ABCF->ab-angle b

Percentage Accurate: 18.9% → 35.6%

Time: 16.2s

Alternatives: 12

Speedup: 10.2×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 35.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_2 := t\_0 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ t_4 := \frac{\sqrt{t\_1 \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right)\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+150}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-132}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_2 (- t_0 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt (* t_1 (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2))
        (t_4 (/ (sqrt (* t_1 (+ A (fma -0.5 (/ (* B_m B_m) C) A)))) t_2)))
   (if (<= t_3 -2e+150)
     t_4
     (if (<= t_3 -1e-132)
       (-
        (*
         (sqrt
          (/
           (* F (- (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m)))))
           (fma B_m B_m (* -4.0 (* A C)))))
         (sqrt 2.0)))
       (if (<= t_3 -1e-218)
         (*
          (sqrt (* F (- C (sqrt (fma B_m B_m (* C C))))))
          (/ (sqrt 2.0) (- B_m)))
         t_4)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_2 = t_0 - pow(B_m, 2.0);
	double t_3 = sqrt((t_1 * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double t_4 = sqrt((t_1 * (A + fma(-0.5, ((B_m * B_m) / C), A)))) / t_2;
	double tmp;
	if (t_3 <= -2e+150) {
		tmp = t_4;
	} else if (t_3 <= -1e-132) {
		tmp = -(sqrt(((F * ((A + C) - sqrt(fma((A - C), (A - C), (B_m * B_m))))) / fma(B_m, B_m, (-4.0 * (A * C))))) * sqrt(2.0));
	} else if (t_3 <= -1e-218) {
		tmp = sqrt((F * (C - sqrt(fma(B_m, B_m, (C * C)))))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	t_4 = Float64(sqrt(Float64(t_1 * Float64(A + fma(-0.5, Float64(Float64(B_m * B_m) / C), A)))) / t_2)
	tmp = 0.0
	if (t_3 <= -2e+150)
		tmp = t_4;
	elseif (t_3 <= -1e-132)
		tmp = Float64(-Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))))) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))) * sqrt(2.0)));
	elseif (t_3 <= -1e-218)
		tmp = Float64(sqrt(Float64(F * Float64(C - sqrt(fma(B_m, B_m, Float64(C * C)))))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$1 * N[(A + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+150], t$95$4, If[LessEqual[t$95$3, -1e-132], (-N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$3, -1e-218], N[(N[Sqrt[N[(F * N[(C - N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]

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))) < -1.99999999999999996e150 or -1e-218 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 6.2%

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

   3. Taylor expanded in C around inf

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

      1. 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. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 14.1

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

   5. Simplified 14.1%

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

   if -1.99999999999999996e150 < (/.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))) < -9.9999999999999999e-133

   1. Initial program 97.1%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. 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. Simplified 99.3%

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

   if -9.9999999999999999e-133 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 99.0%

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

   3. 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. Simplified 40.9%

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

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+150}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-132}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 36.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_2 := t\_0 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ t_4 := \frac{\sqrt{t\_1 \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right)\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_2 (- t_0 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt (* t_1 (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2))
        (t_4 (/ (sqrt (* t_1 (+ A (fma -0.5 (/ (* B_m B_m) C) A)))) t_2)))
   (if (<= t_3 (- INFINITY)) t_4 (if (<= t_3 -1e-218) t_3 t_4))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_2 = t_0 - pow(B_m, 2.0);
	double t_3 = sqrt((t_1 * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double t_4 = sqrt((t_1 * (A + fma(-0.5, ((B_m * B_m) / C), A)))) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -1e-218) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	t_4 = Float64(sqrt(Float64(t_1 * Float64(A + fma(-0.5, Float64(Float64(B_m * B_m) / C), A)))) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -1e-218)
		tmp = t_3;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$1 * N[(A + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -1e-218], t$95$3, t$95$4]]]]]]]

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))) < -inf.0 or -1e-218 < (/.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 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

