ABCF->ab-angle a

Percentage Accurate: 19.3% → 56.7%

Time: 17.6s

Alternatives: 14

Speedup: 16.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\\ t_3 := t\_1 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{t\_2 \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-193}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot t\_0\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)\right)}}{-t\_0}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right)}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 C (* A -4.0) (* B_m B_m)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (* 2.0 (* (- (pow B_m 2.0) t_1) F)))
        (t_3 (- t_1 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt (* t_2 (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
     (if (<= t_4 -5e-193)
       (/
        (sqrt
         (*
          2.0
          (* (* F t_0) (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m)))))))
        (- t_0))
       (if (<= t_4 INFINITY)
         (/ (sqrt (* t_2 (fma 2.0 C (* -0.5 (/ (* B_m B_m) A))))) t_3)
         (/ (- (sqrt F)) (sqrt (* B_m 0.5))))))))

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(C, (A * -4.0), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = 2.0 * ((pow(B_m, 2.0) - t_1) * F);
	double t_3 = t_1 - pow(B_m, 2.0);
	double t_4 = sqrt((t_2 * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else if (t_4 <= -5e-193) {
		tmp = sqrt((2.0 * ((F * t_0) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m))))))) / -t_0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * fma(2.0, C, (-0.5 * ((B_m * B_m) / A))))) / t_3;
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	}
	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(C, Float64(A * -4.0), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F))
	t_3 = Float64(t_1 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(t_2 * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	elseif (t_4 <= -5e-193)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))))))) / Float64(-t_0));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(t_2 * fma(2.0, C, Float64(-0.5 * Float64(Float64(B_m * B_m) / A))))) / t_3);
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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[(C * N[(A * -4.0), $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[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$2 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$4, -5e-193], N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * 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]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(t$95$2 * N[(2.0 * C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Simplified 21.4%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 28.0

         \[\leadsto \sqrt{\frac{\color{blue}{F \cdot -0.5}}{A}} \cdot \left(-\sqrt{2}\right) \]

   8. Simplified 28.0%

      \[\leadsto \sqrt{\color{blue}{\frac{F \cdot -0.5}{A}}} \cdot \left(-\sqrt{2}\right) \]

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000005e-193

   1. Initial program 98.7%

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

   4. Applied egg-rr 98.7%

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

   if -5.0000000000000005e-193 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 13.1%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 28.7

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

   5. Simplified 28.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 17.2

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

   5. Simplified 17.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. pow1/2 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 21.2

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

   7. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-undiv N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 21.2

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

   9. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

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

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

       5. lift-sqrt.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. clear-num N/A

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

       8. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. div-inv N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 21.2

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-193}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := F \cdot t\_0\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-193}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t\_1 \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)\right)}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(t\_1 \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B\_m \cdot B\_m\right)}{A}\right)\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 C (* A -4.0) (* B_m B_m)))
        (t_1 (* F t_0))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 (- INFINITY))
     (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
     (if (<= t_3 -5e-193)
       (/
        (sqrt
         (* 2.0 (* t_1 (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m)))))))
        (- t_0))
       (if (<= t_3 INFINITY)
         (*
          (sqrt (* 2.0 (* t_1 (fma 2.0 C (/ (* -0.5 (* B_m B_m)) A)))))
          (/ -1.0 t_0))
         (/ (- (sqrt F)) (sqrt (* B_m 0.5))))))))

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(C, (A * -4.0), (B_m * B_m));
	double t_1 = F * t_0;
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else if (t_3 <= -5e-193) {
		tmp = sqrt((2.0 * (t_1 * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m))))))) / -t_0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * (t_1 * fma(2.0, C, ((-0.5 * (B_m * B_m)) / A))))) * (-1.0 / t_0);
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	}
	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(C, Float64(A * -4.0), Float64(B_m * B_m))
	t_1 = Float64(F * t_0)
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	elseif (t_3 <= -5e-193)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_1 * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))))))) / Float64(-t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_1 * fma(2.0, C, Float64(Float64(-0.5 * Float64(B_m * B_m)) / A))))) * Float64(-1.0 / t_0));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $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$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, -5e-193], N[(N[Sqrt[N[(2.0 * N[(t$95$1 * 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]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(2.0 * N[(t$95$1 * N[(2.0 * C + N[(N[(-0.5 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Simplified 21.4%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 28.0

         \[\leadsto \sqrt{\frac{\color{blue}{F \cdot -0.5}}{A}} \cdot \left(-\sqrt{2}\right) \]

