ABCF->ab-angle b

Percentage Accurate: 18.9% → 37.1%

Time: 18.1s

Alternatives: 13

Speedup: 10.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 37.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := 2 \cdot \left(\left({B}^{2} - t\_1\right) \cdot F\right)\\ t_3 := t\_1 - {B}^{2}\\ t_4 := \frac{\sqrt{t\_2 \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ t_5 := \left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+52}:\\ \;\;\;\;\left(\sqrt{2 \cdot t\_0} \cdot \sqrt{F \cdot \left(A + A\right)}\right) \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\frac{t\_5}{\sqrt{t\_5}}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)}}{t\_3}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B B)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (* 2.0 (* (- (pow B 2.0) t_1) F)))
        (t_3 (- t_1 (pow B 2.0)))
        (t_4
         (/
          (sqrt (* t_2 (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          t_3))
        (t_5
         (*
          (* (fma (* A C) -4.0 (* B B)) (* 2.0 F))
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B)))))))
   (if (<= t_4 -5e+52)
     (* (* (sqrt (* 2.0 t_0)) (sqrt (* F (+ A A)))) (/ -1.0 t_0))
     (if (<= t_4 -1e-218)
       (/ (/ t_5 (sqrt t_5)) t_3)
       (/ (sqrt (* t_2 (+ A (fma -0.5 (/ (* B B) C) A)))) t_3)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B * B));
	double t_1 = (4.0 * A) * C;
	double t_2 = 2.0 * ((pow(B, 2.0) - t_1) * F);
	double t_3 = t_1 - pow(B, 2.0);
	double t_4 = sqrt((t_2 * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double t_5 = (fma((A * C), -4.0, (B * B)) * (2.0 * F)) * ((A + C) - sqrt(fma((A - C), (A - C), (B * B))));
	double tmp;
	if (t_4 <= -5e+52) {
		tmp = (sqrt((2.0 * t_0)) * sqrt((F * (A + A)))) * (-1.0 / t_0);
	} else if (t_4 <= -1e-218) {
		tmp = (t_5 / sqrt(t_5)) / t_3;
	} else {
		tmp = sqrt((t_2 * (A + fma(-0.5, ((B * B) / C), A)))) / t_3;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(2.0 * Float64(Float64((B ^ 2.0) - t_1) * F))
	t_3 = Float64(t_1 - (B ^ 2.0))
	t_4 = Float64(sqrt(Float64(t_2 * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	t_5 = Float64(Float64(fma(Float64(A * C), -4.0, Float64(B * B)) * Float64(2.0 * F)) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))))
	tmp = 0.0
	if (t_4 <= -5e+52)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * sqrt(Float64(F * Float64(A + A)))) * Float64(-1.0 / t_0));
	elseif (t_4 <= -1e-218)
		tmp = Float64(Float64(t_5 / sqrt(t_5)) / t_3);
	else
		tmp = Float64(sqrt(Float64(t_2 * Float64(A + fma(-0.5, Float64(Float64(B * B) / C), A)))) / t_3);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$2 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+52], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -1e-218], N[(N[(t$95$5 / N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Sqrt[N[(t$95$2 * N[(A + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5e52

