ABCF->ab-angle b

Percentage Accurate: 19.2% → 39.8%

Time: 32.9s

Alternatives: 11

Speedup: 2.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.2% 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: 39.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(2 \cdot F\right) \cdot t_0\\ t_2 := A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\\ \mathbf{if}\;C \leq 4.4 \cdot 10^{-292}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot t_1}}{t_0}\\ \mathbf{elif}\;C \leq 420000:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \cdot \left(-\sqrt{F \cdot t_2}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \left(A + {B}^{2} \cdot \frac{-0.5}{C}\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C 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 (* (* 2.0 F) t_0))
        (t_2 (+ A (- C (hypot B (- A C))))))
   (if (<= C 4.4e-292)
     (/ (- (sqrt (* t_2 t_1))) t_0)
     (if (<= C 420000.0)
       (/
        (* (sqrt (* 2.0 (fma A (* C -4.0) (pow B 2.0)))) (- (sqrt (* F t_2))))
        (- (pow B 2.0) (* 4.0 (* A C))))
       (/ (- (sqrt (* t_1 (+ A (+ A (* (pow B 2.0) (/ -0.5 C))))))) t_0)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (A * (C * -4.0)));
	double t_1 = (2.0 * F) * t_0;
	double t_2 = A + (C - hypot(B, (A - C)));
	double tmp;
	if (C <= 4.4e-292) {
		tmp = -sqrt((t_2 * t_1)) / t_0;
	} else if (C <= 420000.0) {
		tmp = (sqrt((2.0 * fma(A, (C * -4.0), pow(B, 2.0)))) * -sqrt((F * t_2))) / (pow(B, 2.0) - (4.0 * (A * C)));
	} else {
		tmp = -sqrt((t_1 * (A + (A + (pow(B, 2.0) * (-0.5 / C)))))) / t_0;
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(2.0 * F) * t_0)
	t_2 = Float64(A + Float64(C - hypot(B, Float64(A - C))))
	tmp = 0.0
	if (C <= 4.4e-292)
		tmp = Float64(Float64(-sqrt(Float64(t_2 * t_1))) / t_0);
	elseif (C <= 420000.0)
		tmp = Float64(Float64(sqrt(Float64(2.0 * fma(A, Float64(C * -4.0), (B ^ 2.0)))) * Float64(-sqrt(Float64(F * t_2)))) / Float64((B ^ 2.0) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(A + Float64(A + Float64((B ^ 2.0) * Float64(-0.5 / C))))))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(A + N[(C - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, 4.4e-292], N[((-N[Sqrt[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[C, 420000.0], N[(N[(N[Sqrt[N[(2.0 * N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(F * t$95$2), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[Power[B, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(t$95$1 * N[(A + N[(A + N[(N[Power[B, 2.0], $MachinePrecision] * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if C < 4.40000000000000023e-292

   1. Initial program 25.4%

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

   3. Simplified 37.8%

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

   if 4.40000000000000023e-292 < C < 4.2e5

   1. Initial program 29.6%

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

   3. Simplified 29.6%

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

      1. associate--l+ 29.6%

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

      2. unpow2 29.6%

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

      3. unpow2 29.6%

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

      4. hypot-udef 31.5%

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

      5. add-cbrt-cube 26.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      6. *-commutative 26.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      7. pow1 26.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}} \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 26.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      9. pow1 26.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      10. metadata-eval 26.9%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      11. pow-sqr 26.9%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(2 \cdot \frac{2}{2}\right)}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      12. metadata-eval 26.9%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(2 \cdot \color{blue}{1}\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      13. metadata-eval 26.9%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 26.9%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-+r- 26.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      2. +-commutative 26.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      3. associate-+r- 26.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      4. associate-+r- 26.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      5. +-commutative 26.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      6. associate-+r- 27.8%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 27.8%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
   8. Step-by-step derivation