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

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 13.8

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

   5. Simplified 13.8%

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

   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))) < -1e-218

   1. Initial program 97.6%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 28.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+79}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{t\_0}}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{-4 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= t_2 -4e+79)
     (* -2.0 (sqrt (/ (* A F) t_0)))
     (if (<= t_2 -1e-218)
       (/
        (* (sqrt 2.0) (sqrt (* F (- A (sqrt (fma A A (* B_m B_m)))))))
        (- B_m))
       (* -2.0 (/ (sqrt (* -4.0 (* F (* C (* A A))))) t_0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -4e+79) {
		tmp = -2.0 * sqrt(((A * F) / t_0));
	} else if (t_2 <= -1e-218) {
		tmp = (sqrt(2.0) * sqrt((F * (A - sqrt(fma(A, A, (B_m * B_m))))))) / -B_m;
	} else {
		tmp = -2.0 * (sqrt((-4.0 * (F * (C * (A * A))))) / t_0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -4e+79)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / t_0)));
	elseif (t_2 <= -1e-218)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B_m * B_m))))))) / Float64(-B_m));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(-4.0 * Float64(F * Float64(C * Float64(A * A))))) / t_0));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $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$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+79], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-218], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(-4.0 * N[(F * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.99999999999999987e79

   1. Initial program 15.5%

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

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 24.1

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

   5. Simplified 24.1%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 24.2

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

   8. Simplified 24.2%

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

   9. Taylor expanded in F around 0

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

       1. lower-*.f64 N/A

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

       2. associate-*r/ N/A

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

       3. *-rgt-identity N/A

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

       4. lower-/.f64 N/A

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

   11. Simplified 24.1%

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

   12. Taylor expanded in F around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 22.9

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

   14. Simplified 22.9%

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

   if -3.99999999999999987e79 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 99.3%

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

   3. Taylor expanded in C around 0

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

      1. 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. Simplified 41.6%

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

   if -1e-218 < (/.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 5.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 8.6

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

   5. Simplified 8.6%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 9.6

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

   8. Simplified 9.6%

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

   9. Taylor expanded in F around 0

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

       1. lower-*.f64 N/A

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

       2. associate-*r/ N/A

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

       3. *-rgt-identity N/A

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

       4. lower-/.f64 N/A

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

   11. Simplified 8.6%

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

   12. Taylor expanded in A around inf

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

       1. lower-*.f64 N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 7.8

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

   14. Simplified 7.8%

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

4. Final simplification 15.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{+79}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{-4 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 26.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-55}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{t\_0}}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{-4 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= t_2 -2e-55)
     (* -2.0 (sqrt (/ (* A F) t_0)))
     (if (<= t_2 -1e-218)
       (*
        (sqrt (* F (- C (sqrt (fma B_m B_m (* C C))))))
        (/ (sqrt 2.0) (- B_m)))
       (* -2.0 (/ (sqrt (* -4.0 (* F (* C (* A A))))) t_0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -2e-55) {
		tmp = -2.0 * sqrt(((A * F) / t_0));
	} else if (t_2 <= -1e-218) {
		tmp = sqrt((F * (C - sqrt(fma(B_m, B_m, (C * C)))))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = -2.0 * (sqrt((-4.0 * (F * (C * (A * A))))) / t_0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -2e-55)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / t_0)));
	elseif (t_2 <= -1e-218)
		tmp = Float64(sqrt(Float64(F * Float64(C - sqrt(fma(B_m, B_m, Float64(C * C)))))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(-4.0 * Float64(F * Float64(C * Float64(A * A))))) / t_0));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $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$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-55], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-218], N[(N[Sqrt[N[(F * N[(C - N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(-4.0 * N[(F * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.99999999999999999e-55