   8. Simplified 28.0%

      \[\leadsto \sqrt{\color{blue}{\frac{F \cdot -0.5}{A}}} \cdot \left(-\sqrt{2}\right) \]

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000005e-193

   1. Initial program 98.7%

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

   4. Applied egg-rr 98.7%

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

   if -5.0000000000000005e-193 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 13.1%

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

   4. Applied egg-rr 13.0%

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

   5. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 28.6

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

   7. Simplified 28.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 17.2

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

   5. Simplified 17.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. pow1/2 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 21.2

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

   7. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-undiv N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 21.2

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

   9. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

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

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

       5. lift-sqrt.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. clear-num N/A

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

       8. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. div-inv N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 21.2

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-193}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{F}\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{2} \cdot \left(t\_0 \cdot \sqrt{\frac{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}}{\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)}}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot t\_1\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B\_m \cdot B\_m\right)}{A}\right)\right)} \cdot \frac{-1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 (- (sqrt F)))
        (t_1 (fma C (* A -4.0) (* B_m B_m)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 (- INFINITY))
     (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
     (if (<= t_3 -5e-193)
       (*
        (sqrt 2.0)
        (*
         t_0
         (sqrt
          (/
           (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))
           (fma A (* C -4.0) (* B_m B_m))))))
       (if (<= t_3 INFINITY)
         (*
          (sqrt (* 2.0 (* (* F t_1) (fma 2.0 C (/ (* -0.5 (* B_m B_m)) A)))))
          (/ -1.0 t_1))
         (/ t_0 (sqrt (* B_m 0.5))))))))

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 = -sqrt(F);
	double t_1 = fma(C, (A * -4.0), (B_m * B_m));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else if (t_3 <= -5e-193) {
		tmp = sqrt(2.0) * (t_0 * sqrt((((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))) / fma(A, (C * -4.0), (B_m * B_m)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * ((F * t_1) * fma(2.0, C, ((-0.5 * (B_m * B_m)) / A))))) * (-1.0 / t_1);
	} else {
		tmp = t_0 / sqrt((B_m * 0.5));
	}
	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(-sqrt(F))
	t_1 = fma(C, Float64(A * -4.0), Float64(B_m * B_m))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	elseif (t_3 <= -5e-193)
		tmp = Float64(sqrt(2.0) * Float64(t_0 * sqrt(Float64(Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))) / fma(A, Float64(C * -4.0), Float64(B_m * B_m))))));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_1) * fma(2.0, C, Float64(Float64(-0.5 * Float64(B_m * B_m)) / A))))) * Float64(-1.0 / t_1));
	else
		tmp = Float64(t_0 / sqrt(Float64(B_m * 0.5)));
	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[Sqrt[F], $MachinePrecision])}, Block[{t$95$1 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $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$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, -5e-193], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(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] / N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * C + N[(N[(-0.5 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Simplified 21.4%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 28.0

         \[\leadsto \sqrt{\frac{\color{blue}{F \cdot -0.5}}{A}} \cdot \left(-\sqrt{2}\right) \]

   8. Simplified 28.0%

      \[\leadsto \sqrt{\color{blue}{\frac{F \cdot -0.5}{A}}} \cdot \left(-\sqrt{2}\right) \]

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000005e-193

   1. Initial program 98.7%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Simplified 88.8%

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

   7. Applied egg-rr 98.3%

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

   if -5.0000000000000005e-193 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 13.1%

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

   4. Applied egg-rr 13.0%

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

   5. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 28.6

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

   7. Simplified 28.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 17.2

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

   5. Simplified 17.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. pow1/2 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 21.2

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

   7. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-undiv N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 21.2

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

   9. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

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

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

       5. lift-sqrt.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. clear-num N/A

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

       8. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. div-inv N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 21.2

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(-\sqrt{F}\right) \cdot \sqrt{\frac{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot t\_0\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-193}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot t\_1\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B\_m \cdot B\_m\right)}{A}\right)\right)} \cdot \frac{-1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 (- (sqrt 2.0)))
        (t_1 (fma C (* A -4.0) (* B_m B_m)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 (- INFINITY))
     (* (sqrt (/ (* F -0.5) A)) t_0)
     (if (<= t_3 -5e-193)
       (*
        t_0
        (sqrt
         (/
          (* F (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C))))))
          (fma B_m B_m (* -4.0 (* A C))))))
       (if (<= t_3 INFINITY)
         (*
          (sqrt (* 2.0 (* (* F t_1) (fma 2.0 C (/ (* -0.5 (* B_m B_m)) A)))))
          (/ -1.0 t_1))
         (/ (- (sqrt F)) (sqrt (* B_m 0.5))))))))