      1. associate-*l* 27.6%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      2. sqrt-prod 30.3%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 30.3%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot 2}} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      4. cancel-sign-sub-inv 30.3%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left({B}^{2} + \left(-4\right) \cdot \left(A \cdot C\right)\right)} \cdot 2} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      5. metadata-eval 30.3%

         \[\leadsto \frac{-\sqrt{\left({B}^{2} + \color{blue}{-4} \cdot \left(A \cdot C\right)\right) \cdot 2} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      6. *-commutative 30.3%

         \[\leadsto \frac{-\sqrt{\left({B}^{2} + \color{blue}{\left(A \cdot C\right) \cdot -4}\right) \cdot 2} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      7. associate-*r* 30.3%

         \[\leadsto \frac{-\sqrt{\left({B}^{2} + \color{blue}{A \cdot \left(C \cdot -4\right)}\right) \cdot 2} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      8. +-commutative 30.3%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(A \cdot \left(C \cdot -4\right) + {B}^{2}\right)} \cdot 2} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      9. fma-udef 30.3%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \cdot 2} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   9. Applied egg-rr 36.4%

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

       1. *-commutative 36.4%

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

       2. associate-+r- 36.5%

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

       3. +-commutative 36.5%

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

       4. associate--l+ 36.5%

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

   11. Simplified 36.5%

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

   if 4.2e5 < C

   1. Initial program 5.7%

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

   3. Simplified 10.3%

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

   4. Taylor expanded in C around inf 33.1%

      \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({A}^{2} + {B}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 33.1%

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

      2. +-commutative 33.1%

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

      3. mul-1-neg 33.1%

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

   6. Simplified 33.1%

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

   7. Taylor expanded in B around 0 33.1%

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

      1. associate-*r/ 33.1%

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

      2. +-rgt-identity 33.1%

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + \left(A + \frac{-0.5 \cdot \color{blue}{\left({B}^{2} + 0\right)}}{C}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. associate-*l/ 33.1%

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + \left(A + \color{blue}{\frac{-0.5}{C} \cdot \left({B}^{2} + 0\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 33.1%

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + \left(A + \color{blue}{\left({B}^{2} + 0\right) \cdot \frac{-0.5}{C}}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. +-rgt-identity 33.1%

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

   9. Simplified 33.1%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 4.4 \cdot 10^{-292}:\\ \;\;\;\;\frac{-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;C \leq 420000:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \cdot \left(-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(A + \left(A + {B}^{2} \cdot \frac{-0.5}{C}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 2: 39.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)\\ t_1 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_1} \leq -5 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_0} \cdot \left(-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left({B}^{2} \cdot \frac{-0.5}{C} + 2 \cdot A\right)} \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (pow B 2.0)))
        (t_1 (- (pow B 2.0) (* (* 4.0 A) C))))
   (if (<=
        (/
         (-
          (sqrt
           (*
            (* 2.0 (* t_1 F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
         t_1)
        -5e-197)
     (/
      (* (sqrt (* 2.0 t_0)) (- (sqrt (* F (+ A (- C (hypot B (- A C))))))))
      (- (pow B 2.0) (* 4.0 (* A C))))
     (*
      (sqrt (* (* t_0 (* 2.0 F)) (+ (* (pow B 2.0) (/ -0.5 C)) (* 2.0 A))))
      (/ -1.0 t_0)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = fma(A, (C * -4.0), pow(B, 2.0));
	double t_1 = pow(B, 2.0) - ((4.0 * A) * C);
	double tmp;
	if ((-sqrt(((2.0 * (t_1 * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_1) <= -5e-197) {
		tmp = (sqrt((2.0 * t_0)) * -sqrt((F * (A + (C - hypot(B, (A - C))))))) / (pow(B, 2.0) - (4.0 * (A * C)));
	} else {
		tmp = sqrt(((t_0 * (2.0 * F)) * ((pow(B, 2.0) * (-0.5 / C)) + (2.0 * A)))) * (-1.0 / t_0);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = fma(A, Float64(C * -4.0), (B ^ 2.0))
	t_1 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_1) <= -5e-197)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * Float64(-sqrt(Float64(F * Float64(A + Float64(C - hypot(B, Float64(A - C)))))))) / Float64((B ^ 2.0) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(Float64((B ^ 2.0) * Float64(-0.5 / C)) + Float64(2.0 * A)))) * Float64(-1.0 / t_0));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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]) / t$95$1), $MachinePrecision], -5e-197], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(F * N[(A + N[(C - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[Power[B, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[B, 2.0], $MachinePrecision] * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision] + N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -5.0000000000000002e-197

