathsf{fma}\left(2, B \cdot B, -8 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \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{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 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \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 \infty:\\ \;\;\;\;\frac{\sqrt{A \cdot -16} \cdot \sqrt{A \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 39.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := t\_2 - {B}^{2}\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, B \cdot B, -8 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + A\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq 10^{+171}:\\ \;\;\;\;\sqrt{\left(A \cdot F\right) \cdot t\_0} \cdot \frac{-2}{t\_0}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{A \cdot -16} \cdot \sqrt{A \cdot \left(C \cdot F\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B B)))
        (t_1 (fma (* A C) -4.0 (* B B)))
        (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))
     (/ (* (sqrt (fma 2.0 (* B B) (* -8.0 (* A C)))) (sqrt (* F (+ A A)))) t_3)
     (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)
       (if (<= t_4 1e+171)
         (* (sqrt (* (* A F) t_0)) (/ (- 2.0) t_0))
         (if (<= t_4 INFINITY)
           (/ (* (sqrt (* A -16.0)) (sqrt (* A (* C F)))) (- t_1))
           (/ (* (sqrt (* -16.0 (* C F))) (- A)) t_1)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B * B));
	double t_1 = fma((A * C), -4.0, (B * B));
	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 = (sqrt(fma(2.0, (B * B), (-8.0 * (A * C)))) * sqrt((F * (A + A)))) / t_3;
	} 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 if (t_4 <= 1e+171) {
		tmp = sqrt(((A * F) * t_0)) * (-2.0 / t_0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((A * -16.0)) * sqrt((A * (C * F)))) / -t_1;
	} else {
		tmp = (sqrt((-16.0 * (C * F))) * -A) / t_1;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B * B))
	t_1 = fma(Float64(A * C), -4.0, Float64(B * B))
	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(Float64(sqrt(fma(2.0, Float64(B * B), Float64(-8.0 * Float64(A * C)))) * sqrt(Float64(F * Float64(A + A)))) / t_3);
	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);
	elseif (t_4 <= 1e+171)
		tmp = Float64(sqrt(Float64(Float64(A * F) * t_0)) * Float64(Float64(-2.0) / t_0));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(A * -16.0)) * sqrt(Float64(A * Float64(C * F)))) / Float64(-t_1));
	else
		tmp = Float64(Float64(sqrt(Float64(-16.0 * Float64(C * F))) * Float64(-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[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $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[(N[(N[Sqrt[N[(2.0 * N[(B * B), $MachinePrecision] + N[(-8.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $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], If[LessEqual[t$95$4, 1e+171], N[(N[Sqrt[N[(N[(A * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[((-2.0) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(A * -16.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(N[(N[Sqrt[N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-A)), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 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 14.9

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

   5. Applied rewrites 14.9%

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

   7. Applied rewrites 25.8%

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

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

   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))) < 9.99999999999999954e170

   1. Initial program 23.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 27.1

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

   5. Applied rewrites 27.1%

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

   7. Applied rewrites 29.4%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 29.3

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

   9. Applied rewrites 29.3%

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

   10. Taylor expanded in F around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. rem-square-sqrt N/A

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

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

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

   12. Applied rewrites 29.4%

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

   if 9.99999999999999954e170 < (/.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 14.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.6

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

   5. Applied rewrites 25.6%

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

   7. Applied rewrites 25.6%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 8.9

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

   10. Applied rewrites 8.9%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. sqrt-prod N/A

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

       5. pow1/2 N/A

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

       6. metadata-eval N/A

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

       7. lower-*.f64 N/A

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

       8. metadata-eval N/A

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

       9. pow1/2 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. lower-*.f64 36.1

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

   12. Applied rewrites 36.1%

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

   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

   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 3.4

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

   5. Applied rewrites 3.4%

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 4.7

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

   10. Applied rewrites 4.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. sqrt-prod N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. sqrt-pow1 N/A

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

       11. metadata-eval N/A

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

       12. unpow1 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-sqrt.f64 N/A

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

       15. lower-*.f64 5.5

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

   12. Applied rewrites 5.5%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(C \cdot F\right) \cdot -16} \cdot A}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, B \cdot B, -8 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 10^{+171}:\\ \;\;\;\;\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{-2}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{A \cdot -16} \cdot \sqrt{A \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 40.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := t\_2 - {B}^{2}\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, B \cdot B, -8 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + A\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;t\_4 \leq 10^{+171}:\\ \;\;\;\;\sqrt{\left(A \cdot F\right) \cdot t\_1} \cdot \frac{-2}{t\_1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{A \cdot -16} \cdot \sqrt{A \cdot \left(C \cdot F\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B B)))
        (t_1 (fma -4.0 (* A C) (* B B)))
        (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))
     (/ (* (sqrt (fma 2.0 (* B B) (* -8.0 (* A C)))) (sqrt (* F (+ A A)))) t_3)
     (if (<= t_4 -4e-214)
       (/
        (sqrt
         (*
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B))))
          (* (fma B B (* C (* A -4.0))) (* 2.0 F))))
        (fma B (- B) (* A (* 4.0 C))))
       (if (<= t_4 1e+171)
         (* (sqrt (* (* A F) t_1)) (/ (- 2.0) t_1))
         (if (<= t_4 INFINITY)
           (/ (* (sqrt (* A -16.0)) (sqrt (* A (* C F)))) (- t_0))
           (/ (* (sqrt (* -16.0 (* C F))) (- A)) t_0)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B * B));
	double t_1 = fma(-4.0, (A * C), (B * B));
	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 = (sqrt(fma(2.0, (B * B), (-8.0 * (A * C)))) * sqrt((F * (A + A)))) / t_3;
	} else if (t_4 <= -4e-214) {
		tmp = sqrt((((A + C) - sqrt(fma((A - C), (A - C), (B * B)))) * (fma(B, B, (C * (A * -4.0))) * (2.0 * F)))) / fma(B, -B, (A * (4.0 * C)));
	} else if (t_4 <= 1e+171) {
		tmp = sqrt(((A * F) * t_1)) * (-2.0 / t_1);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((A * -16.0)) * sqrt((A * (C * F)))) / -t_0;
	} else {
		tmp = (sqrt((-16.0 * (C * F))) * -A) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B * B))
	t_1 = fma(-4.0, Float64(A * C), Float64(B * B))
	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(Float64(sqrt(fma(2.0, Float64(B * B), Float64(-8.0 * Float64(A * C)))) * sqrt(Float64(F * Float64(A + A)))) / t_3);
	elseif (t_4 <= -4e-214)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))) * Float64(fma(B, B, Float64(C * Float64(A * -4.0))) * Float64(2.0 * F)))) / fma(B, Float64(-B), Float64(A * Float64(4.0 * C))));
	elseif (t_4 <= 1e+171)
		tmp = Float64(sqrt(Float64(Float64(A * F) * t_1)) * Float64(Float64(-2.0) / t_1));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(A * -16.0)) * sqrt(Float64(A * Float64(C * F)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(-16.0 * Float64(C * F))) * Float64(-A)) / 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[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $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[(N[(N[Sqrt[N[(2.0 * N[(B * B), $MachinePrecision] + N[(-8.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -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[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B * (-B) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e+171], N[(N[Sqrt[N[(N[(A * F), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * N[((-2.0) / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(A * -16.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-A)), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 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 14.9