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 = -sqrt(2.0);
	double t_1 = fma(C, (A * -4.0), (B_m * B_m));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt(((F * -0.5) / A)) * t_0;
	} else if (t_3 <= -5e-193) {
		tmp = t_0 * sqrt(((F * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C)))))) / fma(B_m, B_m, (-4.0 * (A * C)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * ((F * t_1) * fma(2.0, C, ((-0.5 * (B_m * B_m)) / A))))) * (-1.0 / t_1);
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	}
	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(-sqrt(2.0))
	t_1 = fma(C, Float64(A * -4.0), Float64(B_m * B_m))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * t_0);
	elseif (t_3 <= -5e-193)
		tmp = Float64(t_0 * sqrt(Float64(Float64(F * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C)))))) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_1) * fma(2.0, C, Float64(Float64(-0.5 * Float64(B_m * B_m)) / A))))) * Float64(-1.0 / t_1));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $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$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$3, -5e-193], N[(t$95$0 * N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(B$95$m * B$95$m + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * C + N[(N[(-0.5 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Simplified 21.4%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 28.0

         \[\leadsto \sqrt{\frac{\color{blue}{F \cdot -0.5}}{A}} \cdot \left(-\sqrt{2}\right) \]

   8. Simplified 28.0%

      \[\leadsto \sqrt{\color{blue}{\frac{F \cdot -0.5}{A}}} \cdot \left(-\sqrt{2}\right) \]

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000005e-193

   1. Initial program 98.7%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Simplified 88.8%

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

   if -5.0000000000000005e-193 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 13.1%

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

   4. Applied egg-rr 13.0%

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

   5. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 28.6

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

   7. Simplified 28.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 17.2

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

   5. Simplified 17.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. pow1/2 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 21.2

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

   7. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-undiv N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 21.2

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

   9. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

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

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

       5. lift-sqrt.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. clear-num N/A

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

       8. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. div-inv N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 21.2

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-193}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-308}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot t\_0\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+133}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 (- (sqrt 2.0))))
   (if (<= (pow B_m 2.0) 1e-308)
     (/
      (sqrt (* 2.0 (* (fma B_m B_m (* A (* C -4.0))) (* F (* 2.0 C)))))
      (- (* (* 4.0 A) C) (* B_m B_m)))
     (if (<= (pow B_m 2.0) 5e-10)
       (* (sqrt (/ (* F -0.5) A)) t_0)
       (if (<= (pow B_m 2.0) 2e+133)
         (*
          t_0
          (sqrt
           (/
            (* F (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C))))))
            (fma B_m B_m (* -4.0 (* A C))))))
         (/ (- (sqrt F)) (sqrt (* B_m 0.5))))))))

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 = -sqrt(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 1e-308) {
		tmp = sqrt((2.0 * (fma(B_m, B_m, (A * (C * -4.0))) * (F * (2.0 * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (pow(B_m, 2.0) <= 5e-10) {
		tmp = sqrt(((F * -0.5) / A)) * t_0;
	} else if (pow(B_m, 2.0) <= 2e+133) {
		tmp = t_0 * sqrt(((F * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C)))))) / fma(B_m, B_m, (-4.0 * (A * C)))));
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	}
	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(-sqrt(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-308)
		tmp = Float64(sqrt(Float64(2.0 * Float64(fma(B_m, B_m, Float64(A * Float64(C * -4.0))) * Float64(F * Float64(2.0 * C))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif ((B_m ^ 2.0) <= 5e-10)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * t_0);
	elseif ((B_m ^ 2.0) <= 2e+133)
		tmp = Float64(t_0 * sqrt(Float64(Float64(F * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C)))))) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-308], N[(N[Sqrt[N[(2.0 * N[(N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-10], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+133], N[(t$95$0 * N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(B$95$m * B$95$m + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999991e-309

   1. Initial program 14.0%

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

   4. Applied egg-rr 13.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 29.6

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

   7. Simplified 29.6%

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

   9. Applied egg-rr 29.7%

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

   if 9.9999999999999991e-309 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000031e-10