   1. Initial program 50.6%

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

   3. Simplified 50.6%

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

      1. associate--l+ 50.6%

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

      2. unpow2 50.6%

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

      3. unpow2 50.6%

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

      4. hypot-udef 61.3%

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

      5. add-cbrt-cube 41.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      6. *-commutative 41.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      7. pow1 41.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}} \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 41.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      9. pow1 41.9%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      10. metadata-eval 41.9%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      11. pow-sqr 41.9%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(2 \cdot \frac{2}{2}\right)}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      12. metadata-eval 41.9%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(2 \cdot \color{blue}{1}\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      13. metadata-eval 41.9%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 41.9%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-+r- 41.7%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      2. +-commutative 41.7%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      3. associate-+r- 41.7%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      4. associate-+r- 41.7%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      5. +-commutative 41.7%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      6. associate-+r- 42.1%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 42.1%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
   8. Step-by-step derivation

      1. associate-*l* 40.1%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      2. sqrt-prod 44.7%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 44.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot 2}} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      4. cancel-sign-sub-inv 44.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left({B}^{2} + \left(-4\right) \cdot \left(A \cdot C\right)\right)} \cdot 2} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      5. metadata-eval 44.7%

         \[\leadsto \frac{-\sqrt{\left({B}^{2} + \color{blue}{-4} \cdot \left(A \cdot C\right)\right) \cdot 2} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      6. *-commutative 44.7%

         \[\leadsto \frac{-\sqrt{\left({B}^{2} + \color{blue}{\left(A \cdot C\right) \cdot -4}\right) \cdot 2} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      7. associate-*r* 44.7%

         \[\leadsto \frac{-\sqrt{\left({B}^{2} + \color{blue}{A \cdot \left(C \cdot -4\right)}\right) \cdot 2} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      8. +-commutative 44.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(A \cdot \left(C \cdot -4\right) + {B}^{2}\right)} \cdot 2} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      9. fma-udef 44.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \cdot 2} \cdot \sqrt{F \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   9. Applied egg-rr 74.3%

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

       1. *-commutative 74.3%

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

       2. associate-+r- 73.1%

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

       3. +-commutative 73.1%

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

       4. associate--l+ 73.8%

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

   11. Simplified 73.8%

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

   if -5.0000000000000002e-197 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))

   1. Initial program 6.5%

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

   3. Simplified 12.7%

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

   4. Taylor expanded in C around inf 11.5%

      \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({A}^{2} + {B}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 11.5%

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

      2. +-commutative 11.5%

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

      3. mul-1-neg 11.5%

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

   6. Simplified 11.5%

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

   7. Taylor expanded in B around 0 14.1%

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

      1. associate-*r/ 14.1%

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

      2. +-rgt-identity 14.1%

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + \left(A + \frac{-0.5 \cdot \color{blue}{\left({B}^{2} + 0\right)}}{C}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. associate-*l/ 14.1%

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + \left(A + \color{blue}{\frac{-0.5}{C} \cdot \left({B}^{2} + 0\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 14.1%

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + \left(A + \color{blue}{\left({B}^{2} + 0\right) \cdot \frac{-0.5}{C}}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. +-rgt-identity 14.1%

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

   9. Simplified 14.1%

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

       1. div-inv 14.1%

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

       2. *-commutative 14.1%

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

       3. fma-udef 14.1%

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

       4. unpow2 14.1%

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

       5. +-commutative 14.1%

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

       6. fma-udef 14.1%

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

       7. fma-udef 14.1%

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

       8. unpow2 14.1%

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

       9. +-commutative 14.1%

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

       10. fma-udef 14.1%

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

       11. +-commutative 14.1%

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

       12. +-commutative 14.1%

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

   11. Applied egg-rr 14.1%

       \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \cdot \left(-\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right) \cdot \left(F \cdot 2\right)\right) \cdot \left({B}^{2} \cdot \frac{-0.5}{C} + A \cdot 2\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -5 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \cdot \left(-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right) \cdot \left(2 \cdot F\right)\right) \cdot \left({B}^{2} \cdot \frac{-0.5}{C} + 2 \cdot A\right)} \cdot \frac{-1}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \end{array} \]