   1. Initial program 16.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 25.4

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

   5. Simplified 25.4%

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

   7. Applied egg-rr 25.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

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

      13. pow1/2 N/A

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

      14. lower-*.f64 N/A

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

   9. Applied egg-rr 32.2%

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

   if -5e52 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 99.2%

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

   4. Applied egg-rr 99.4%

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

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

   1. Initial program 5.4%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 11.0

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

   5. Simplified 11.0%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+52}:\\ \;\;\;\;\left(\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{F \cdot \left(A + A\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}{\sqrt{\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 35.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\\ t_2 := \frac{-1}{t\_0}\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3 - {B}^{2}}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+120}:\\ \;\;\;\;\left(\sqrt{2 \cdot t\_0} \cdot \sqrt{F \cdot \left(A + A\right)}\right) \cdot t\_2\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}{t\_1}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(2 \cdot \sqrt{\left(A \cdot F\right) \cdot t\_1}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B B)))
        (t_1 (fma B B (* -4.0 (* A C))))
        (t_2 (/ -1.0 t_0))
        (t_3 (* (* 4.0 A) C))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_3) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_3 (pow B 2.0)))))
   (if (<= t_4 -2e+120)
     (* (* (sqrt (* 2.0 t_0)) (sqrt (* F (+ A A)))) t_2)
     (if (<= t_4 -1e-132)
       (*
        (sqrt (/ (* F (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B))))) t_1))
        (- (sqrt 2.0)))
       (if (<= t_4 -1e-218)
         (* (sqrt (* F (- C (sqrt (fma B B (* C C)))))) (/ (sqrt 2.0) (- B)))
         (* t_2 (* 2.0 (sqrt (* (* A F) t_1)))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B * B));
	double t_1 = fma(B, B, (-4.0 * (A * C)));
	double t_2 = -1.0 / t_0;
	double t_3 = (4.0 * A) * C;
	double t_4 = sqrt(((2.0 * ((pow(B, 2.0) - t_3) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_3 - pow(B, 2.0));
	double tmp;
	if (t_4 <= -2e+120) {
		tmp = (sqrt((2.0 * t_0)) * sqrt((F * (A + A)))) * t_2;
	} else if (t_4 <= -1e-132) {
		tmp = sqrt(((F * ((A + C) - sqrt(fma((A - C), (A - C), (B * B))))) / t_1)) * -sqrt(2.0);
	} else if (t_4 <= -1e-218) {
		tmp = sqrt((F * (C - sqrt(fma(B, B, (C * C)))))) * (sqrt(2.0) / -B);
	} else {
		tmp = t_2 * (2.0 * sqrt(((A * F) * t_1)));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_1 = fma(B, B, Float64(-4.0 * Float64(A * C)))
	t_2 = Float64(-1.0 / t_0)
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_3) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_3 - (B ^ 2.0)))
	tmp = 0.0
	if (t_4 <= -2e+120)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * sqrt(Float64(F * Float64(A + A)))) * t_2);
	elseif (t_4 <= -1e-132)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))) / t_1)) * Float64(-sqrt(2.0)));
	elseif (t_4 <= -1e-218)
		tmp = Float64(sqrt(Float64(F * Float64(C - sqrt(fma(B, B, Float64(C * C)))))) * Float64(sqrt(2.0) / Float64(-B)));
	else
		tmp = Float64(t_2 * Float64(2.0 * sqrt(Float64(Float64(A * F) * t_1))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+120], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$4, -1e-132], N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$4, -1e-218], N[(N[Sqrt[N[(F * N[(C - N[Sqrt[N[(B * B + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B)), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2e120