   1. Initial program 28.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Simplified 24.5%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 27.9

         \[\leadsto \sqrt{\frac{\color{blue}{F \cdot -0.5}}{A}} \cdot \left(-\sqrt{2}\right) \]

   8. Simplified 27.9%

      \[\leadsto \sqrt{\color{blue}{\frac{F \cdot -0.5}{A}}} \cdot \left(-\sqrt{2}\right) \]

   if 5.00000000000000031e-10 < (pow.f64 B #s(literal 2 binary64)) < 2e133

   1. Initial program 39.3%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Simplified 44.4%

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

   if 2e133 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.9%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 26.4

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

   5. Simplified 26.4%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. pow1/2 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 30.6

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

   7. Applied egg-rr 30.6%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-undiv N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 30.5

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

   9. Applied egg-rr 30.5%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

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

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

       5. lift-sqrt.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. clear-num N/A

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

       8. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. div-inv N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 30.6

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 30.6%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-308}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-308}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 1e-308)
   (/
    (sqrt (* 2.0 (* (fma B_m B_m (* A (* C -4.0))) (* F (* 2.0 C)))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= (pow B_m 2.0) 4e-28)
     (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
     (/ (- (sqrt F)) (sqrt (* B_m 0.5))))))

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 (pow(B_m, 2.0) <= 1e-308) {
		tmp = sqrt((2.0 * (fma(B_m, B_m, (A * (C * -4.0))) * (F * (2.0 * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (pow(B_m, 2.0) <= 4e-28) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	}
	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 ^ 2.0) <= 1e-308)
		tmp = Float64(sqrt(Float64(2.0 * Float64(fma(B_m, B_m, Float64(A * Float64(C * -4.0))) * Float64(F * Float64(2.0 * C))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif ((B_m ^ 2.0) <= 4e-28)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-308], N[(N[Sqrt[N[(2.0 * N[(N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-28], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999991e-309

   1. Initial program 14.0%

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

   4. Applied egg-rr 13.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 29.6

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

   7. Simplified 29.6%

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

   9. Applied egg-rr 29.7%

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

   if 9.9999999999999991e-309 < (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999988e-28

   1. Initial program 28.2%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Simplified 23.9%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 27.6

         \[\leadsto \sqrt{\frac{\color{blue}{F \cdot -0.5}}{A}} \cdot \left(-\sqrt{2}\right) \]

   8. Simplified 27.6%

      \[\leadsto \sqrt{\color{blue}{\frac{F \cdot -0.5}{A}}} \cdot \left(-\sqrt{2}\right) \]

   if 3.99999999999999988e-28 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 18.2%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 24.2

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

   5. Simplified 24.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. pow1/2 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 27.2

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

   7. Applied egg-rr 27.2%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-undiv N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 27.2

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

   9. Applied egg-rr 27.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

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

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

       5. lift-sqrt.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. clear-num N/A

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

       8. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. div-inv N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 27.3

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 27.3%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-308}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.55 \cdot 10^{-219}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)} \cdot \left(2 \cdot \sqrt{C \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\right)}\right)\\ \mathbf{elif}\;B\_m \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 2.55e-219)
   (*
    (/ -1.0 (fma C (* A -4.0) (* B_m B_m)))
    (* 2.0 (sqrt (* C (* F (fma -4.0 (* A C) (* B_m B_m)))))))
   (if (<= B_m 6.2e-12)
     (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
     (/ (- (sqrt F)) (sqrt (* B_m 0.5))))))

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 <= 2.55e-219) {
		tmp = (-1.0 / fma(C, (A * -4.0), (B_m * B_m))) * (2.0 * sqrt((C * (F * fma(-4.0, (A * C), (B_m * B_m))))));
	} else if (B_m <= 6.2e-12) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	}
	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 <= 2.55e-219)
		tmp = Float64(Float64(-1.0 / fma(C, Float64(A * -4.0), Float64(B_m * B_m))) * Float64(2.0 * sqrt(Float64(C * Float64(F * fma(-4.0, Float64(A * C), Float64(B_m * B_m)))))));
	elseif (B_m <= 6.2e-12)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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, 2.55e-219], N[(N[(-1.0 / N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sqrt[N[(C * N[(F * N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6.2e-12], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.5499999999999999e-219

   1. Initial program 17.3%

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

   4. Applied egg-rr 17.2%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 16.4

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

   7. Simplified 16.4%

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

   8. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-*.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 16.4