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

   5. Applied rewrites 14.9%

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

   7. Applied rewrites 25.8%

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

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

   4. Applied rewrites 96.1%

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

   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))) < 9.99999999999999954e170

   1. Initial program 23.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 27.1

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

   5. Applied rewrites 27.1%

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

   7. Applied rewrites 29.4%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 29.3

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

   9. Applied rewrites 29.3%

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

   10. Taylor expanded in F around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. rem-square-sqrt N/A

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

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

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

   12. Applied rewrites 29.4%

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

   if 9.99999999999999954e170 < (/.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 14.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.6

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

   5. Applied rewrites 25.6%

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

   7. Applied rewrites 25.6%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 8.9

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

   10. Applied rewrites 8.9%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. sqrt-prod N/A

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

       5. pow1/2 N/A

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

       6. metadata-eval N/A

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

       7. lower-*.f64 N/A

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

       8. metadata-eval N/A

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

       9. pow1/2 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. lower-*.f64 36.1

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

   12. Applied rewrites 36.1%

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

   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

   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 3.4

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

   5. Applied rewrites 3.4%

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 4.7

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

   10. Applied rewrites 4.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. sqrt-prod N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. sqrt-pow1 N/A

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

       11. metadata-eval N/A

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

       12. unpow1 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-sqrt.f64 N/A

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

       15. lower-*.f64 5.5

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

   12. Applied rewrites 5.5%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(C \cdot F\right) \cdot -16} \cdot A}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 33.8%

   \[\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{\sqrt{\mathsf{fma}\left(2, B \cdot B, -8 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \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(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\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 10^{+171}:\\ \;\;\;\;\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{-2}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\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 \infty:\\ \;\;\;\;\frac{\sqrt{A \cdot -16} \cdot \sqrt{A \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{t\_0}}\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+171}:\\ \;\;\;\;\sqrt{\left(A \cdot F\right) \cdot t\_0} \cdot \frac{-2}{t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{A \cdot -16} \cdot \sqrt{A \cdot \left(C \cdot F\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B B)))
        (t_1 (fma (* A C) -4.0 (* B B)))
        (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))
     (* -2.0 (sqrt (/ (* A F) t_0)))
     (if (<= t_3 -4e-214)
       (/
        (sqrt
         (*
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B))))
          (* (fma B B (* C (* A -4.0))) (* 2.0 F))))
        (fma B (- B) (* A (* 4.0 C))))
       (if (<= t_3 1e+171)
         (* (sqrt (* (* A F) t_0)) (/ (- 2.0) t_0))
         (if (<= t_3 INFINITY)
           (/ (* (sqrt (* A -16.0)) (sqrt (* A (* C F)))) (- t_1))
           (/ (* (sqrt (* -16.0 (* C F))) (- A)) t_1)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B * B));
	double t_1 = fma((A * C), -4.0, (B * B));
	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 = -2.0 * sqrt(((A * F) / t_0));
	} else if (t_3 <= -4e-214) {
		tmp = sqrt((((A + C) - sqrt(fma((A - C), (A - C), (B * B)))) * (fma(B, B, (C * (A * -4.0))) * (2.0 * F)))) / fma(B, -B, (A * (4.0 * C)));
	} else if (t_3 <= 1e+171) {
		tmp = sqrt(((A * F) * t_0)) * (-2.0 / t_0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((A * -16.0)) * sqrt((A * (C * F)))) / -t_1;
	} else {
		tmp = (sqrt((-16.0 * (C * F))) * -A) / t_1;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B * B))
	t_1 = fma(Float64(A * C), -4.0, Float64(B * B))
	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(-2.0 * sqrt(Float64(Float64(A * F) / t_0)));
	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(fma(B, B, Float64(C * Float64(A * -4.0))) * Float64(2.0 * F)))) / fma(B, Float64(-B), Float64(A * Float64(4.0 * C))));
	elseif (t_3 <= 1e+171)
		tmp = Float64(sqrt(Float64(Float64(A * F) * t_0)) * Float64(Float64(-2.0) / t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(A * -16.0)) * sqrt(Float64(A * Float64(C * F)))) / Float64(-t_1));
	else
		tmp = Float64(Float64(sqrt(Float64(-16.0 * Float64(C * F))) * Float64(-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[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $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[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / t$95$0), $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[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B * (-B) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+171], N[(N[Sqrt[N[(N[(A * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[((-2.0) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(A * -16.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(N[(N[Sqrt[N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-A)), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 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 14.9

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

   5. Applied rewrites 14.9%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 22.1

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

   8. Applied rewrites 22.1%

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

   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))) < -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

   4. Applied rewrites 96.1%

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

   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))) < 9.99999999999999954e170

   1. Initial program 23.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 27.1

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

   5. Applied rewrites 27.1%

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

   7. Applied rewrites 29.4%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 29.3

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

   9. Applied rewrites 29.3%

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

   10. Taylor expanded in F around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. rem-square-sqrt N/A

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

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

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

   12. Applied rewrites 29.4%

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

   if 9.99999999999999954e170 < (/.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 14.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.6