   1. Initial program 10.4%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 23.9

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

   5. Simplified 23.9%

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

   7. Applied egg-rr 23.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

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

      13. pow1/2 N/A

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

      14. lower-*.f64 N/A

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

   9. Applied egg-rr 31.4%

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

   if -2e120 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999999e-133

   1. Initial program 97.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Simplified 99.3%

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

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

   1. Initial program 99.0%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Simplified 40.9%

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

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

   1. Initial program 5.4%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 8.6

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

   5. Simplified 8.6%

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

   7. Applied egg-rr 8.6%

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

   8. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 9.5

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

   10. Simplified 9.5%

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

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+120}:\\ \;\;\;\;\left(\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{F \cdot \left(A + A\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 35.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := t\_1 - {B}^{2}\\ t_3 := {B}^{2} - t\_1\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ t_5 := \left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+52}:\\ \;\;\;\;\left(\sqrt{2 \cdot t\_0} \cdot \sqrt{F \cdot \left(A + A\right)}\right) \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\frac{t\_5}{\sqrt{t\_5}}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(A \cdot F\right)} \cdot -2}{t\_3}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B B)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- t_1 (pow B 2.0)))
        (t_3 (- (pow B 2.0) t_1))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          t_2))
        (t_5
         (*
          (* (fma (* A C) -4.0 (* B B)) (* 2.0 F))
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B)))))))
   (if (<= t_4 -5e+52)
     (* (* (sqrt (* 2.0 t_0)) (sqrt (* F (+ A A)))) (/ -1.0 t_0))
     (if (<= t_4 -1e-218)
       (/ (/ t_5 (sqrt t_5)) t_2)
       (/ (* (sqrt (* t_0 (* A F))) -2.0) t_3)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B * B));
	double t_1 = (4.0 * A) * C;
	double t_2 = t_1 - pow(B, 2.0);
	double t_3 = pow(B, 2.0) - t_1;
	double t_4 = sqrt(((2.0 * (t_3 * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double t_5 = (fma((A * C), -4.0, (B * B)) * (2.0 * F)) * ((A + C) - sqrt(fma((A - C), (A - C), (B * B))));
	double tmp;
	if (t_4 <= -5e+52) {
		tmp = (sqrt((2.0 * t_0)) * sqrt((F * (A + A)))) * (-1.0 / t_0);
	} else if (t_4 <= -1e-218) {
		tmp = (t_5 / sqrt(t_5)) / t_2;
	} else {
		tmp = (sqrt((t_0 * (A * F))) * -2.0) / t_3;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(t_1 - (B ^ 2.0))
	t_3 = Float64((B ^ 2.0) - t_1)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	t_5 = Float64(Float64(fma(Float64(A * C), -4.0, Float64(B * B)) * Float64(2.0 * F)) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))))
	tmp = 0.0
	if (t_4 <= -5e+52)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * sqrt(Float64(F * Float64(A + A)))) * Float64(-1.0 / t_0));
	elseif (t_4 <= -1e-218)
		tmp = Float64(Float64(t_5 / sqrt(t_5)) / t_2);
	else
		tmp = Float64(Float64(sqrt(Float64(t_0 * Float64(A * F))) * -2.0) / t_3);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+52], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -1e-218], N[(N[(t$95$5 / N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[Sqrt[N[(t$95$0 * N[(A * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5e52