Alternative 3: 26.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-21}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(A \cdot \left(F \cdot \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 3 \cdot 10^{+262}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C - B\right)} \cdot \left(B \cdot \left(-\sqrt{2}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= (pow B 2.0) 1e-21)
   (/
    (- (sqrt (* 4.0 (* A (* F (+ (pow B 2.0) (* -4.0 (* A C))))))))
    (fma B B (* A (* C -4.0))))
   (if (<= (pow B 2.0) 3e+262)
     (/
      (* (sqrt (* F (- C B))) (* B (- (sqrt 2.0))))
      (- (pow B 2.0) (* 4.0 (* A C))))
     (* -2.0 (/ (sqrt (* A F)) B)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (pow(B, 2.0) <= 1e-21) {
		tmp = -sqrt((4.0 * (A * (F * (pow(B, 2.0) + (-4.0 * (A * C))))))) / fma(B, B, (A * (C * -4.0)));
	} else if (pow(B, 2.0) <= 3e+262) {
		tmp = (sqrt((F * (C - B))) * (B * -sqrt(2.0))) / (pow(B, 2.0) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * (sqrt((A * F)) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if ((B ^ 2.0) <= 1e-21)
		tmp = Float64(Float64(-sqrt(Float64(4.0 * Float64(A * Float64(F * Float64((B ^ 2.0) + Float64(-4.0 * Float64(A * C)))))))) / fma(B, B, Float64(A * Float64(C * -4.0))));
	elseif ((B ^ 2.0) <= 3e+262)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(C - B))) * Float64(B * Float64(-sqrt(2.0)))) / Float64((B ^ 2.0) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e-21], N[((-N[Sqrt[N[(4.0 * N[(A * N[(F * N[(N[Power[B, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 3e+262], N[(N[(N[Sqrt[N[(F * N[(C - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(B * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / N[(N[Power[B, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 9.99999999999999908e-22

   1. Initial program 23.6%

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

   3. Simplified 38.2%

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

   4. Taylor expanded in C around inf 24.9%

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

   5. Taylor expanded in F around 0 24.8%

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

   if 9.99999999999999908e-22 < (pow.f64 B 2) < 3e262

   1. Initial program 42.5%

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

   3. Simplified 42.5%

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

      1. associate--l+ 42.7%

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

      2. unpow2 42.7%

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

      3. unpow2 42.7%

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

      4. hypot-udef 43.3%

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

      5. add-cbrt-cube 34.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      6. *-commutative 34.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      7. pow1 34.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}} \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 34.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      9. pow1 34.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      10. metadata-eval 34.5%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      11. pow-sqr 34.5%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(2 \cdot \frac{2}{2}\right)}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      12. metadata-eval 34.5%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(2 \cdot \color{blue}{1}\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      13. metadata-eval 34.5%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 34.5%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-+r- 34.3%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      2. +-commutative 34.3%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      3. associate-+r- 34.3%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      4. associate-+r- 34.3%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      5. +-commutative 34.3%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      6. associate-+r- 34.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 34.5%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around 0 26.9%

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

   9. Taylor expanded in B around inf 26.4%

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

   if 3e262 < (pow.f64 B 2)

   1. Initial program 0.2%

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

   3. Simplified 1.7%

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

   4. Taylor expanded in C around inf 1.7%

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

   5. Taylor expanded in B around inf 4.8%

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

      1. *-commutative 4.8%

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

   7. Simplified 4.8%

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

   8. Taylor expanded in B around 0 4.8%

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

      1. associate-*r/ 4.9%

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

      2. *-rgt-identity 4.9%

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

   10. Simplified 4.9%

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

4. Final simplification 19.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-21}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(A \cdot \left(F \cdot \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 3 \cdot 10^{+262}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C - B\right)} \cdot \left(B \cdot \left(-\sqrt{2}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \]