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

   5. Applied rewrites 25.6%

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

   7. Applied rewrites 25.6%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 8.9

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

   10. Applied rewrites 8.9%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. sqrt-prod N/A

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

       5. pow1/2 N/A

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

       6. metadata-eval N/A

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

       7. lower-*.f64 N/A

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

       8. metadata-eval N/A

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

       9. pow1/2 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. lower-*.f64 36.1

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

   12. Applied rewrites 36.1%

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

   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

   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 3.4

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

   5. Applied rewrites 3.4%

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 4.7

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

   10. Applied rewrites 4.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. sqrt-prod N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. sqrt-pow1 N/A

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

       11. metadata-eval N/A

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

       12. unpow1 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-sqrt.f64 N/A

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

       15. lower-*.f64 5.5

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

   12. Applied rewrites 5.5%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(C \cdot F\right) \cdot -16} \cdot A}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 10^{+171}:\\ \;\;\;\;\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{-2}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{A \cdot -16} \cdot \sqrt{A \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 36.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B}^{2}}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+168}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{t\_1}}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{t\_0}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq 10^{+171}:\\ \;\;\;\;\sqrt{\left(A \cdot F\right) \cdot t\_1} \cdot \frac{-2}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{A \cdot -16} \cdot \sqrt{A \cdot \left(C \cdot F\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B B)))
        (t_1 (fma -4.0 (* A C) (* B B)))
        (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 -5e+168)
     (* -2.0 (sqrt (/ (* A F) t_1)))
     (if (<= t_3 -2e-153)
       (*
        (sqrt (/ (* F (- (+ A C) (sqrt (fma B B (* (- A C) (- A C)))))) t_0))
        (- (sqrt 2.0)))
       (if (<= t_3 1e+171)
         (* (sqrt (* (* A F) t_1)) (/ (- 2.0) t_1))
         (if (<= t_3 INFINITY)
           (/ (* (sqrt (* A -16.0)) (sqrt (* A (* C F)))) (- t_0))
           (/ (* (sqrt (* -16.0 (* C F))) (- A)) t_0)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B * B));
	double t_1 = fma(-4.0, (A * C), (B * B));
	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 <= -5e+168) {
		tmp = -2.0 * sqrt(((A * F) / t_1));
	} else if (t_3 <= -2e-153) {
		tmp = sqrt(((F * ((A + C) - sqrt(fma(B, B, ((A - C) * (A - C)))))) / t_0)) * -sqrt(2.0);
	} else if (t_3 <= 1e+171) {
		tmp = sqrt(((A * F) * t_1)) * (-2.0 / t_1);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((A * -16.0)) * sqrt((A * (C * F)))) / -t_0;
	} else {
		tmp = (sqrt((-16.0 * (C * F))) * -A) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B * B))
	t_1 = fma(-4.0, Float64(A * C), Float64(B * B))
	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 <= -5e+168)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / t_1)));
	elseif (t_3 <= -2e-153)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) - sqrt(fma(B, B, Float64(Float64(A - C) * Float64(A - C)))))) / t_0)) * Float64(-sqrt(2.0)));
	elseif (t_3 <= 1e+171)
		tmp = Float64(sqrt(Float64(Float64(A * F) * t_1)) * Float64(Float64(-2.0) / t_1));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(A * -16.0)) * sqrt(Float64(A * Float64(C * F)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(-16.0 * Float64(C * F))) * Float64(-A)) / 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[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $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, -5e+168], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-153], N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(B * B + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, 1e+171], N[(N[Sqrt[N[(N[(A * F), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * N[((-2.0) / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(A * -16.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-A)), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 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))) < -4.99999999999999967e168

   1. Initial program 8.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 20.3

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

   5. Applied rewrites 20.3%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 21.1

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

   8. Applied rewrites 21.1%

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

   if -4.99999999999999967e168 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.00000000000000008e-153

   1. Initial program 96.3%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 98.5%

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

   if -2.00000000000000008e-153 < (/.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))) < 9.99999999999999954e170

   1. Initial program 33.0%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 24.0

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

   5. Applied rewrites 24.0%

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

   7. Applied rewrites 26.0%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 25.9

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

   9. Applied rewrites 25.9%

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

   10. Taylor expanded in F around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. rem-square-sqrt N/A

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

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

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

   12. Applied rewrites 26.0%

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

   if 9.99999999999999954e170 < (/.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 14.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.6

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

   5. Applied rewrites 25.6%

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

   7. Applied rewrites 25.6%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 8.9

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

   10. Applied rewrites 8.9%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. sqrt-prod N/A

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

       5. pow1/2 N/A

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

       6. metadata-eval N/A

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

       7. lower-*.f64 N/A

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

       8. metadata-eval N/A

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

       9. pow1/2 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. lower-*.f64 36.1

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

   12. Applied rewrites 36.1%

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

   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

   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 3.4

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

   5. Applied rewrites 3.4%

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 4.7

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

   10. Applied rewrites 4.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. sqrt-prod N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. sqrt-pow1 N/A

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

       11. metadata-eval N/A

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

       12. unpow1 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-sqrt.f64 N/A

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

       15. lower-*.f64 5.5

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

   12. Applied rewrites 5.5%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(C \cdot F\right) \cdot -16} \cdot A}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+168}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 10^{+171}:\\ \;\;\;\;\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{-2}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{A \cdot -16} \cdot \sqrt{A \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\\ t_1 := -t\_0\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B}^{2}}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-30}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(\left(A \cdot -16\right) \cdot \left(C \cdot F\right)\right)}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{A \cdot -16} \cdot \sqrt{A \cdot \left(C \cdot F\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B B)))
        (t_1 (- t_0))
        (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 -5e-30)
     (* -2.0 (sqrt (/ (* A F) (fma -4.0 (* A C) (* B B)))))
     (if (<= t_3 -4e-214)
       (/ (* (sqrt 2.0) (sqrt (* F (- A (sqrt (fma B B (* A A))))))) (- B))
       (if (<= t_3 2e+63)
         (/ (sqrt (* A (* (* A -16.0) (* C F)))) t_1)
         (if (<= t_3 INFINITY)
           (/ (* (sqrt (* A -16.0)) (sqrt (* A (* C F)))) t_1)
           (/ (* (sqrt (* -16.0 (* C F))) (- A)) t_0)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B * B));
	double t_1 = -t_0;
	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 <= -5e-30) {
		tmp = -2.0 * sqrt(((A * F) / fma(-4.0, (A * C), (B * B))));
	} else if (t_3 <= -4e-214) {
		tmp = (sqrt(2.0) * sqrt((F * (A - sqrt(fma(B, B, (A * A))))))) / -B;
	} else if (t_3 <= 2e+63) {
		tmp = sqrt((A * ((A * -16.0) * (C * F)))) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((A * -16.0)) * sqrt((A * (C * F)))) / t_1;
	} else {
		tmp = (sqrt((-16.0 * (C * F))) * -A) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B * B))
	t_1 = Float64(-t_0)
	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 <= -5e-30)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / fma(-4.0, Float64(A * C), Float64(B * B)))));
	elseif (t_3 <= -4e-214)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(A - sqrt(fma(B, B, Float64(A * A))))))) / Float64(-B));
	elseif (t_3 <= 2e+63)
		tmp = Float64(sqrt(Float64(A * Float64(Float64(A * -16.0) * Float64(C * F)))) / t_1);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(A * -16.0)) * sqrt(Float64(A * Float64(C * F)))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(-16.0 * Float64(C * F))) * Float64(-A)) / 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[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, 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, -5e-30], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -4e-214], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B)), $MachinePrecision], If[LessEqual[t$95$3, 2e+63], N[(N[Sqrt[N[(A * N[(N[(A * -16.0), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(A * -16.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-A)), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 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))) < -4.99999999999999972e-30