   1. Initial program 28.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.2

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

   5. Simplified 25.2%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 25.3

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

   8. Simplified 25.3%

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

   9. Taylor expanded in F around 0

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

       1. lower-*.f64 N/A

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

       2. associate-*r/ N/A

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

       3. *-rgt-identity N/A

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

       4. lower-/.f64 N/A

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

   11. Simplified 25.2%

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

   12. Taylor expanded in F around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 24.2

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

   14. Simplified 24.2%

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

   if -1.99999999999999999e-55 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 99.3%

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

   3. Taylor expanded in 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. Simplified 49.2%

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

   if -1e-218 < (/.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 5.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 8.6

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

   5. Simplified 8.6%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 9.6

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

   8. Simplified 9.6%

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

   9. Taylor expanded in F around 0

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

       1. lower-*.f64 N/A

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

       2. associate-*r/ N/A

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

       3. *-rgt-identity N/A

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

       4. lower-/.f64 N/A

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

   11. Simplified 8.6%

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

   12. Taylor expanded in A around inf

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

       1. lower-*.f64 N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 7.8

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

   14. Simplified 7.8%

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

4. Final simplification 16.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-55}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{-4 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 24.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{-4 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m))) (t_1 (* (* 4.0 A) C)))
   (if (<=
        (/
         (sqrt
          (*
           (* 2.0 (* (- (pow B_m 2.0) t_1) F))
           (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
         (- t_1 (pow B_m 2.0)))
        -1e-218)
     (* -2.0 (sqrt (/ (* A F) t_0)))
     (* -2.0 (/ (sqrt (* -4.0 (* F (* C (* A A))))) t_0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double tmp;
	if ((sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0))) <= -1e-218) {
		tmp = -2.0 * sqrt(((A * F) / t_0));
	} else {
		tmp = -2.0 * (sqrt((-4.0 * (F * (C * (A * A))))) / t_0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0))) <= -1e-218)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / t_0)));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(-4.0 * Float64(F * Float64(C * Float64(A * A))))) / t_0));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-218], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(-4.0 * N[(F * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $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))) < -1e-218

   1. Initial program 46.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 21.8

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

   5. Simplified 21.8%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 22.0

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

   8. Simplified 22.0%

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

   9. Taylor expanded in F around 0

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

       1. lower-*.f64 N/A

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

       2. associate-*r/ N/A

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

       3. *-rgt-identity N/A

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

       4. lower-/.f64 N/A

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

   11. Simplified 21.8%

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

   12. Taylor expanded in F around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 21.2

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

   14. Simplified 21.2%

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

   if -1e-218 < (/.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 5.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 8.6

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

   5. Simplified 8.6%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 9.6

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

   8. Simplified 9.6%

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

   9. Taylor expanded in F around 0

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

       1. lower-*.f64 N/A

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

       2. associate-*r/ N/A

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

       3. *-rgt-identity N/A

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

       4. lower-/.f64 N/A

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

   11. Simplified 8.6%

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

   12. Taylor expanded in A around inf

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

       1. lower-*.f64 N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 7.8

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

   14. Simplified 7.8%

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

4. Final simplification 12.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{-4 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B\_m}^{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(A + A\right)}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)}\right)} \cdot \left(B\_m \cdot \sqrt{2}\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A \cdot F} \cdot -2}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)) (t_1 (- t_0 (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 5e-60)
     (/ (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (+ A A))) t_1)
     (if (<= (pow B_m 2.0) 5e+298)
       (/
        (* (sqrt (* F (- A (sqrt (fma A A (* B_m B_m)))))) (* B_m (sqrt 2.0)))
        t_1)
       (/ (* (sqrt (* A F)) -2.0) B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B_m, 2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 5e-60) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (A + A))) / t_1;
	} else if (pow(B_m, 2.0) <= 5e+298) {
		tmp = (sqrt((F * (A - sqrt(fma(A, A, (B_m * B_m)))))) * (B_m * sqrt(2.0))) / t_1;
	} else {
		tmp = (sqrt((A * F)) * -2.0) / B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-60)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(A + A))) / t_1);
	elseif ((B_m ^ 2.0) <= 5e+298)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B_m * B_m)))))) * Float64(B_m * sqrt(2.0))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(A * F)) * -2.0) / B_m);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-60], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+298], N[(N[(N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] / B$95$m), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000001e-60

