ABCF->ab-angle b

Percentage Accurate: 19.3% → 39.3%

Time: 14.5s

Alternatives: 13

Speedup: 6.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 39.3% accurate, 0.4× speedup

Math

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

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

C

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

Julia

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

   1. Initial program 2.8%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 14.9

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

   5. Simplified 14.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

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

   7. Applied egg-rr 25.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.99999999999999965e-214

   1. Initial program 96.0%

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

   4. Applied egg-rr 97.6%

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

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

   1. Initial program 8.4%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 14.0

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

   5. Simplified 14.0%

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

4. Final simplification 33.1%

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

Alternative 2: 37.8% accurate, 0.4× speedup

Math

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

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

C

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

Julia

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

   1. Initial program 2.8%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 14.9

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

   5. Simplified 14.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

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

   7. Applied egg-rr 25.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.99999999999999965e-214

   1. Initial program 96.0%

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

   4. Applied egg-rr 97.6%

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

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

   1. Initial program 8.4%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 12.7

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

   5. Simplified 12.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. count-2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied egg-rr 13.3%

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

   8. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 13.3

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

   10. Simplified 13.3%

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

4. Final simplification 32.6%

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

Alternative 3: 38.1% accurate, 0.4× speedup

Math

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

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

C

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

Julia

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

   1. Initial program 2.8%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 14.9

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

   5. Simplified 14.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

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

   7. Applied egg-rr 25.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.99999999999999965e-214

   1. Initial program 96.0%

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

   4. Applied egg-rr 96.1%

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

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

   1. Initial program 8.4%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 12.7

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

   5. Simplified 12.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. count-2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied egg-rr 13.3%

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

   8. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 13.3

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

   10. Simplified 13.3%

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

4. Final simplification 32.3%

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

Alternative 4: 38.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 2.8%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 14.9

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

   5. Simplified 14.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. count-2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied egg-rr 15.1%

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

   8. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 15.1

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

   10. Simplified 15.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. sqrt-prod N/A

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

       6. pow1/2 N/A

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

       7. lift-fma.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. lift-*.f64 N/A

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

       12. lift-fma.f64 N/A

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

       13. pow1/2 N/A

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

       14. associate-*r* N/A

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

       15. lower-*.f64 N/A

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

   12. Applied egg-rr 25.7%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.99999999999999965e-214

   1. Initial program 96.0%

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

   4. Applied egg-rr 96.1%

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

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

   1. Initial program 8.4%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 12.7

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

   5. Simplified 12.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. count-2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied egg-rr 13.3%

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

   8. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 13.3

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

   10. Simplified 13.3%

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

4. Final simplification 32.3%

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

Alternative 5: 37.9% accurate, 0.4× speedup

Math

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

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

Julia

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

   1. Initial program 2.8%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 14.9

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

   5. Simplified 14.9%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 19.0

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

   8. Simplified 19.0%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.99999999999999965e-214

   1. Initial program 96.0%

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

   4. Applied egg-rr 96.1%

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

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

   1. Initial program 8.4%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 12.7

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

   5. Simplified 12.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. count-2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied egg-rr 13.3%

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

   8. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 13.3

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

   10. Simplified 13.3%

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

4. Final simplification 31.1%

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

Alternative 6: 37.5% accurate, 0.4× speedup

Math

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

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* (* A C) -4.0)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B 2.0)))))
   (if (<= t_2 (- INFINITY))
     (* -2.0 (sqrt (* A (/ F (fma -4.0 (* A C) (* B B))))))
     (if (<= t_2 -4e-214)
       (/
        (sqrt
         (*
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B))))
          (* (fma (* A C) -4.0 (* B B)) (* 2.0 F))))
        (fma B (- B) (* 4.0 (* A C))))
       (* (sqrt (* (* F t_0) (* 2.0 (+ A A)))) (/ -1.0 t_0))))))