Alternative 4: 36.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(2 \cdot F\right) \cdot t_0\\ \mathbf{if}\;C \leq 25000:\\ \;\;\;\;\frac{-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot t_1}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \left(A + {B}^{2} \cdot \frac{-0.5}{C}\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C 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 (* (* 2.0 F) t_0)))
   (if (<= C 25000.0)
     (/ (- (sqrt (* (+ A (- C (hypot B (- A C)))) t_1))) t_0)
     (/ (- (sqrt (* t_1 (+ A (+ A (* (pow B 2.0) (/ -0.5 C))))))) t_0))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (A * (C * -4.0)));
	double t_1 = (2.0 * F) * t_0;
	double tmp;
	if (C <= 25000.0) {
		tmp = -sqrt(((A + (C - hypot(B, (A - C)))) * t_1)) / t_0;
	} else {
		tmp = -sqrt((t_1 * (A + (A + (pow(B, 2.0) * (-0.5 / C)))))) / t_0;
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(2.0 * F) * t_0)
	tmp = 0.0
	if (C <= 25000.0)
		tmp = Float64(Float64(-sqrt(Float64(Float64(A + Float64(C - hypot(B, Float64(A - C)))) * t_1))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(A + Float64(A + Float64((B ^ 2.0) * Float64(-0.5 / C))))))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[C, 25000.0], N[((-N[Sqrt[N[(N[(A + N[(C - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(t$95$1 * N[(A + N[(A + N[(N[Power[B, 2.0], $MachinePrecision] * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if C < 25000

   1. Initial program 27.0%

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

   3. Simplified 35.9%

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

   if 25000 < C

   1. Initial program 5.7%

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

   3. Simplified 10.2%

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

   4. Taylor expanded in C around inf 32.7%

      \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({A}^{2} + {B}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 32.7%

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

      2. +-commutative 32.7%

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

      3. mul-1-neg 32.7%

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

   6. Simplified 32.7%

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

   7. Taylor expanded in B around 0 32.8%

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

      1. associate-*r/ 32.8%

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

      2. +-rgt-identity 32.8%

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + \left(A + \frac{-0.5 \cdot \color{blue}{\left({B}^{2} + 0\right)}}{C}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. associate-*l/ 32.8%

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + \left(A + \color{blue}{\frac{-0.5}{C} \cdot \left({B}^{2} + 0\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. *-commutative 32.8%

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + \left(A + \color{blue}{\left({B}^{2} + 0\right) \cdot \frac{-0.5}{C}}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. +-rgt-identity 32.8%

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

   9. Simplified 32.8%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 25000:\\ \;\;\;\;\frac{-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(A + \left(A + {B}^{2} \cdot \frac{-0.5}{C}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 5: 28.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot t_0\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(B \cdot \sqrt{2}\right) \cdot \left(-\sqrt{F \cdot \left(C - \left(B + 0.5 \cdot \frac{{C}^{2}}{B}\right)\right)}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C 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 4.2e-15)
     (/ (- (sqrt (* (* (* 2.0 F) t_0) (+ A A)))) t_0)
     (if (<= B 1.9e+131)
       (/
        (*
         (* B (sqrt 2.0))
         (- (sqrt (* F (- C (+ B (* 0.5 (/ (pow C 2.0) B))))))))
        (- (pow B 2.0) (* 4.0 (* A C))))
       (* -2.0 (/ (sqrt (* A F)) B))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (B <= 4.2e-15) {
		tmp = -sqrt((((2.0 * F) * t_0) * (A + A))) / t_0;
	} else if (B <= 1.9e+131) {
		tmp = ((B * sqrt(2.0)) * -sqrt((F * (C - (B + (0.5 * (pow(C, 2.0) / B))))))) / (pow(B, 2.0) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * (sqrt((A * F)) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B <= 4.2e-15)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * F) * t_0) * Float64(A + A)))) / t_0);
	elseif (B <= 1.9e+131)
		tmp = Float64(Float64(Float64(B * sqrt(2.0)) * Float64(-sqrt(Float64(F * Float64(C - Float64(B + Float64(0.5 * Float64((C ^ 2.0) / B)))))))) / Float64((B ^ 2.0) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 4.2e-15], N[((-N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 1.9e+131], N[(N[(N[(B * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(F * N[(C - N[(B + N[(0.5 * N[(N[Power[C, 2.0], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[Power[B, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.19999999999999962e-15