   1. Initial program 18.7%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 20.2

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

   5. Simplified 20.2%

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

   if 5.0000000000000001e-60 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e298

   1. Initial program 35.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 18.9

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

   5. Simplified 18.9%

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

   if 5.0000000000000003e298 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.7%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 1.8

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

   5. Simplified 1.8%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 1.8

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

   8. Simplified 1.8%

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

   9. Taylor expanded in B around inf

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

       1. mul-1-neg N/A

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

       2. associate-*r/ N/A

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

       3. distribute-neg-frac N/A

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

       4. mul-1-neg N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

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

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

       8. unpow2 N/A

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

       9. rem-square-sqrt N/A

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

       10. metadata-eval N/A

           \[\leadsto \frac{\sqrt{A \cdot F} \cdot \color{blue}{-2}}{B} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{\color{blue}{\sqrt{A \cdot F} \cdot -2}}{B} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \frac{\color{blue}{\sqrt{A \cdot F}} \cdot -2}{B} \]

       13. lower-*.f64 5.6

           \[\leadsto \frac{\sqrt{\color{blue}{A \cdot F}} \cdot -2}{B} \]

   11. Simplified 5.6%

       \[\leadsto \color{blue}{\frac{\sqrt{A \cdot F} \cdot -2}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 16.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)} \cdot \left(B \cdot \sqrt{2}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A \cdot F} \cdot -2}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-60}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot \left(F \cdot t\_0\right)}}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)}\right)} \cdot \left(B\_m \cdot \sqrt{2}\right)}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A \cdot F} \cdot -2}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 5e-60)
     (* -2.0 (/ (sqrt (* A (* F t_0))) t_0))
     (if (<= (pow B_m 2.0) 5e+298)
       (/
        (* (sqrt (* F (- A (sqrt (fma A A (* B_m B_m)))))) (* B_m (sqrt 2.0)))
        (- (* (* 4.0 A) C) (pow B_m 2.0)))
       (/ (* (sqrt (* A F)) -2.0) B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-60) {
		tmp = -2.0 * (sqrt((A * (F * t_0))) / t_0);
	} else if (pow(B_m, 2.0) <= 5e+298) {
		tmp = (sqrt((F * (A - sqrt(fma(A, A, (B_m * B_m)))))) * (B_m * sqrt(2.0))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else {
		tmp = (sqrt((A * F)) * -2.0) / B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-60)
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * Float64(F * t_0))) / t_0));
	elseif ((B_m ^ 2.0) <= 5e+298)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B_m * B_m)))))) * Float64(B_m * sqrt(2.0))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(A * F)) * -2.0) / B_m);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-60], N[(-2.0 * N[(N[Sqrt[N[(A * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+298], N[(N[(N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] / B$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000001e-60

   1. Initial program 18.7%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 20.2

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

   5. Simplified 20.2%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 21.0

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

   8. Simplified 21.0%

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

   9. Taylor expanded in F around 0

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

       1. lower-*.f64 N/A

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

       2. associate-*r/ N/A

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

       3. *-rgt-identity N/A

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

       4. lower-/.f64 N/A

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

   11. Simplified 20.2%

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

   if 5.0000000000000001e-60 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e298

   1. Initial program 35.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 18.9

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

   5. Simplified 18.9%

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

   if 5.0000000000000003e298 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.7%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 1.8