   1. Initial program 36.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 29.3

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

   5. Applied rewrites 29.3%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 29.8

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

   8. Applied rewrites 29.8%

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

   if -4.99999999999999972e-30 < (/.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 96.4%

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

   3. Taylor expanded in C around 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. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 36.8%

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

   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))) < 2.00000000000000012e63

   1. Initial program 16.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 24.6

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

   5. Applied rewrites 24.6%

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

   7. Applied rewrites 24.6%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 24.8

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

   10. Applied rewrites 24.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 24.9

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

   12. Applied rewrites 24.9%

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

   if 2.00000000000000012e63 < (/.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 30.0%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 30.5

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

   5. Applied rewrites 30.5%

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

   7. Applied rewrites 30.5%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 16.8

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

   10. Applied rewrites 16.8%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. sqrt-prod N/A

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

       5. pow1/2 N/A

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

       6. metadata-eval N/A

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

       7. lower-*.f64 N/A

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

       8. metadata-eval N/A

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

       9. pow1/2 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. lower-*.f64 39.0

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

   12. Applied rewrites 39.0%

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

   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

   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 3.4

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

   5. Applied rewrites 3.4%

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 4.7

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

   10. Applied rewrites 4.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. sqrt-prod N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. sqrt-pow1 N/A

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

       11. metadata-eval N/A

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

       12. unpow1 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-sqrt.f64 N/A

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

       15. lower-*.f64 5.5

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

   12. Applied rewrites 5.5%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(C \cdot F\right) \cdot -16} \cdot A}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-30}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(\left(A \cdot -16\right) \cdot \left(C \cdot F\right)\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{A \cdot -16} \cdot \sqrt{A \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 29.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B}^{2}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-30}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B B)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B 2.0)))))
   (if (<= t_2 -5e-30)
     (* -2.0 (sqrt (/ (* A F) (fma -4.0 (* A C) (* B B)))))
     (if (<= t_2 -4e-214)
       (/ (* (sqrt 2.0) (sqrt (* F (- A (sqrt (fma B B (* A A))))))) (- B))
       (if (<= t_2 INFINITY)
         (/ (sqrt (* (* F t_0) (* 2.0 (+ A A)))) (* 4.0 (* A C)))
         (/ (* (sqrt (* -16.0 (* C F))) (- A)) t_0))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B * B));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B, 2.0));
	double tmp;
	if (t_2 <= -5e-30) {
		tmp = -2.0 * sqrt(((A * F) / fma(-4.0, (A * C), (B * B))));
	} else if (t_2 <= -4e-214) {
		tmp = (sqrt(2.0) * sqrt((F * (A - sqrt(fma(B, B, (A * A))))))) / -B;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + A)))) / (4.0 * (A * C));
	} else {
		tmp = (sqrt((-16.0 * (C * F))) * -A) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B * B))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -5e-30)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / fma(-4.0, Float64(A * C), Float64(B * B)))));
	elseif (t_2 <= -4e-214)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(A - sqrt(fma(B, B, Float64(A * A))))))) / Float64(-B));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + A)))) / Float64(4.0 * Float64(A * C)));
	else
		tmp = Float64(Float64(sqrt(Float64(-16.0 * Float64(C * F))) * Float64(-A)) / 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[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $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$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-30], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-214], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B)), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-A)), $MachinePrecision] / t$95$0), $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))) < -4.99999999999999972e-30

   1. Initial program 36.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 29.3

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

   5. Applied rewrites 29.3%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 29.8

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

   8. Applied rewrites 29.8%

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

   if -4.99999999999999972e-30 < (/.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 96.4%

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

   3. Taylor expanded in C around 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. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 36.8%

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

   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))) < +inf.0

   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

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 26.7

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

   5. Applied rewrites 26.7%

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

   7. Applied rewrites 26.7%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 26.5

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

   10. Applied rewrites 26.5%

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

   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

   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 3.4

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

   5. Applied rewrites 3.4%

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 4.7

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

   10. Applied rewrites 4.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. sqrt-prod N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. sqrt-pow1 N/A