   1. Initial program 16.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 25.4

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

   5. Simplified 25.4%

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

   7. Applied egg-rr 25.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

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

      13. pow1/2 N/A

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

      14. lower-*.f64 N/A

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

   9. Applied egg-rr 32.2%

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

   if -5e52 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 99.2%

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

   4. Applied egg-rr 99.4%

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

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

   1. Initial program 5.4%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 8.6

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

   5. Simplified 8.6%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 9.6

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

   8. Simplified 9.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   10. Applied egg-rr 9.6%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+52}:\\ \;\;\;\;\left(\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{F \cdot \left(A + A\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}{\sqrt{\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(A \cdot F\right)} \cdot -2}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := {B}^{2} - t\_1\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B}^{2}}\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{+79}:\\ \;\;\;\;\left(\sqrt{2 \cdot t\_0} \cdot \sqrt{F \cdot \left(A + A\right)}\right) \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\mathsf{fma}\left(B, -B, 4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(A \cdot F\right)} \cdot -2}{t\_2}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B B)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- (pow B 2.0) t_1))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B 2.0)))))
   (if (<= t_3 -4e+79)
     (* (* (sqrt (* 2.0 t_0)) (sqrt (* F (+ A A)))) (/ -1.0 t_0))
     (if (<= t_3 -1e-218)
       (/
        (sqrt
         (*
          (* (fma (* A C) -4.0 (* B B)) (* 2.0 F))
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B))))))
        (fma B (- B) (* 4.0 (* A C))))
       (/ (* (sqrt (* t_0 (* A F))) -2.0) t_2)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B * B));
	double t_1 = (4.0 * A) * C;
	double t_2 = pow(B, 2.0) - t_1;
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B, 2.0));
	double tmp;
	if (t_3 <= -4e+79) {
		tmp = (sqrt((2.0 * t_0)) * sqrt((F * (A + A)))) * (-1.0 / t_0);
	} else if (t_3 <= -1e-218) {
		tmp = sqrt(((fma((A * C), -4.0, (B * B)) * (2.0 * F)) * ((A + C) - sqrt(fma((A - C), (A - C), (B * B)))))) / fma(B, -B, (4.0 * (A * C)));
	} else {
		tmp = (sqrt((t_0 * (A * F))) * -2.0) / t_2;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64((B ^ 2.0) - t_1)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -4e+79)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * sqrt(Float64(F * Float64(A + A)))) * Float64(-1.0 / t_0));
	elseif (t_3 <= -1e-218)
		tmp = Float64(sqrt(Float64(Float64(fma(Float64(A * C), -4.0, Float64(B * B)) * Float64(2.0 * F)) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))))) / fma(B, Float64(-B), Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(Float64(sqrt(Float64(t_0 * Float64(A * F))) * -2.0) / t_2);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -4e+79], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-218], N[(N[Sqrt[N[(N[(N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B * (-B) + N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(t$95$0 * N[(A * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.5%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 24.1

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

   5. Simplified 24.1%

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

   7. Applied egg-rr 24.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

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

      13. pow1/2 N/A

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

      14. lower-*.f64 N/A

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

   9. Applied egg-rr 31.0%

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

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

   1. Initial program 99.3%

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

   4. Applied egg-rr 99.3%

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

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

   1. Initial program 5.4%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 8.6

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

   5. Simplified 8.6%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 9.6

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

   8. Simplified 9.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   10. Applied egg-rr 9.6%

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

4. Final simplification 26.3%

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

Alternative 5: 36.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \frac{-1}{t\_0}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B}^{2}}\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{+79}:\\ \;\;\;\;\left(\sqrt{2 \cdot t\_0} \cdot \sqrt{F \cdot \left(A + A\right)}\right) \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\mathsf{fma}\left(B, -B, 4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B B)))
        (t_1 (/ -1.0 t_0))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_2) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B 2.0)))))
   (if (<= t_3 -4e+79)
     (* (* (sqrt (* 2.0 t_0)) (sqrt (* F (+ A A)))) t_1)
     (if (<= t_3 -1e-218)
       (/
        (sqrt
         (*
          (* (fma (* A C) -4.0 (* B B)) (* 2.0 F))
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B))))))
        (fma B (- B) (* 4.0 (* A C))))
       (* t_1 (* 2.0 (sqrt (* (* A F) (fma B B (* -4.0 (* A C)))))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B * B));
	double t_1 = -1.0 / t_0;
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B, 2.0) - t_2) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B, 2.0));
	double tmp;
	if (t_3 <= -4e+79) {
		tmp = (sqrt((2.0 * t_0)) * sqrt((F * (A + A)))) * t_1;
	} else if (t_3 <= -1e-218) {
		tmp = sqrt(((fma((A * C), -4.0, (B * B)) * (2.0 * F)) * ((A + C) - sqrt(fma((A - C), (A - C), (B * B)))))) / fma(B, -B, (4.0 * (A * C)));
	} else {
		tmp = t_1 * (2.0 * sqrt(((A * F) * fma(B, B, (-4.0 * (A * C))))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_1 = Float64(-1.0 / t_0)
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -4e+79)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * sqrt(Float64(F * Float64(A + A)))) * t_1);
	elseif (t_3 <= -1e-218)
		tmp = Float64(sqrt(Float64(Float64(fma(Float64(A * C), -4.0, Float64(B * B)) * Float64(2.0 * F)) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))))) / fma(B, Float64(-B), Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(t_1 * Float64(2.0 * sqrt(Float64(Float64(A * F) * fma(B, B, Float64(-4.0 * Float64(A * C)))))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / 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, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -4e+79], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -1e-218], N[(N[Sqrt[N[(N[(N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B * (-B) + N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] * N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.5%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 24.1