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

   5. Simplified 1.8%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 1.8

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

   8. Simplified 1.8%

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

   9. Taylor expanded in B around inf

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

       1. mul-1-neg N/A

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

       2. associate-*r/ N/A

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

       3. distribute-neg-frac N/A

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

       4. mul-1-neg N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

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

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

       8. unpow2 N/A

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

       9. rem-square-sqrt N/A

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

       10. metadata-eval N/A

           \[\leadsto \frac{\sqrt{A \cdot F} \cdot \color{blue}{-2}}{B} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{\color{blue}{\sqrt{A \cdot F} \cdot -2}}{B} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \frac{\color{blue}{\sqrt{A \cdot F}} \cdot -2}{B} \]

       13. lower-*.f64 5.6

           \[\leadsto \frac{\sqrt{\color{blue}{A \cdot F}} \cdot -2}{B} \]

   11. Simplified 5.6%

       \[\leadsto \color{blue}{\frac{\sqrt{A \cdot F} \cdot -2}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 16.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-60}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)} \cdot \left(B \cdot \sqrt{2}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A \cdot F} \cdot -2}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-60}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot \left(F \cdot t\_0\right)}}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A \cdot F} \cdot -2}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 5e-60)
     (* -2.0 (/ (sqrt (* A (* F t_0))) t_0))
     (if (<= (pow B_m 2.0) 5e+298)
       (/
        (* (sqrt 2.0) (sqrt (* F (- A (sqrt (fma A A (* B_m B_m)))))))
        (- B_m))
       (/ (* (sqrt (* A F)) -2.0) B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-60) {
		tmp = -2.0 * (sqrt((A * (F * t_0))) / t_0);
	} else if (pow(B_m, 2.0) <= 5e+298) {
		tmp = (sqrt(2.0) * sqrt((F * (A - sqrt(fma(A, A, (B_m * B_m))))))) / -B_m;
	} else {
		tmp = (sqrt((A * F)) * -2.0) / B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-60)
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * Float64(F * t_0))) / t_0));
	elseif ((B_m ^ 2.0) <= 5e+298)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B_m * B_m))))))) / Float64(-B_m));
	else
		tmp = Float64(Float64(sqrt(Float64(A * F)) * -2.0) / B_m);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-60], N[(-2.0 * N[(N[Sqrt[N[(A * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+298], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] / B$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000001e-60

   1. Initial program 18.7%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 20.2

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

   5. Simplified 20.2%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 21.0

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

   8. Simplified 21.0%

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

   9. Taylor expanded in F around 0

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

       1. lower-*.f64 N/A

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

       2. associate-*r/ N/A

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

       3. *-rgt-identity N/A

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

       4. lower-/.f64 N/A

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

   11. Simplified 20.2%

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

   if 5.0000000000000001e-60 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e298

   1. Initial program 35.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. Simplified 19.0%

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

   if 5.0000000000000003e298 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.7%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 1.8

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

   5. Simplified 1.8%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 1.8

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

   8. Simplified 1.8%

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

   9. Taylor expanded in B around inf

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

       1. mul-1-neg N/A

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

       2. associate-*r/ N/A

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

       3. distribute-neg-frac N/A

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

       4. mul-1-neg N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

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

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

       8. unpow2 N/A

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

       9. rem-square-sqrt N/A

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

       10. metadata-eval N/A

           \[\leadsto \frac{\sqrt{A \cdot F} \cdot \color{blue}{-2}}{B} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{\color{blue}{\sqrt{A \cdot F} \cdot -2}}{B} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \frac{\color{blue}{\sqrt{A \cdot F}} \cdot -2}{B} \]

       13. lower-*.f64 5.6

           \[\leadsto \frac{\sqrt{\color{blue}{A \cdot F}} \cdot -2}{B} \]