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

       11. metadata-eval N/A

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

       12. unpow1 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-sqrt.f64 N/A

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

       15. lower-*.f64 5.5

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

   12. Applied rewrites 5.5%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(C \cdot F\right) \cdot -16} \cdot A}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-30}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B}^{2}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-92}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B B)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B 2.0)))))
   (if (<= t_2 -5e-92)
     (* -2.0 (sqrt (/ (* A F) (fma -4.0 (* A C) (* B B)))))
     (if (<= t_2 -4e-214)
       (* (sqrt (* F (- C (sqrt (fma B B (* C C)))))) (- (/ (sqrt 2.0) B)))
       (if (<= t_2 INFINITY)
         (/ (sqrt (* (* F t_0) (* 2.0 (+ A A)))) (* 4.0 (* A C)))
         (/ (* (sqrt (* -16.0 (* C F))) (- A)) t_0))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B * B));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B, 2.0));
	double tmp;
	if (t_2 <= -5e-92) {
		tmp = -2.0 * sqrt(((A * F) / fma(-4.0, (A * C), (B * B))));
	} else if (t_2 <= -4e-214) {
		tmp = sqrt((F * (C - sqrt(fma(B, B, (C * C)))))) * -(sqrt(2.0) / B);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + A)))) / (4.0 * (A * C));
	} else {
		tmp = (sqrt((-16.0 * (C * F))) * -A) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B * B))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -5e-92)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / fma(-4.0, Float64(A * C), Float64(B * B)))));
	elseif (t_2 <= -4e-214)
		tmp = Float64(sqrt(Float64(F * Float64(C - sqrt(fma(B, B, Float64(C * C)))))) * Float64(-Float64(sqrt(2.0) / B)));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + A)))) / Float64(4.0 * Float64(A * C)));
	else
		tmp = Float64(Float64(sqrt(Float64(-16.0 * Float64(C * F))) * Float64(-A)) / 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[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $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$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-92], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-214], N[(N[Sqrt[N[(F * N[(C - N[Sqrt[N[(B * B + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-A)), $MachinePrecision] / t$95$0), $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))) < -5.00000000000000011e-92

   1. Initial program 46.1%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 31.0

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 31.3

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

   8. Applied rewrites 31.3%

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

   if -5.00000000000000011e-92 < (/.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 95.9%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 47.3%

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

   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))) < +inf.0

   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

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 26.7

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

   5. Applied rewrites 26.7%

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

   7. Applied rewrites 26.7%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 26.5

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

   10. Applied rewrites 26.5%

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

   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

   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 3.4

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

   5. Applied rewrites 3.4%

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 4.7

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

   10. Applied rewrites 4.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. sqrt-prod N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. sqrt-pow1 N/A

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

       11. metadata-eval N/A

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

       12. unpow1 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-sqrt.f64 N/A

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

       15. lower-*.f64 5.5

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

   12. Applied rewrites 5.5%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(C \cdot F\right) \cdot -16} \cdot A}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-92}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 28.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B}^{2}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-172}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B B)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B 2.0)))))
   (if (<= t_2 -5e-172)
     (* -2.0 (sqrt (/ (* A F) (fma -4.0 (* A C) (* B B)))))
     (if (<= t_2 INFINITY)
       (/ (sqrt (* (* F t_0) (* 2.0 (+ A A)))) (* 4.0 (* A C)))
       (/ (* (sqrt (* -16.0 (* C F))) (- A)) t_0)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B * B));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B, 2.0));
	double tmp;
	if (t_2 <= -5e-172) {
		tmp = -2.0 * sqrt(((A * F) / fma(-4.0, (A * C), (B * B))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + A)))) / (4.0 * (A * C));
	} else {
		tmp = (sqrt((-16.0 * (C * F))) * -A) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B * B))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -5e-172)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / fma(-4.0, Float64(A * C), Float64(B * B)))));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + A)))) / Float64(4.0 * Float64(A * C)));
	else
		tmp = Float64(Float64(sqrt(Float64(-16.0 * Float64(C * F))) * Float64(-A)) / 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[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $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$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-172], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-A)), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999999e-172

   1. Initial program 50.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 28.8

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

   5. Applied rewrites 28.8%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 29.3

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

   8. Applied rewrites 29.3%

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

   if -4.9999999999999999e-172 < (/.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 27.3%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 24.7

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

   5. Applied rewrites 24.7%

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

   7. Applied rewrites 24.7%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 24.4

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

   10. Applied rewrites 24.4%

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

   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

   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 3.4

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

   5. Applied rewrites 3.4%

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 4.7

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

   10. Applied rewrites 4.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. sqrt-prod N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. sqrt-pow1 N/A

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

       11. metadata-eval N/A

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

       12. unpow1 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-sqrt.f64 N/A

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

       15. lower-*.f64 5.5

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

   12. Applied rewrites 5.5%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(C \cdot F\right) \cdot -16} \cdot A}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 19.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-172}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot F\right)} \cdot \left(-A\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B}^{2}} \leq -5 \cdot 10^{-172}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(\left(A \cdot -16\right) \cdot \left(C \cdot F\right)\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)))
   (if (<=
        (/
         (sqrt
          (*
           (* 2.0 (* (- (pow B 2.0) t_0) F))
           (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
         (- t_0 (pow B 2.0)))
        -5e-172)
     (* -2.0 (sqrt (/ (* A F) (fma -4.0 (* A C) (* B B)))))
     (/ (sqrt (* A (* (* A -16.0) (* C F)))) (- (fma (* A C) -4.0 (* B B)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if ((sqrt(((2.0 * ((pow(B, 2.0) - t_0) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B, 2.0))) <= -5e-172) {
		tmp = -2.0 * sqrt(((A * F) / fma(-4.0, (A * C), (B * B))));
	} else {
		tmp = sqrt((A * ((A * -16.0) * (C * F)))) / -fma((A * C), -4.0, (B * B));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B ^ 2.0))) <= -5e-172)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / fma(-4.0, Float64(A * C), Float64(B * B)))));
	else
		tmp = Float64(sqrt(Float64(A * Float64(Float64(A * -16.0) * Float64(C * F)))) / Float64(-fma(Float64(A * C), -4.0, Float64(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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $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$0 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-172], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(A * N[(N[(A * -16.0), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]]

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))) < -4.9999999999999999e-172

   1. Initial program 50.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 28.8

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

   5. Applied rewrites 28.8%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 29.3

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

   8. Applied rewrites 29.3%

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

   if -4.9999999999999999e-172 < (/.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 11.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 12.3