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

   10. Simplified 16.4%

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

   if 2.5499999999999999e-219 < B < 6.2000000000000002e-12

   1. Initial program 25.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Simplified 25.4%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 18.0

         \[\leadsto \sqrt{\frac{\color{blue}{F \cdot -0.5}}{A}} \cdot \left(-\sqrt{2}\right) \]

   8. Simplified 18.0%

      \[\leadsto \sqrt{\color{blue}{\frac{F \cdot -0.5}{A}}} \cdot \left(-\sqrt{2}\right) \]

   if 6.2000000000000002e-12 < B

   1. Initial program 19.8%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 53.4

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

   5. Simplified 53.4%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. pow1/2 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 62.8

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

   7. Applied egg-rr 62.8%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-undiv N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 62.8

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

   9. Applied egg-rr 62.8%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

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

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

       5. lift-sqrt.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. clear-num N/A

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

       8. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. div-inv N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 62.9

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 62.9%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.55 \cdot 10^{-219}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \cdot \left(2 \cdot \sqrt{C \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}\right)\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.55 \cdot 10^{-219}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)} \cdot \sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}\\ \mathbf{elif}\;B\_m \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 2.55e-219)
   (*
    (/ -1.0 (fma C (* A -4.0) (* B_m B_m)))
    (sqrt (* 2.0 (* (* 2.0 C) (* -4.0 (* A (* C F)))))))
   (if (<= B_m 6.2e-12)
     (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
     (/ (- (sqrt F)) (sqrt (* B_m 0.5))))))

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 <= 2.55e-219) {
		tmp = (-1.0 / fma(C, (A * -4.0), (B_m * B_m))) * sqrt((2.0 * ((2.0 * C) * (-4.0 * (A * (C * F))))));
	} else if (B_m <= 6.2e-12) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	}
	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 <= 2.55e-219)
		tmp = Float64(Float64(-1.0 / fma(C, Float64(A * -4.0), Float64(B_m * B_m))) * sqrt(Float64(2.0 * Float64(Float64(2.0 * C) * Float64(-4.0 * Float64(A * Float64(C * F)))))));
	elseif (B_m <= 6.2e-12)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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, 2.55e-219], N[(N[(-1.0 / N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(2.0 * C), $MachinePrecision] * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6.2e-12], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.5499999999999999e-219

   1. Initial program 17.3%

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

   4. Applied egg-rr 17.2%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 16.4

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

   7. Simplified 16.4%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 15.4

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

   10. Simplified 15.4%

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

   if 2.5499999999999999e-219 < B < 6.2000000000000002e-12

   1. Initial program 25.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Simplified 25.4%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 18.0

         \[\leadsto \sqrt{\frac{\color{blue}{F \cdot -0.5}}{A}} \cdot \left(-\sqrt{2}\right) \]

   8. Simplified 18.0%

      \[\leadsto \sqrt{\color{blue}{\frac{F \cdot -0.5}{A}}} \cdot \left(-\sqrt{2}\right) \]

   if 6.2000000000000002e-12 < B

   1. Initial program 19.8%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 53.4

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

   5. Simplified 53.4%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. pow1/2 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 62.8

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

   7. Applied egg-rr 62.8%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-undiv N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 62.8

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

   9. Applied egg-rr 62.8%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

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

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

       5. lift-sqrt.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. clear-num N/A

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

       8. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. div-inv N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 62.9

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 62.9%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.55 \cdot 10^{-219}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \cdot \sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.4% accurate, 9.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 6.2e-12)
   (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
   (/ (- (sqrt F)) (sqrt (* B_m 0.5)))))

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 <= 6.2e-12) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (b_m <= 6.2d-12) then
        tmp = sqrt(((f * (-0.5d0)) / a)) * -sqrt(2.0d0)
    else
        tmp = -sqrt(f) / sqrt((b_m * 0.5d0))
    end if
    code = tmp
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) {
	double tmp;
	if (B_m <= 6.2e-12) {
		tmp = Math.sqrt(((F * -0.5) / A)) * -Math.sqrt(2.0);
	} else {
		tmp = -Math.sqrt(F) / Math.sqrt((B_m * 0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 6.2e-12:
		tmp = math.sqrt(((F * -0.5) / A)) * -math.sqrt(2.0)
	else:
		tmp = -math.sqrt(F) / math.sqrt((B_m * 0.5))
	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 <= 6.2e-12)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 6.2e-12)
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	else
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	end
	tmp_2 = 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, 6.2e-12], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.2000000000000002e-12