   11. Simplified 5.6%

       \[\leadsto \color{blue}{\frac{\sqrt{A \cdot F} \cdot -2}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 27.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7 \cdot 10^{-59}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{-4 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 6.5 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A \cdot F} \cdot -2}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7e-59)
   (*
    -2.0
    (/ (sqrt (* -4.0 (* F (* C (* A A))))) (fma -4.0 (* A C) (* B_m B_m))))
   (if (<= B_m 6.5e+149)
     (* (sqrt 2.0) (/ (sqrt (* F (- A (sqrt (fma A A (* B_m B_m)))))) (- B_m)))
     (/ (* (sqrt (* A F)) -2.0) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7e-59) {
		tmp = -2.0 * (sqrt((-4.0 * (F * (C * (A * A))))) / fma(-4.0, (A * C), (B_m * B_m)));
	} else if (B_m <= 6.5e+149) {
		tmp = sqrt(2.0) * (sqrt((F * (A - sqrt(fma(A, A, (B_m * B_m)))))) / -B_m);
	} else {
		tmp = (sqrt((A * F)) * -2.0) / B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7e-59)
		tmp = Float64(-2.0 * Float64(sqrt(Float64(-4.0 * Float64(F * Float64(C * Float64(A * A))))) / fma(-4.0, Float64(A * C), Float64(B_m * B_m))));
	elseif (B_m <= 6.5e+149)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B_m * B_m)))))) / Float64(-B_m)));
	else
		tmp = Float64(Float64(sqrt(Float64(A * F)) * -2.0) / B_m);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 7e-59], N[(-2.0 * N[(N[Sqrt[N[(-4.0 * N[(F * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6.5e+149], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] / B$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 7.0000000000000002e-59

   1. Initial program 17.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 15.5

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

   5. Simplified 15.5%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 16.1

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

   8. Simplified 16.1%

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

   9. Taylor expanded in F around 0

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

       1. lower-*.f64 N/A

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

       2. associate-*r/ N/A

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

       3. *-rgt-identity N/A

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

       4. lower-/.f64 N/A

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

   11. Simplified 15.5%

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

   12. Taylor expanded in A around inf

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

       1. lower-*.f64 N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 11.5

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

   14. Simplified 11.5%

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

   if 7.0000000000000002e-59 < B < 6.50000000000000015e149

   1. Initial program 39.1%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. 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. Simplified 40.2%

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

   6. Taylor expanded in C around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 36.0

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

   8. Simplified 36.0%

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

   if 6.50000000000000015e149 < B

   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 2.9

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

   5. Simplified 2.9%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 2.9

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

   8. Simplified 2.9%

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

   9. Taylor expanded in B around inf

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

       1. mul-1-neg N/A

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

       2. associate-*r/ N/A

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

       3. distribute-neg-frac N/A

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

       4. mul-1-neg N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

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

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

       8. unpow2 N/A

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

       9. rem-square-sqrt N/A

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

       10. metadata-eval N/A

           \[\leadsto \frac{\sqrt{A \cdot F} \cdot \color{blue}{-2}}{B} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{\color{blue}{\sqrt{A \cdot F} \cdot -2}}{B} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \frac{\color{blue}{\sqrt{A \cdot F}} \cdot -2}{B} \]

       13. lower-*.f64 7.4

           \[\leadsto \frac{\sqrt{\color{blue}{A \cdot F}} \cdot -2}{B} \]

   11. Simplified 7.4%

       \[\leadsto \color{blue}{\frac{\sqrt{A \cdot F} \cdot -2}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 16.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7 \cdot 10^{-59}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{-4 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A \cdot F} \cdot -2}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 19.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* -2.0 (sqrt (/ (* A F) (fma -4.0 (* A C) (* B_m B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -2.0 * sqrt(((A * F) / fma(-4.0, (A * C), (B_m * B_m))));
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-2.0 * sqrt(Float64(Float64(A * F) / fma(-4.0, Float64(A * C), Float64(B_m * B_m)))))
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.1%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 13.3