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

   5. Applied rewrites 12.3%

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

   7. Applied rewrites 12.3%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 11.3

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

   10. Applied rewrites 11.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 12.7

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

   12. Applied rewrites 12.7%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(C \cdot F\right) \cdot \left(A \cdot -16\right)\right) \cdot A}}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-172}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(\left(A \cdot -16\right) \cdot \left(C \cdot F\right)\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 24.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B}^{2}} \leq -5 \cdot 10^{-172}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot F\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)))
   (if (<=
        (/
         (sqrt
          (*
           (* 2.0 (* (- (pow B 2.0) t_0) F))
           (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
         (- t_0 (pow B 2.0)))
        -5e-172)
     (* -2.0 (sqrt (/ (* A F) (fma -4.0 (* A C) (* B B)))))
     (/ (sqrt (* (* C F) (* -16.0 (* A A)))) (* 4.0 (* A C))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if ((sqrt(((2.0 * ((pow(B, 2.0) - t_0) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B, 2.0))) <= -5e-172) {
		tmp = -2.0 * sqrt(((A * F) / fma(-4.0, (A * C), (B * B))));
	} else {
		tmp = sqrt(((C * F) * (-16.0 * (A * A)))) / (4.0 * (A * C));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B ^ 2.0))) <= -5e-172)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / fma(-4.0, Float64(A * C), Float64(B * B)))));
	else
		tmp = Float64(sqrt(Float64(Float64(C * F) * Float64(-16.0 * Float64(A * A)))) / Float64(4.0 * Float64(A * C)));
	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 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $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$0 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-172], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(C * F), $MachinePrecision] * N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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))) < -4.9999999999999999e-172

   1. Initial program 50.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 28.8

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

   5. Applied rewrites 28.8%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 29.3

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

   8. Applied rewrites 29.3%

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

   if -4.9999999999999999e-172 < (/.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 11.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 12.3

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

   5. Applied rewrites 12.3%

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

   7. Applied rewrites 12.3%

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

   8. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 11.3

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

   10. Applied rewrites 11.3%

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

   11. Taylor expanded in A around inf

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-*.f64 11.8

          \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}{4 \cdot \color{blue}{\left(A \cdot C\right)}} \]

   13. Applied rewrites 11.8%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-172}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot F\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 29.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(A \cdot F\right) \cdot t\_0} \cdot \frac{-2}{t\_0}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+306}:\\ \;\;\;\;\frac{B \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)\right)}}{C \cdot 0 - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

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 (* A C) (* B B))))
   (if (<= (pow B 2.0) 2e+147)
     (* (sqrt (* (* A F) t_0)) (/ (- 2.0) t_0))
     (if (<= (pow B 2.0) 1e+306)
       (/
        (* B (sqrt (* 2.0 (* F (- A (sqrt (fma B B (* A A))))))))
        (- (* C 0.0) (* B B)))
       (* -2.0 (/ (sqrt (* A F)) 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, (A * C), (B * B));
	double tmp;
	if (pow(B, 2.0) <= 2e+147) {
		tmp = sqrt(((A * F) * t_0)) * (-2.0 / t_0);
	} else if (pow(B, 2.0) <= 1e+306) {
		tmp = (B * sqrt((2.0 * (F * (A - sqrt(fma(B, B, (A * A)))))))) / ((C * 0.0) - (B * B));
	} else {
		tmp = -2.0 * (sqrt((A * F)) / 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(A * C), Float64(B * B))
	tmp = 0.0
	if ((B ^ 2.0) <= 2e+147)
		tmp = Float64(sqrt(Float64(Float64(A * F) * t_0)) * Float64(Float64(-2.0) / t_0));
	elseif ((B ^ 2.0) <= 1e+306)
		tmp = Float64(Float64(B * sqrt(Float64(2.0 * Float64(F * Float64(A - sqrt(fma(B, B, Float64(A * A)))))))) / Float64(Float64(C * 0.0) - Float64(B * B)));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / 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[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e+147], N[(N[Sqrt[N[(N[(A * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[((-2.0) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e+306], N[(N[(B * N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(C * 0.0), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e147

   1. Initial program 31.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 27.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 27.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 27.8%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \sqrt{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{{2}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. sqr-pow N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{{\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{{\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot {\color{blue}{\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}}^{2}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. metadata-eval 27.8

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot {\left({2}^{\color{blue}{0.25}}\right)}^{2}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 27.8%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{{\left({2}^{0.25}\right)}^{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   10. Taylor expanded in F around 0

       \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)} \]

       2. lower-neg.f64 N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot F\right)} \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, \color{blue}{A \cdot C}, {B}^{2}\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       13. rem-square-sqrt N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{\color{blue}{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       14. cancel-sign-sub-inv N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{2}{\color{blue}{{B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}}\right) \]

   12. Applied rewrites 27.9%

       \[\leadsto \color{blue}{-\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{2}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]

   if 2e147 < (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000002e306

   1. Initial program 33.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(B \cdot \sqrt{2}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \left(B \cdot \sqrt{2}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \left(B \cdot \sqrt{2}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \left(A - \color{blue}{\sqrt{{A}^{2} + {B}^{2}}}\right)} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{\color{blue}{{B}^{2} + {A}^{2}}}\right)} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{\color{blue}{\mathsf{fma}\left(B, B, {A}^{2}\right)}}\right)} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \color{blue}{\left(B \cdot \sqrt{2}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      13. lower-sqrt.f64 34.8

          \[\leadsto \frac{-\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(B \cdot \color{blue}{\sqrt{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 34.8%

      \[\leadsto \frac{-\color{blue}{\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(B \cdot \sqrt{2}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\frac{B \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)\right)}}{C \cdot 0 - B \cdot B}} \]

   if 1.00000000000000002e306 < (pow.f64 B #s(literal 2 binary64))

   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

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 0.0

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 0.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]

      2. associate-*r/ N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

      3. *-rgt-identity N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

      6. lower-*.f64 4.8

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{A \cdot F}}}{B} \]

   8. Applied rewrites 4.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{A \cdot F}}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{-2}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+306}:\\ \;\;\;\;\frac{B \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)\right)}}{C \cdot 0 - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 29.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(A \cdot F\right) \cdot t\_0} \cdot \frac{-2}{t\_0}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+306}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