C

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

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(Float64(A * C) * -4.0))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B ^ 2.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * sqrt(Float64(A * Float64(F / fma(-4.0, Float64(A * C), Float64(B * B))))));
	elseif (t_2 <= -4e-214)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))) * Float64(fma(Float64(A * C), -4.0, Float64(B * B)) * Float64(2.0 * F)))) / fma(B, Float64(-B), Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + A)))) * Float64(-1.0 / t_0));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[Sqrt[N[(A * N[(F / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-214], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B * (-B) + N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 2.8%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 14.9

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

   5. Simplified 14.9%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 19.0

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

   8. Simplified 19.0%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.99999999999999965e-214

   1. Initial program 96.0%

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

   4. Applied egg-rr 96.1%

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

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

   1. Initial program 8.4%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 12.7

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

   5. Simplified 12.7%

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

   7. Applied egg-rr 12.6%

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

4. Final simplification 30.7%

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

Alternative 7: 20.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+267}:\\ \;\;\;\;-2 \cdot \sqrt{A \cdot \frac{F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= (pow B 2.0) 2e+267)
   (* -2.0 (sqrt (* A (/ F (fma -4.0 (* A C) (* B B))))))
   (* -2.0 (* (sqrt (* A F)) (/ 1.0 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) <= 2e+267) {
		tmp = -2.0 * sqrt((A * (F / fma(-4.0, (A * C), (B * B)))));
	} else {
		tmp = -2.0 * (sqrt((A * F)) * (1.0 / 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) <= 2e+267)
		tmp = Float64(-2.0 * sqrt(Float64(A * Float64(F / fma(-4.0, Float64(A * C), Float64(B * B))))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / 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], 2e+267], N[(-2.0 * N[Sqrt[N[(A * N[(F / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e267

   1. Initial program 32.1%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 24.2

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

   5. Simplified 24.2%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 16.5

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

   8. Simplified 16.5%

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

   if 1.9999999999999999e267 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 4.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 0.5

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

   5. Simplified 0.5%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 4.4

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

   8. Simplified 4.4%

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

Alternative 8: 30.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\\ \mathbf{if}\;B \leq 4.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{-t\_0}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* (* A C) -4.0))))
   (if (<= B 4.8e+74)
     (/ (sqrt (* (* F t_0) (* 2.0 (+ A A)))) (- t_0))
     (if (<= B 1.35e+154)
       (/ (* (sqrt 2.0) (sqrt (* F (- A (sqrt (fma A A (* B B))))))) (- B))
       (* -2.0 (* (sqrt (* A F)) (/ 1.0 B)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, ((A * C) * -4.0));
	double tmp;
	if (B <= 4.8e+74) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + A)))) / -t_0;
	} else if (B <= 1.35e+154) {
		tmp = (sqrt(2.0) * sqrt((F * (A - sqrt(fma(A, A, (B * B))))))) / -B;
	} else {
		tmp = -2.0 * (sqrt((A * F)) * (1.0 / B));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(Float64(A * C) * -4.0))
	tmp = 0.0
	if (B <= 4.8e+74)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + A)))) / Float64(-t_0));
	elseif (B <= 1.35e+154)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B * B))))))) / Float64(-B));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / 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[(B * B + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 4.8e+74], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B, 1.35e+154], 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[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.80000000000000017e74

   1. Initial program 27.6%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 22.1

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

   5. Simplified 22.1%

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

   7. Applied egg-rr 22.1%

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

   if 4.80000000000000017e74 < B < 1.35000000000000003e154

   1. Initial program 36.1%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Simplified 57.5%

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

   if 1.35000000000000003e154 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 6.4

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

   8. Simplified 6.4%

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

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 19.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1.95 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot \sqrt{A \cdot \frac{F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}}{-B}\\ \end{array} \end{array} \]