   1. Initial program 21.6%

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

   3. Simplified 31.8%

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

   4. Taylor expanded in C around inf 17.4%

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

   if 4.19999999999999962e-15 < B < 1.9000000000000002e131

   1. Initial program 38.3%

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

   3. Simplified 38.3%

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

      1. associate--l+ 38.6%

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

      2. unpow2 38.6%

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

      3. unpow2 38.6%

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

      4. hypot-udef 39.6%

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

      5. add-cbrt-cube 30.7%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      6. *-commutative 30.7%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      7. pow1 30.7%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}} \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 30.7%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      9. pow1 30.7%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      10. metadata-eval 30.7%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      11. pow-sqr 30.7%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(2 \cdot \frac{2}{2}\right)}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      12. metadata-eval 30.7%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(2 \cdot \color{blue}{1}\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      13. metadata-eval 30.7%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 30.7%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-+r- 30.4%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      2. +-commutative 30.4%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      3. associate-+r- 30.4%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      4. associate-+r- 30.4%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      5. +-commutative 30.4%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      6. associate-+r- 30.7%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 30.7%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around 0 43.3%

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

   9. Taylor expanded in B around inf 43.5%

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

   if 1.9000000000000002e131 < B

   1. Initial program 0.2%

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

   3. Simplified 0.3%

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

   4. Taylor expanded in C around inf 0.2%

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

   5. Taylor expanded in B around inf 8.4%

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

      1. *-commutative 8.4%

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

   7. Simplified 8.4%

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

   8. Taylor expanded in B around 0 8.4%

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

      1. associate-*r/ 8.4%

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

      2. *-rgt-identity 8.4%

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

   10. Simplified 8.4%

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

4. Final simplification 19.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(A + A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(B \cdot \sqrt{2}\right) \cdot \left(-\sqrt{F \cdot \left(C - \left(B + 0.5 \cdot \frac{{C}^{2}}{B}\right)\right)}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \]

Alternative 6: 24.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(2 \cdot \left(C \cdot \left(A \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= (pow B 2.0) 5e+115)
   (/
    (- (sqrt (* -8.0 (* A (* 2.0 (* C (* A F)))))))
    (fma A (* C -4.0) (pow B 2.0)))
   (* -2.0 (/ (sqrt (* A F)) B))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (pow(B, 2.0) <= 5e+115) {
		tmp = -sqrt((-8.0 * (A * (2.0 * (C * (A * F)))))) / fma(A, (C * -4.0), pow(B, 2.0));
	} else {
		tmp = -2.0 * (sqrt((A * F)) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if ((B ^ 2.0) <= 5e+115)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(2.0 * Float64(C * Float64(A * F))))))) / fma(A, Float64(C * -4.0), (B ^ 2.0)));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e+115], N[((-N[Sqrt[N[(-8.0 * N[(A * N[(2.0 * N[(C * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.00000000000000008e115

   1. Initial program 28.4%

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

   3. Simplified 35.1%

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

   4. Taylor expanded in C around inf 21.1%

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

      1. mul-1-neg 21.1%

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

   6. Simplified 21.1%

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

   7. Taylor expanded in C around 0 18.1%

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

      1. *-commutative 18.1%

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

      2. associate-*l* 21.1%

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

   9. Simplified 21.1%

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

   if 5.00000000000000008e115 < (pow.f64 B 2)

   1. Initial program 10.4%

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

   3. Simplified 11.5%

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

   4. Taylor expanded in C around inf 1.8%

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

   5. Taylor expanded in B around inf 4.1%

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

      1. *-commutative 4.1%

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

   7. Simplified 4.1%

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

   8. Taylor expanded in B around 0 4.1%

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

      1. associate-*r/ 4.1%

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

      2. *-rgt-identity 4.1%

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

   10. Simplified 4.1%

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

4. Final simplification 14.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(2 \cdot \left(C \cdot \left(A \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \]

Alternative 7: 28.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq 5.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot t_0\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C - B\right)} \cdot \left(B \cdot \left(-\sqrt{2}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C 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 5.7e-11)
     (/ (- (sqrt (* (* (* 2.0 F) t_0) (+ A A)))) t_0)
     (if (<= B 1.9e+131)
       (/
        (* (sqrt (* F (- C B))) (* B (- (sqrt 2.0))))
        (- (pow B 2.0) (* 4.0 (* A C))))
       (* -2.0 (/ (sqrt (* A F)) B))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (B <= 5.7e-11) {
		tmp = -sqrt((((2.0 * F) * t_0) * (A + A))) / t_0;
	} else if (B <= 1.9e+131) {
		tmp = (sqrt((F * (C - B))) * (B * -sqrt(2.0))) / (pow(B, 2.0) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * (sqrt((A * F)) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B <= 5.7e-11)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * F) * t_0) * Float64(A + A)))) / t_0);
	elseif (B <= 1.9e+131)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(C - B))) * Float64(B * Float64(-sqrt(2.0)))) / Float64((B ^ 2.0) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 5.7e-11], N[((-N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 1.9e+131], N[(N[(N[Sqrt[N[(F * N[(C - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(B * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / N[(N[Power[B, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 5.6999999999999997e-11