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 (* A C) (* B B))))
   (if (<= (pow B 2.0) 2e+147)
     (* (sqrt (* (* A F) t_0)) (/ (- 2.0) t_0))
     (if (<= (pow B 2.0) 1e+306)
       (/ (* (sqrt 2.0) (sqrt (* F (- A (sqrt (fma B B (* A A))))))) (- B))
       (* -2.0 (/ (sqrt (* A F)) 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, (A * C), (B * B));
	double tmp;
	if (pow(B, 2.0) <= 2e+147) {
		tmp = sqrt(((A * F) * t_0)) * (-2.0 / t_0);
	} else if (pow(B, 2.0) <= 1e+306) {
		tmp = (sqrt(2.0) * sqrt((F * (A - sqrt(fma(B, B, (A * A))))))) / -B;
	} else {
		tmp = -2.0 * (sqrt((A * F)) / 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(A * C), Float64(B * B))
	tmp = 0.0
	if ((B ^ 2.0) <= 2e+147)
		tmp = Float64(sqrt(Float64(Float64(A * F) * t_0)) * Float64(Float64(-2.0) / t_0));
	elseif ((B ^ 2.0) <= 1e+306)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(A - sqrt(fma(B, B, Float64(A * A))))))) / Float64(-B));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / 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[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e+147], N[(N[Sqrt[N[(N[(A * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[((-2.0) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e+306], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B)), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e147

   1. Initial program 31.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 27.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 27.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 27.8%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \sqrt{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{{2}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. sqr-pow N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{{\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{{\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot {\color{blue}{\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}}^{2}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. metadata-eval 27.8

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot {\left({2}^{\color{blue}{0.25}}\right)}^{2}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 27.8%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{{\left({2}^{0.25}\right)}^{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   10. Taylor expanded in F around 0

       \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)} \]

       2. lower-neg.f64 N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot F\right)} \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, \color{blue}{A \cdot C}, {B}^{2}\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       13. rem-square-sqrt N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{\color{blue}{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       14. cancel-sign-sub-inv N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{2}{\color{blue}{{B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}}\right) \]

   12. Applied rewrites 27.9%

       \[\leadsto \color{blue}{-\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{2}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]

   if 2e147 < (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000002e306

   1. Initial program 33.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}{-1 \cdot B}} \]

   5. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}} \]

   if 1.00000000000000002e306 < (pow.f64 B #s(literal 2 binary64))

   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

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 0.0

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 0.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]

      2. associate-*r/ N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

      3. *-rgt-identity N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

      6. lower-*.f64 4.8

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{A \cdot F}}}{B} \]

   8. Applied rewrites 4.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{A \cdot F}}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{-2}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+306}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 20.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+267}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= (pow B 2.0) 2e+267)
   (* -2.0 (sqrt (/ (* A F) (fma -4.0 (* A C) (* B B)))))
   (* -2.0 (/ (sqrt (* A F)) B))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (pow(B, 2.0) <= 2e+267) {
		tmp = -2.0 * sqrt(((A * F) / fma(-4.0, (A * C), (B * B))));
	} else {
		tmp = -2.0 * (sqrt((A * F)) / B);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if ((B ^ 2.0) <= 2e+267)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / fma(-4.0, Float64(A * C), Float64(B * B)))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e+267], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e267

   1. Initial program 32.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 24.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 24.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{A \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{A \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\sqrt{\frac{A \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\color{blue}{\frac{A \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      4. lower-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{\color{blue}{A \cdot F}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto -2 \cdot \sqrt{\frac{A \cdot F}{\color{blue}{{B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}}} \]

      6. metadata-eval N/A

         \[\leadsto -2 \cdot \sqrt{\frac{A \cdot F}{{B}^{2} + \color{blue}{-4} \cdot \left(A \cdot C\right)}} \]

      7. +-commutative N/A

         \[\leadsto -2 \cdot \sqrt{\frac{A \cdot F}{\color{blue}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{A \cdot F}{\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}} \]

      9. lower-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, \color{blue}{A \cdot C}, {B}^{2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto -2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)}} \]

      11. lower-*.f64 17.0

          \[\leadsto -2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)}} \]

   8. Applied rewrites 17.0%

      \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}} \]

   if 1.9999999999999999e267 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 4.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 0.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 0.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]

      2. associate-*r/ N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

      3. *-rgt-identity N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

      6. lower-*.f64 4.4

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{A \cdot F}}}{B} \]

   8. Applied rewrites 4.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{A \cdot F}}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 29.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ \mathbf{if}\;B \leq 7 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\left(A \cdot F\right) \cdot t\_0} \cdot \frac{-2}{t\_0}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(B \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(C, 0, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

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 (* A C) (* B B))))
   (if (<= B 7e+73)
     (* (sqrt (* (* A F) t_0)) (/ (- 2.0) t_0))
     (if (<= B 1.35e+154)
       (*
        (* B (sqrt (* 2.0 (* F (- A (sqrt (fma B B (* A A))))))))
        (/ -1.0 (fma C 0.0 (* B B))))
       (* -2.0 (/ (sqrt (* A F)) 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, (A * C), (B * B));
	double tmp;
	if (B <= 7e+73) {
		tmp = sqrt(((A * F) * t_0)) * (-2.0 / t_0);
	} else if (B <= 1.35e+154) {
		tmp = (B * sqrt((2.0 * (F * (A - sqrt(fma(B, B, (A * A)))))))) * (-1.0 / fma(C, 0.0, (B * B)));
	} else {
		tmp = -2.0 * (sqrt((A * F)) / 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(A * C), Float64(B * B))
	tmp = 0.0
	if (B <= 7e+73)
		tmp = Float64(sqrt(Float64(Float64(A * F) * t_0)) * Float64(Float64(-2.0) / t_0));
	elseif (B <= 1.35e+154)
		tmp = Float64(Float64(B * sqrt(Float64(2.0 * Float64(F * Float64(A - sqrt(fma(B, B, Float64(A * A)))))))) * Float64(-1.0 / fma(C, 0.0, Float64(B * B))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / 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[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 7e+73], N[(N[Sqrt[N[(N[(A * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[((-2.0) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.35e+154], N[(N[(B * N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(C * 0.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 7.00000000000000004e73

   1. Initial program 27.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 22.1

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 22.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 22.6%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \sqrt{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{{2}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. sqr-pow N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{{\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{{\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot {\color{blue}{\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}}^{2}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. metadata-eval 22.5

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot {\left({2}^{\color{blue}{0.25}}\right)}^{2}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 22.5%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \color{blue}{{\left({2}^{0.25}\right)}^{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   10. Taylor expanded in F around 0

       \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)} \]

       2. lower-neg.f64 N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot F\right)} \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, \color{blue}{A \cdot C}, {B}^{2}\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       13. rem-square-sqrt N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{\color{blue}{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right) \]

       14. cancel-sign-sub-inv N/A

           \[\leadsto \mathsf{neg}\left(\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{2}{\color{blue}{{B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}}\right) \]