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

   5. Simplified 24.1%

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

   7. Applied egg-rr 24.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

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

      13. pow1/2 N/A

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

      14. lower-*.f64 N/A

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

   9. Applied egg-rr 31.0%

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

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

   1. Initial program 99.3%

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

   4. Applied egg-rr 99.3%

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

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

   1. Initial program 5.4%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 8.6

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

   5. Simplified 8.6%

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

   7. Applied egg-rr 8.6%

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

   8. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 9.5

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

   10. Simplified 9.5%

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

4. Final simplification 26.2%

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

Alternative 6: 30.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \frac{-1}{t\_0}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B}^{2}}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+52}:\\ \;\;\;\;\left(\sqrt{2 \cdot t\_0} \cdot \sqrt{F \cdot \left(A + A\right)}\right) \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B B)))
        (t_1 (/ -1.0 t_0))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_2) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B 2.0)))))
   (if (<= t_3 -5e+52)
     (* (* (sqrt (* 2.0 t_0)) (sqrt (* F (+ A A)))) t_1)
     (if (<= t_3 -1e-218)
       (/ (* (sqrt 2.0) (sqrt (* F (- A (sqrt (fma A A (* B B))))))) (- B))
       (* t_1 (* 2.0 (sqrt (* (* A F) (fma B B (* -4.0 (* A C)))))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B * B));
	double t_1 = -1.0 / t_0;
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B, 2.0) - t_2) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B, 2.0));
	double tmp;
	if (t_3 <= -5e+52) {
		tmp = (sqrt((2.0 * t_0)) * sqrt((F * (A + A)))) * t_1;
	} else if (t_3 <= -1e-218) {
		tmp = (sqrt(2.0) * sqrt((F * (A - sqrt(fma(A, A, (B * B))))))) / -B;
	} else {
		tmp = t_1 * (2.0 * sqrt(((A * F) * fma(B, B, (-4.0 * (A * C))))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_1 = Float64(-1.0 / t_0)
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -5e+52)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * sqrt(Float64(F * Float64(A + A)))) * t_1);
	elseif (t_3 <= -1e-218)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B * B))))))) / Float64(-B));
	else
		tmp = Float64(t_1 * Float64(2.0 * sqrt(Float64(Float64(A * F) * fma(B, B, Float64(-4.0 * Float64(A * C)))))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / 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, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+52], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -1e-218], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B)), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] * N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5e52