   1. Initial program 21.5%

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

   3. Simplified 31.7%

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

   4. Taylor expanded in C around inf 17.3%

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

   if 5.6999999999999997e-11 < B < 1.9000000000000002e131

   1. Initial program 39.4%

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

   3. Simplified 39.4%

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

      1. associate--l+ 39.6%

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

      2. unpow2 39.6%

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

      3. unpow2 39.6%

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

      4. hypot-udef 40.6%

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

      5. add-cbrt-cube 31.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      6. *-commutative 31.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      7. pow1 31.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}} \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 31.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      9. pow1 31.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      10. metadata-eval 31.5%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      11. pow-sqr 31.5%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(2 \cdot \frac{2}{2}\right)}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      12. metadata-eval 31.5%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(2 \cdot \color{blue}{1}\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      13. metadata-eval 31.5%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 31.5%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-+r- 31.2%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      2. +-commutative 31.2%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      3. associate-+r- 31.2%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      4. associate-+r- 31.2%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      5. +-commutative 31.2%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      6. associate-+r- 31.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 31.5%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around 0 44.6%

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

   9. Taylor expanded in B around inf 44.9%

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

   if 1.9000000000000002e131 < B

   1. Initial program 0.2%

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

   3. Simplified 0.3%

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

   4. Taylor expanded in C around inf 0.2%

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

   5. Taylor expanded in B around inf 8.4%

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

      1. *-commutative 8.4%

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

   7. Simplified 8.4%

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

   8. Taylor expanded in B around 0 8.4%

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

      1. associate-*r/ 8.4%

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

      2. *-rgt-identity 8.4%

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

   10. Simplified 8.4%

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

4. Final simplification 19.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(A + A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C - B\right)} \cdot \left(B \cdot \left(-\sqrt{2}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \]

Alternative 8: 26.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(2 \cdot \left(C \cdot \left(A \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C - B\right)} \cdot \left(B \cdot \left(-\sqrt{2}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 5e-11)
   (/
    (- (sqrt (* -8.0 (* A (* 2.0 (* C (* A F)))))))
    (fma A (* C -4.0) (pow B 2.0)))
   (if (<= B 1.9e+131)
     (/
      (* (sqrt (* F (- C B))) (* B (- (sqrt 2.0))))
      (- (pow B 2.0) (* 4.0 (* A C))))
     (* -2.0 (/ (sqrt (* A F)) B)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 5e-11) {
		tmp = -sqrt((-8.0 * (A * (2.0 * (C * (A * F)))))) / fma(A, (C * -4.0), pow(B, 2.0));
	} else if (B <= 1.9e+131) {
		tmp = (sqrt((F * (C - B))) * (B * -sqrt(2.0))) / (pow(B, 2.0) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * (sqrt((A * F)) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 5e-11)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(2.0 * Float64(C * Float64(A * F))))))) / fma(A, Float64(C * -4.0), (B ^ 2.0)));
	elseif (B <= 1.9e+131)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(C - B))) * Float64(B * Float64(-sqrt(2.0)))) / Float64((B ^ 2.0) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 5e-11], N[((-N[Sqrt[N[(-8.0 * N[(A * N[(2.0 * N[(C * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.9e+131], N[(N[(N[Sqrt[N[(F * N[(C - B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(B * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / N[(N[Power[B, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 5.00000000000000018e-11