   12. Applied rewrites 22.6%

       \[\leadsto \color{blue}{-\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{2}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]

   if 7.00000000000000004e73 < B < 1.35000000000000003e154

   1. Initial program 36.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(B \cdot \sqrt{2}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \left(B \cdot \sqrt{2}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \left(B \cdot \sqrt{2}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \left(A - \color{blue}{\sqrt{{A}^{2} + {B}^{2}}}\right)} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{\color{blue}{{B}^{2} + {A}^{2}}}\right)} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{\color{blue}{\mathsf{fma}\left(B, B, {A}^{2}\right)}}\right)} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \cdot \left(B \cdot \sqrt{2}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \color{blue}{\left(B \cdot \sqrt{2}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      13. lower-sqrt.f64 57.5

          \[\leadsto \frac{-\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(B \cdot \color{blue}{\sqrt{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 57.5%

      \[\leadsto \frac{-\color{blue}{\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(B \cdot \sqrt{2}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\left(B \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(C, 0, B \cdot B\right)}} \]

   if 1.35000000000000003e154 < B

   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

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 0.0

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 0.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]

      2. associate-*r/ N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

      3. *-rgt-identity N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

      6. lower-*.f64 6.4

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{A \cdot F}}}{B} \]

   8. Applied rewrites 6.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{A \cdot F}}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \frac{-2}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(B \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(C, 0, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\\ \mathbf{if}\;B \leq 3.4 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{A \cdot \left(F \cdot t\_0\right)} \cdot \frac{-2}{t\_0}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

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 B B (* (* A C) -4.0))))
   (if (<= B 3.4e+73)
     (* (sqrt (* A (* F t_0))) (/ (- 2.0) t_0))
     (if (<= B 1.35e+154)
       (/ (* (sqrt 2.0) (sqrt (* F (- A (sqrt (fma B B (* A A))))))) (- B))
       (* -2.0 (/ (sqrt (* A F)) B))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, ((A * C) * -4.0));
	double tmp;
	if (B <= 3.4e+73) {
		tmp = sqrt((A * (F * t_0))) * (-2.0 / t_0);
	} else if (B <= 1.35e+154) {
		tmp = (sqrt(2.0) * sqrt((F * (A - sqrt(fma(B, B, (A * A))))))) / -B;
	} else {
		tmp = -2.0 * (sqrt((A * F)) / B);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(Float64(A * C) * -4.0))
	tmp = 0.0
	if (B <= 3.4e+73)
		tmp = Float64(sqrt(Float64(A * Float64(F * t_0))) * Float64(Float64(-2.0) / t_0));
	elseif (B <= 1.35e+154)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(A - sqrt(fma(B, B, Float64(A * A))))))) / Float64(-B));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / 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[(B * B + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 3.4e+73], N[(N[Sqrt[N[(A * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-2.0) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.35e+154], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B)), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.4000000000000002e73

   1. Initial program 27.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 22.1

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 22.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 22.6%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \sqrt{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right) \]

   10. Applied rewrites 22.0%

       \[\leadsto \color{blue}{-\sqrt{A \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \frac{2}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \]

   if 3.4000000000000002e73 < B < 1.35000000000000003e154

   1. Initial program 36.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}{-1 \cdot B}} \]

   5. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}} \]

   if 1.35000000000000003e154 < B

   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

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 0.0

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 0.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]

      2. associate-*r/ N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

      3. *-rgt-identity N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

      6. lower-*.f64 6.4

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{A \cdot F}}}{B} \]

   8. Applied rewrites 6.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{A \cdot F}}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.4 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{A \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)} \cdot \frac{-2}{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 5.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ -2 \cdot \frac{\sqrt{A \cdot F}}{B} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (* -2.0 (/ (sqrt (* A F)) B)))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return -2.0 * (sqrt((A * F)) / B);
}

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 = (-2.0d0) * (sqrt((a * f)) / b)
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return -2.0 * (Math.sqrt((A * F)) / B);
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return -2.0 * (math.sqrt((A * F)) / B)

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = -2.0 * (sqrt((A * F)) / 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[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in C around inf

   \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. *-lft-identity N/A

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. lower-+.f64 18.2

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

5. Applied rewrites 18.2%

   \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

6. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]

   2. associate-*r/ N/A

      \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

   3. *-rgt-identity N/A

      \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

   4. lower-/.f64 N/A

      \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

   6. lower-*.f64 4.0

      \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{A \cdot F}}}{B} \]

8. Applied rewrites 4.0%

   \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{A \cdot F}}{B}} \]
9. Add Preprocessing

Alternative 18: 2.0% accurate, 18.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \sqrt{F \cdot \frac{2}{B}} \end{array} \]

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 (* F (/ 2.0 B))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return sqrt((F * (2.0 / B)));
}

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((f * (2.0d0 / b)))
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((F * (2.0 / B)));
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return math.sqrt((F * (2.0 / B)))

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return sqrt(Float64(F * Float64(2.0 / B)))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = sqrt((F * (2.0 / 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[Sqrt[N[(F * N[(2.0 / B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 25.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in 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. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B}}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{F}{B}}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

   6. lower-neg.f64 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \color{blue}{\left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   7. unpow2 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{2}\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \sqrt{2}}\right)\right) \]

   10. lower-sqrt.f64 1.8

       \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(--1 \cdot \color{blue}{\sqrt{2}}\right) \]

5. Applied rewrites 1.8%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(--1 \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{F}{B}}} \cdot \left(\mathsf{neg}\left(-1 \cdot \sqrt{2}\right)\right) \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(-1 \cdot \color{blue}{\sqrt{2}}\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right)}\right)\right) \]

   4. remove-double-neg N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}} \]

   6. sqrt-unprod N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   8. lower-*.f64 1.8

      \[\leadsto \sqrt{\color{blue}{\frac{F}{B} \cdot 2}} \]

7. Applied rewrites 1.8%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \sqrt{\color{blue}{\frac{F \cdot 2}{B}}} \]

   2. associate-/l* N/A

      \[\leadsto \sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]

   4. lower-/.f64 1.8

      \[\leadsto \sqrt{F \cdot \color{blue}{\frac{2}{B}}} \]

9. Applied rewrites 1.8%

   \[\leadsto \sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
10. 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))))