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 (<= A -1.95e-45)
   (* -2.0 (sqrt (* A (/ F (fma -4.0 (* A C) (* B B))))))
   (/ (* (sqrt 2.0) (sqrt (* F (- A (sqrt (fma A A (* B B))))))) (- B))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -1.95e-45) {
		tmp = -2.0 * sqrt((A * (F / fma(-4.0, (A * C), (B * B)))));
	} else {
		tmp = (sqrt(2.0) * sqrt((F * (A - sqrt(fma(A, A, (B * B))))))) / -B;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (A <= -1.95e-45)
		tmp = Float64(-2.0 * sqrt(Float64(A * Float64(F / fma(-4.0, Float64(A * C), Float64(B * B))))));
	else
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B * B))))))) / Float64(-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[A, -1.95e-45], N[(-2.0 * N[Sqrt[N[(A * N[(F / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if A < -1.95e-45

   1. Initial program 34.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 41.1

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

   5. Simplified 41.1%

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

   6. Taylor expanded in F around 0

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

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.9

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

   8. Simplified 33.9%

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

   if -1.95e-45 < A

   1. Initial program 21.1%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Simplified 10.7%

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

Alternative 10: 19.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1.95 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot \sqrt{A \cdot \frac{F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \end{array} \]

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 (<= A -1.95e-45)
   (* -2.0 (sqrt (* A (/ F (fma -4.0 (* A C) (* B B))))))
   (* (sqrt (* F (- A (sqrt (fma A A (* B B)))))) (/ (sqrt 2.0) (- B)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -1.95e-45) {
		tmp = -2.0 * sqrt((A * (F / fma(-4.0, (A * C), (B * B)))));
	} else {
		tmp = sqrt((F * (A - sqrt(fma(A, A, (B * B)))))) * (sqrt(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 (A <= -1.95e-45)
		tmp = Float64(-2.0 * sqrt(Float64(A * Float64(F / fma(-4.0, Float64(A * C), Float64(B * B))))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B * B)))))) * Float64(sqrt(2.0) / Float64(-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[A, -1.95e-45], N[(-2.0 * N[Sqrt[N[(A * N[(F / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.95e-45

   1. Initial program 34.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 41.1

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

   5. Simplified 41.1%

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

   6. Taylor expanded in F around 0

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

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.9

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

   8. Simplified 33.9%

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

   if -1.95e-45 < A

   1. Initial program 21.1%

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

   4. Applied egg-rr 13.2%

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

   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 10.7

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

   7. Simplified 10.7%

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

4. Final simplification 17.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.95 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot \sqrt{A \cdot \frac{F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 5.2% accurate, 13.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ -2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right) \end{array} \]

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)) (/ 1.0 B))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return -2.0 * (sqrt((A * F)) * (1.0 / 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 = (-2.0d0) * (sqrt((a * f)) * (1.0d0 / b))
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return -2.0 * (Math.sqrt((A * F)) * (1.0 / B));
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return -2.0 * (math.sqrt((A * F)) * (1.0 / B))

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(-2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = -2.0 * (sqrt((A * F)) * (1.0 / 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[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.2%

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

3. Taylor expanded in C around inf

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower-+.f64 18.2

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

5. Simplified 18.2%

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

6. Taylor expanded in B around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 4.0

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

8. Simplified 4.0%

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

Alternative 12: 2.0% accurate, 14.9× speedup

Math

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

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 (/ 2.0 (/ B F))))

C

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

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((2.0d0 / (b / f)))
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((2.0 / (B / F)));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 25.2%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 1.8

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

5. Simplified 1.8%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. mul-1-neg N/A

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

   4. remove-double-neg N/A

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

   5. lift-sqrt.f64 N/A

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

   6. sqrt-unprod N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-*.f64 1.8

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

7. Applied egg-rr 1.8%

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

   1. clear-num N/A

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

   2. associate-*l/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 1.9

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

9. Applied egg-rr 1.9%

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

Alternative 13: 2.0% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.2%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 1.8

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

5. Simplified 1.8%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. mul-1-neg N/A

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

   4. remove-double-neg N/A

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

   5. lift-sqrt.f64 N/A

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

   6. sqrt-unprod N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-*.f64 1.8

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

7. Applied egg-rr 1.8%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 1.8

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

9. Applied egg-rr 1.8%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))