   1. Initial program 18.8%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Simplified 17.8%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 17.9

         \[\leadsto \sqrt{\frac{\color{blue}{F \cdot -0.5}}{A}} \cdot \left(-\sqrt{2}\right) \]

   8. Simplified 17.9%

      \[\leadsto \sqrt{\color{blue}{\frac{F \cdot -0.5}{A}}} \cdot \left(-\sqrt{2}\right) \]

   if 6.2000000000000002e-12 < B

   1. Initial program 19.8%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 53.4

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

   5. Simplified 53.4%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. pow1/2 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 62.8

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

   7. Applied egg-rr 62.8%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-undiv N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 62.8

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

   9. Applied egg-rr 62.8%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

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

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

       5. lift-sqrt.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. clear-num N/A

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

       8. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. div-inv N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 62.9

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 62.9%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 33.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 3.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m + A\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B\_m}}\\ \end{array} \end{array} \]

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 (<= F 3.9e-58)
   (/ (sqrt (* 2.0 (* F (+ B_m A)))) (- B_m))
   (- (sqrt (/ (* 2.0 F) 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 (F <= 3.9e-58) {
		tmp = sqrt((2.0 * (F * (B_m + A)))) / -B_m;
	} else {
		tmp = -sqrt(((2.0 * F) / B_m));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (f <= 3.9d-58) then
        tmp = sqrt((2.0d0 * (f * (b_m + a)))) / -b_m
    else
        tmp = -sqrt(((2.0d0 * f) / b_m))
    end if
    code = tmp
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) {
	double tmp;
	if (F <= 3.9e-58) {
		tmp = Math.sqrt((2.0 * (F * (B_m + A)))) / -B_m;
	} else {
		tmp = -Math.sqrt(((2.0 * F) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 3.9e-58:
		tmp = math.sqrt((2.0 * (F * (B_m + A)))) / -B_m
	else:
		tmp = -math.sqrt(((2.0 * F) / 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 (F <= 3.9e-58)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m + A)))) / Float64(-B_m));
	else
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) / B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 3.9e-58)
		tmp = sqrt((2.0 * (F * (B_m + A)))) / -B_m;
	else
		tmp = -sqrt(((2.0 * F) / B_m));
	end
	tmp_2 = 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[F, 3.9e-58], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < 3.89999999999999992e-58

   1. Initial program 22.6%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Simplified 9.1%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 14.4

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

   8. Simplified 14.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(B\right)}{\sqrt{2} \cdot \sqrt{F \cdot \left(B \cdot \left(1 + \frac{A}{B}\right)\right)}}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(B\right)}{\sqrt{2} \cdot \sqrt{F \cdot \left(B \cdot \left(1 + \frac{A}{B}\right)\right)}}}} \]

      11. lower-/.f64 14.4

          \[\leadsto \frac{1}{\color{blue}{\frac{-B}{\sqrt{2} \cdot \sqrt{F \cdot \left(B \cdot \left(1 + \frac{A}{B}\right)\right)}}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{neg}\left(B\right)}{\color{blue}{\sqrt{2} \cdot \sqrt{F \cdot \left(B \cdot \left(1 + \frac{A}{B}\right)\right)}}}} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{neg}\left(B\right)}{\color{blue}{\sqrt{2}} \cdot \sqrt{F \cdot \left(B \cdot \left(1 + \frac{A}{B}\right)\right)}}} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{neg}\left(B\right)}{\sqrt{2} \cdot \color{blue}{\sqrt{F \cdot \left(B \cdot \left(1 + \frac{A}{B}\right)\right)}}}} \]

      15. sqrt-unprod N/A

          \[\leadsto \frac{1}{\frac{\mathsf{neg}\left(B\right)}{\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(1 + \frac{A}{B}\right)\right)\right)}}}} \]

   10. Applied egg-rr 14.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{-B}{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(A, 1, B\right)\right)}}}} \]
   11. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{neg}\left(B\right)}{\sqrt{2 \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(A, 1, B\right)}\right)}}} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{neg}\left(B\right)}{\sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(A, 1, B\right)\right)}}}} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{neg}\left(B\right)}{\sqrt{\color{blue}{2 \cdot \left(F \cdot \mathsf{fma}\left(A, 1, B\right)\right)}}}} \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{neg}\left(B\right)}{\color{blue}{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(A, 1, B\right)\right)}}}} \]