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

5. Simplified 13.3%

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

6. Taylor expanded in F around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

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

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 14.0

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

8. Simplified 14.0%

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

9. Taylor expanded in F around 0

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

    1. lower-*.f64 N/A

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

    2. associate-*r/ N/A

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

    3. *-rgt-identity N/A

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

    4. lower-/.f64 N/A

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

11. Simplified 13.3%

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

12. Taylor expanded in F around 0

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

    1. lower-*.f64 N/A

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

    2. lower-sqrt.f64 N/A

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

    3. lower-/.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-fma.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f64 10.5

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

14. Simplified 10.5%

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

Alternative 11: 9.2% accurate, 15.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{A \cdot F} \cdot -2}{B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (/ (* (sqrt (* A F)) -2.0) B_m))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return (sqrt((A * F)) * -2.0) / B_m;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (sqrt((a * f)) * (-2.0d0)) / b_m
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return (Math.sqrt((A * F)) * -2.0) / B_m;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return (math.sqrt((A * F)) * -2.0) / B_m

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(Float64(sqrt(Float64(A * F)) * -2.0) / B_m)
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = (sqrt((A * F)) * -2.0) / B_m;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] / B$95$m), $MachinePrecision]

Derivation

1. Initial program 20.1%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 13.3

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

5. Simplified 13.3%

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

6. Taylor expanded in B around inf

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 7.9

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

8. Simplified 7.9%

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

9. Taylor expanded in B around inf

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

    1. mul-1-neg N/A

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

    2. associate-*r/ N/A

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

    3. distribute-neg-frac N/A

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

    4. mul-1-neg N/A

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

    5. lower-/.f64 N/A

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

    6. mul-1-neg N/A

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

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

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

    8. unpow2 N/A

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

    9. rem-square-sqrt N/A

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

    10. metadata-eval N/A

        \[\leadsto \frac{\sqrt{A \cdot F} \cdot \color{blue}{-2}}{B} \]

    11. lower-*.f64 N/A

        \[\leadsto \frac{\color{blue}{\sqrt{A \cdot F} \cdot -2}}{B} \]

    12. lower-sqrt.f64 N/A

        \[\leadsto \frac{\color{blue}{\sqrt{A \cdot F}} \cdot -2}{B} \]

    13. lower-*.f64 3.5

        \[\leadsto \frac{\sqrt{\color{blue}{A \cdot F}} \cdot -2}{B} \]

11. Simplified 3.5%

    \[\leadsto \color{blue}{\frac{\sqrt{A \cdot F} \cdot -2}{B}} \]
12. Add Preprocessing

Alternative 12: 0.7% accurate, 15.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{2}{B\_m} \cdot \sqrt{C \cdot F} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (* (/ 2.0 B_m) (sqrt (* C F))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return (2.0 / B_m) * sqrt((C * F));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (2.0d0 / b_m) * sqrt((c * f))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return (2.0 / B_m) * Math.sqrt((C * F));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return (2.0 / B_m) * math.sqrt((C * F))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(Float64(2.0 / B_m) * sqrt(Float64(C * F)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = (2.0 / B_m) * sqrt((C * F));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[(2.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.1%

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

3. Taylor expanded in F around 0

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

   1. mul-1-neg N/A

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

   2. 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. Simplified 20.1%

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

6. Taylor expanded in B around 0

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

   1. lower--.f64 9.8

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

8. Simplified 9.8%

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

9. Taylor expanded in B around -inf

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

    1. lower-*.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. unpow2 N/A

       \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{B} \cdot \sqrt{C \cdot F} \]

    4. rem-square-sqrt N/A

       \[\leadsto \frac{\color{blue}{2}}{B} \cdot \sqrt{C \cdot F} \]

    5. lower-sqrt.f64 N/A

       \[\leadsto \frac{2}{B} \cdot \color{blue}{\sqrt{C \cdot F}} \]

    6. lower-*.f64 3.0

       \[\leadsto \frac{2}{B} \cdot \sqrt{\color{blue}{C \cdot F}} \]

11. Simplified 3.0%

    \[\leadsto \color{blue}{\frac{2}{B} \cdot \sqrt{C \cdot F}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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))))