   1. Initial program 16.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 25.4

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

   5. Simplified 25.4%

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

   7. Applied egg-rr 25.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

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

      13. pow1/2 N/A

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

      14. lower-*.f64 N/A

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

   9. Applied egg-rr 32.2%

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

   if -5e52 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-218

   1. Initial program 99.2%

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

   3. Taylor expanded in C around 0

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

      1. 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 39.9%

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

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

   1. Initial program 5.4%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 8.6

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

   5. Simplified 8.6%

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

   7. Applied egg-rr 8.6%

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

   8. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 9.5

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

   10. Simplified 9.5%

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

4. Final simplification 18.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+52}:\\ \;\;\;\;\left(\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{F \cdot \left(A + A\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 28.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;{B}^{2} \leq 10^{-19}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{-t\_0}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{1}{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B B))))
   (if (<= (pow B 2.0) 1e-19)
     (/ (sqrt (* (+ A A) (* t_0 (* 2.0 F)))) (- t_0))
     (if (<= (pow B 2.0) 5e+298)
       (/
        -1.0
        (*
         (/ B (sqrt 2.0))
         (sqrt (/ 1.0 (* F (- A (sqrt (fma A A (* B B)))))))))
       (* (sqrt (* A F)) (/ -2.0 B))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B * B));
	double tmp;
	if (pow(B, 2.0) <= 1e-19) {
		tmp = sqrt(((A + A) * (t_0 * (2.0 * F)))) / -t_0;
	} else if (pow(B, 2.0) <= 5e+298) {
		tmp = -1.0 / ((B / sqrt(2.0)) * sqrt((1.0 / (F * (A - sqrt(fma(A, A, (B * B))))))));
	} else {
		tmp = sqrt((A * F)) * (-2.0 / B);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B * B))
	tmp = 0.0
	if ((B ^ 2.0) <= 1e-19)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(t_0 * Float64(2.0 * F)))) / Float64(-t_0));
	elseif ((B ^ 2.0) <= 5e+298)
		tmp = Float64(-1.0 / Float64(Float64(B / sqrt(2.0)) * sqrt(Float64(1.0 / Float64(F * Float64(A - sqrt(fma(A, A, Float64(B * B)))))))));
	else
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-2.0 / B));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e-19], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e+298], N[(-1.0 / N[(N[(B / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(F * N[(A - N[Sqrt[N[(A * A + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999998e-20

   1. Initial program 21.3%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 21.2

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

   5. Simplified 21.2%

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

   7. Applied egg-rr 21.2%

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

   if 9.9999999999999998e-20 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e298

   1. Initial program 33.7%

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

   4. Applied egg-rr 33.7%

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

   5. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 18.5

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

   7. Simplified 18.5%

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

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

   1. Initial program 1.7%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B}} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{2}}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}}\right) \]

   7. Simplified 0.1%

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

   8. Taylor expanded in A around -inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{A \cdot F} \cdot \frac{-1 \cdot \color{blue}{2}}{B} \]

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 5.6

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

   10. Simplified 5.6%

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

Alternative 8: 27.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 18.7%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 20.2

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

   5. Simplified 20.2%

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

   7. Applied egg-rr 20.2%

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

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

   1. Initial program 35.1%

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

   4. Applied egg-rr 1.4%

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

   5. Taylor expanded in C around 0

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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 19.0

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

   7. Simplified 19.0%

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

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

   1. Initial program 1.7%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B}} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{2}}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}}\right) \]

   7. Simplified 0.1%

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

   8. Taylor expanded in A around -inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{A \cdot F} \cdot \frac{-1 \cdot \color{blue}{2}}{B} \]

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 5.6

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

   10. Simplified 5.6%

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

4. Final simplification 16.4%

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

Alternative 9: 25.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)} \cdot \frac{1}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= (pow B 2.0) 4e-121)
   (*
    (sqrt (* (+ A A) (* (fma A (* C -4.0) (* B B)) (* 2.0 F))))
    (/ 1.0 (* 4.0 (* A C))))
   (if (<= (pow B 2.0) 5e+298)
     (* (/ (sqrt 2.0) (- B)) (sqrt (* F (- A (sqrt (fma A A (* B B)))))))
     (* (sqrt (* A F)) (/ -2.0 B)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (pow(B, 2.0) <= 4e-121) {
		tmp = sqrt(((A + A) * (fma(A, (C * -4.0), (B * B)) * (2.0 * F)))) * (1.0 / (4.0 * (A * C)));
	} else if (pow(B, 2.0) <= 5e+298) {
		tmp = (sqrt(2.0) / -B) * sqrt((F * (A - sqrt(fma(A, A, (B * B))))));
	} else {
		tmp = sqrt((A * F)) * (-2.0 / B);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if ((B ^ 2.0) <= 4e-121)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(fma(A, Float64(C * -4.0), Float64(B * B)) * Float64(2.0 * F)))) * Float64(1.0 / Float64(4.0 * Float64(A * C))));
	elseif ((B ^ 2.0) <= 5e+298)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B)) * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B * B)))))));
	else
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-2.0 / B));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 4e-121], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e+298], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B)), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.9999999999999999e-121