       5. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{neg}\left(B\right)}}{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(A, 1, B\right)\right)}}} \]

       6. lift-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(B\right)}{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(A, 1, B\right)\right)}}}} \]

       7. /-rgt-identity N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\mathsf{neg}\left(B\right)}{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(A, 1, B\right)\right)}}}{1}}} \]

       8. /-rgt-identity N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(B\right)}{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(A, 1, B\right)\right)}}}} \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(B\right)}{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(A, 1, B\right)\right)}}}} \]

       10. clear-num N/A

           \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(A, 1, B\right)\right)}}{\mathsf{neg}\left(B\right)}} \]

       11. lower-/.f64 14.6

           \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(A, 1, B\right)\right)}}{-B}} \]

       12. lift-fma.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot \left(F \cdot \color{blue}{\left(A \cdot 1 + B\right)}\right)}}{\mathsf{neg}\left(B\right)} \]

       13. *-rgt-identity N/A

           \[\leadsto \frac{\sqrt{2 \cdot \left(F \cdot \left(\color{blue}{A} + B\right)\right)}}{\mathsf{neg}\left(B\right)} \]

       14. +-commutative N/A

           \[\leadsto \frac{\sqrt{2 \cdot \left(F \cdot \color{blue}{\left(B + A\right)}\right)}}{\mathsf{neg}\left(B\right)} \]

       15. lower-+.f64 14.6

           \[\leadsto \frac{\sqrt{2 \cdot \left(F \cdot \color{blue}{\left(B + A\right)}\right)}}{-B} \]

   12. Applied egg-rr 14.6%

       \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(B + A\right)\right)}}{-B}} \]

   if 3.89999999999999992e-58 < F

   1. Initial program 15.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

      7. lower-/.f64 21.1

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 21.1%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \color{blue}{\frac{F}{B}}}\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      7. lower-*.f64 21.2

         \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

   7. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{\frac{2 \cdot F}{B}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 35.5% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}} \end{array} \]

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 F)) (sqrt (* B_m 0.5))))

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(F) / sqrt((B_m * 0.5));
}

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(f) / sqrt((b_m * 0.5d0))
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(F) / Math.sqrt((B_m * 0.5));
}

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(F) / math.sqrt((B_m * 0.5))

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(F)) / sqrt(Float64(B_m * 0.5)))
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(F) / sqrt((B_m * 0.5));
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[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.1%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

   7. lower-/.f64 16.3

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 16.3%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}\right) \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F}{B}}} \cdot \sqrt{2}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{F}{B}}} \cdot \sqrt{2}\right) \]

   7. sqrt-div N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \cdot \sqrt{2}\right) \]

   8. pow1/2 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}}}{\sqrt{B}} \cdot \sqrt{2}\right) \]

   9. associate-*l/ N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

   11. pow1/2 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

   12. lift-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt{F} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

   13. sqrt-unprod N/A

       \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

   14. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

   16. lower-sqrt.f64 18.0

       \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

7. Applied egg-rr 18.0%

   \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

   3. sqrt-prod N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F} \cdot \sqrt{2}}}{\sqrt{B}}\right) \]

   4. pow1/2 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\color{blue}{\sqrt{B}}}\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot \frac{\sqrt{2}}{\sqrt{B}}}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot \frac{\sqrt{2}}{\sqrt{B}}}\right) \]

   9. pow1/2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{2}}{\sqrt{B}}\right) \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{2}}{\sqrt{B}}\right) \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \frac{\color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

   12. lift-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \frac{\sqrt{2}}{\color{blue}{\sqrt{B}}}\right) \]

   13. sqrt-undiv N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

   14. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

   15. lower-/.f64 18.0

       \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

9. Applied egg-rr 18.0%

   \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
10. Step-by-step derivation

    1. lift-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \sqrt{\frac{2}{B}}\right) \]

    2. lift-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}}\right) \]

    3. lift-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

    4. distribute-lft-neg-in N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{F}\right)\right) \cdot \sqrt{\frac{2}{B}}} \]

    5. lift-sqrt.f64 N/A

       \[\leadsto \left(\mathsf{neg}\left(\sqrt{F}\right)\right) \cdot \color{blue}{\sqrt{\frac{2}{B}}} \]

    6. lift-/.f64 N/A

       \[\leadsto \left(\mathsf{neg}\left(\sqrt{F}\right)\right) \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

    7. clear-num N/A

       \[\leadsto \left(\mathsf{neg}\left(\sqrt{F}\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{B}{2}}}} \]

    8. sqrt-div N/A

       \[\leadsto \left(\mathsf{neg}\left(\sqrt{F}\right)\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{B}{2}}}} \]

    9. metadata-eval N/A

       \[\leadsto \left(\mathsf{neg}\left(\sqrt{F}\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{B}{2}}} \]

    10. un-div-inv N/A

        \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{\frac{B}{2}}}} \]

    11. lower-/.f64 N/A

        \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{\frac{B}{2}}}} \]

    12. lower-neg.f64 N/A

        \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{F}\right)}}{\sqrt{\frac{B}{2}}} \]

    13. lower-sqrt.f64 N/A

        \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\color{blue}{\sqrt{\frac{B}{2}}}} \]

    14. div-inv N/A

        \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{\color{blue}{B \cdot \frac{1}{2}}}} \]

    15. metadata-eval N/A

        \[\leadsto \frac{\mathsf{neg}\left(\sqrt{F}\right)}{\sqrt{B \cdot \color{blue}{\frac{1}{2}}}} \]

    16. lower-*.f64 18.1

        \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

11. Applied egg-rr 18.1%

    \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
12. Add Preprocessing

Alternative 12: 35.5% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}} \end{array} \]

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 F)) (sqrt (/ 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(F) * sqrt((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(f) * sqrt((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(F) * Math.sqrt((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(F) * math.sqrt((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(F)) * sqrt(Float64(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(F) * sqrt((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[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.1%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

   7. lower-/.f64 16.3

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 16.3%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}\right) \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F}{B}}} \cdot \sqrt{2}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{F}{B}}} \cdot \sqrt{2}\right) \]

   7. sqrt-div N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \cdot \sqrt{2}\right) \]

   8. pow1/2 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}}}{\sqrt{B}} \cdot \sqrt{2}\right) \]

   9. associate-*l/ N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

   11. pow1/2 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

   12. lift-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt{F} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

   13. sqrt-unprod N/A

       \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

   14. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

   16. lower-sqrt.f64 18.0

       \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

7. Applied egg-rr 18.0%

   \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

   3. sqrt-prod N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F} \cdot \sqrt{2}}}{\sqrt{B}}\right) \]

   4. pow1/2 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\color{blue}{\sqrt{B}}}\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot \frac{\sqrt{2}}{\sqrt{B}}}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot \frac{\sqrt{2}}{\sqrt{B}}}\right) \]

   9. pow1/2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{2}}{\sqrt{B}}\right) \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{2}}{\sqrt{B}}\right) \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \frac{\color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

   12. lift-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \frac{\sqrt{2}}{\color{blue}{\sqrt{B}}}\right) \]

   13. sqrt-undiv N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

   14. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

   15. lower-/.f64 18.0

       \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

9. Applied egg-rr 18.0%

   \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]

10. Final simplification 18.0%

    \[\leadsto \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}} \]
11. Add Preprocessing

Alternative 13: 26.9% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{\frac{2 \cdot F}{B\_m}} \end{array} \]

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 (/ (* 2.0 F) 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(((2.0 * F) / 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(((2.0d0 * f) / 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(((2.0 * F) / 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(((2.0 * F) / 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(-sqrt(Float64(Float64(2.0 * F) / 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(((2.0 * F) / 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[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 19.1%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

   7. lower-/.f64 16.3

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 16.3%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

   2. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \color{blue}{\frac{F}{B}}}\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   7. lower-*.f64 16.4

      \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

7. Applied egg-rr 16.4%

   \[\leadsto -\color{blue}{\sqrt{\frac{2 \cdot F}{B}}} \]
8. Add Preprocessing

Alternative 14: 26.9% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

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 (* 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((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((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((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((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(-sqrt(Float64(F * Float64(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((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[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 19.1%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

   7. lower-/.f64 16.3

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 16.3%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

   2. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \color{blue}{\frac{F}{B}}}\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   7. lower-*.f64 16.4

      \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

7. Applied egg-rr 16.4%

   \[\leadsto -\color{blue}{\sqrt{\frac{2 \cdot F}{B}}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

   4. lower-/.f64 16.4

      \[\leadsto -\sqrt{F \cdot \color{blue}{\frac{2}{B}}} \]

9. Applied egg-rr 16.4%

   \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :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))))