   1. Initial program 15.5%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 20.2

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

   5. Simplified 20.2%

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

   7. Applied egg-rr 20.2%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 20.1

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

   10. Simplified 20.1%

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

   if 3.9999999999999999e-121 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e298

   1. Initial program 35.6%

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

   4. Applied egg-rr 1.1%

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

   5. Taylor expanded in C around 0

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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 20.3

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

   7. Simplified 20.3%

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

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

   1. Initial program 1.7%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B}} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{2}}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}}\right) \]

   7. Simplified 0.1%

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

   8. Taylor expanded in A around -inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{A \cdot F} \cdot \frac{-1 \cdot \color{blue}{2}}{B} \]

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 5.6

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

   10. Simplified 5.6%

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

4. Final simplification 16.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)} \cdot \frac{1}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 19.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-90}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= (pow B 2.0) 1e-90)
   (* -2.0 (sqrt (/ (* A F) (fma B B (* -4.0 (* A C))))))
   (if (<= (pow B 2.0) 5e+298)
     (* (/ (sqrt 2.0) (- B)) (sqrt (* F (- A (sqrt (fma A A (* B B)))))))
     (* (sqrt (* A F)) (/ -2.0 B)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (pow(B, 2.0) <= 1e-90) {
		tmp = -2.0 * sqrt(((A * F) / fma(B, B, (-4.0 * (A * C)))));
	} else if (pow(B, 2.0) <= 5e+298) {
		tmp = (sqrt(2.0) / -B) * sqrt((F * (A - sqrt(fma(A, A, (B * B))))));
	} else {
		tmp = sqrt((A * F)) * (-2.0 / B);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if ((B ^ 2.0) <= 1e-90)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / fma(B, B, Float64(-4.0 * Float64(A * C))))));
	elseif ((B ^ 2.0) <= 5e+298)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B)) * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B * B)))))));
	else
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-2.0 / B));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e-90], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e+298], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B)), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999995e-91

   1. Initial program 17.4%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 20.9

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

   5. Simplified 20.9%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 14.8

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

   8. Simplified 14.8%

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

   if 9.99999999999999995e-91 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e298

   1. Initial program 35.0%

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

   4. Applied egg-rr 1.2%

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

   5. Taylor expanded in C around 0

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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 19.6

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

   7. Simplified 19.6%

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

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

   1. Initial program 1.7%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B}} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{2}}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}}\right) \]

   7. Simplified 0.1%

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

   8. Taylor expanded in A around -inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{A \cdot F} \cdot \frac{-1 \cdot \color{blue}{2}}{B} \]

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 5.6

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

   10. Simplified 5.6%

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

4. Final simplification 14.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-90}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 19.8% accurate, 10.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.1%

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

3. Taylor expanded in C around inf

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower-+.f64 13.3

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

5. Simplified 13.3%

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

6. Taylor expanded in F around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

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

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

   6. unpow2 N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 10.6

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

8. Simplified 10.6%

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

Alternative 12: 5.4% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.1%

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

4. Applied egg-rr 14.0%

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

5. Taylor expanded in C around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B}} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{2}}}{B} \cdot \sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{\frac{F \cdot \left({A}^{3} - \sqrt{{\left({A}^{2} + {B}^{2}\right)}^{3}}\right)}{2 \cdot {A}^{2} + \left(A \cdot \sqrt{{A}^{2} + {B}^{2}} + {B}^{2}\right)}}}\right) \]

7. Simplified 4.1%

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

8. Taylor expanded in A around -inf

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. rem-square-sqrt N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{-1 \cdot \color{blue}{2}}{B} \]

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 3.5

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

10. Simplified 3.5%

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

Alternative 13: 1.9% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.1%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 1.9

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

5. Simplified 1.9%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. mul-1-neg N/A

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

   4. remove-double-neg N/A

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

   5. lift-sqrt.f64 N/A

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

   6. sqrt-unprod N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-*.f64 1.9

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

7. Applied egg-rr 1.9%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 1.9

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

9. Applied egg-rr 1.9%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))