   1. Initial program 21.5%

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

   3. Simplified 25.8%

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

   4. Taylor expanded in C around inf 17.0%

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

      1. mul-1-neg 17.0%

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

   6. Simplified 17.0%

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

   7. Taylor expanded in C around 0 14.6%

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

      1. *-commutative 14.6%

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

      2. associate-*l* 17.0%

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

   9. Simplified 17.0%

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

   if 5.00000000000000018e-11 < B < 1.9000000000000002e131

   1. Initial program 39.4%

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

   3. Simplified 39.4%

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

      1. associate--l+ 39.6%

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

      2. unpow2 39.6%

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

      3. unpow2 39.6%

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

      4. hypot-udef 40.6%

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

      5. add-cbrt-cube 31.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      6. *-commutative 31.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      7. pow1 31.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}} \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 31.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      9. pow1 31.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      10. metadata-eval 31.5%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left({\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      11. pow-sqr 31.5%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \color{blue}{{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(2 \cdot \frac{2}{2}\right)}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      12. metadata-eval 31.5%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(2 \cdot \color{blue}{1}\right)}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      13. metadata-eval 31.5%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\color{blue}{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 31.5%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-+r- 31.2%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      2. +-commutative 31.2%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right) \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      3. associate-+r- 31.2%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot {\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      4. associate-+r- 31.2%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      5. +-commutative 31.2%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

      6. associate-+r- 31.5%

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}^{2}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 31.5%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot {\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{2}}}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around 0 44.6%

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

   9. Taylor expanded in B around inf 44.9%

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

   if 1.9000000000000002e131 < B

   1. Initial program 0.2%

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

   3. Simplified 0.3%

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

   4. Taylor expanded in C around inf 0.2%

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

   5. Taylor expanded in B around inf 8.4%

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

      1. *-commutative 8.4%

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

   7. Simplified 8.4%

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

   8. Taylor expanded in B around 0 8.4%

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

      1. associate-*r/ 8.4%

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

      2. *-rgt-identity 8.4%

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

   10. Simplified 8.4%

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

4. Final simplification 19.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(2 \cdot \left(C \cdot \left(A \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C - B\right)} \cdot \left(B \cdot \left(-\sqrt{2}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \]

Alternative 9: 21.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 2.9 \cdot 10^{+54}:\\ \;\;\;\;\frac{-\sqrt{\left(A + A\right) \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 2.9e+54)
   (/ (- (sqrt (* (+ A A) (* -8.0 (* A (* C F)))))) (fma B B (* A (* C -4.0))))
   (* -2.0 (/ (sqrt (* A F)) B))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 2.9e+54) {
		tmp = -sqrt(((A + A) * (-8.0 * (A * (C * F))))) / fma(B, B, (A * (C * -4.0)));
	} else {
		tmp = -2.0 * (sqrt((A * F)) / B);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 2.9e+54)
		tmp = Float64(Float64(-sqrt(Float64(Float64(A + A) * Float64(-8.0 * Float64(A * Float64(C * F)))))) / fma(B, B, Float64(A * Float64(C * -4.0))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 2.9e+54], N[((-N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(-8.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.8999999999999999e54

   1. Initial program 23.4%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in C around inf 16.4%

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

   5. Taylor expanded in B around 0 14.2%

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

      1. *-commutative 14.2%

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

   7. Simplified 14.2%

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

   if 2.8999999999999999e54 < B

   1. Initial program 11.6%

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

   3. Simplified 11.7%

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

   4. Taylor expanded in C around inf 0.9%

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

   5. Taylor expanded in B around inf 6.6%

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

      1. *-commutative 6.6%

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

   7. Simplified 6.6%

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

   8. Taylor expanded in B around 0 6.6%

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

      1. associate-*r/ 6.6%

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

      2. *-rgt-identity 6.6%

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

   10. Simplified 6.6%

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

4. Final simplification 12.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.9 \cdot 10^{+54}:\\ \;\;\;\;\frac{-\sqrt{\left(A + A\right) \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \]

Alternative 10: 5.4% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.3%

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

3. Simplified 29.0%

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

4. Taylor expanded in C around inf 13.7%

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

5. Taylor expanded in B around inf 2.6%

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

   1. *-commutative 2.6%

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

7. Simplified 2.6%

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

8. Taylor expanded in B around 0 2.6%

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

   1. associate-*r/ 2.6%

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

   2. *-rgt-identity 2.6%

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

10. Simplified 2.6%

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

11. Final simplification 2.6%

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

Alternative 11: 3.8% accurate, 634.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ 0 \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 0.0)

C

assert(A < C);
double code(double A, double B, double C, double F) {
	return 0.0;
}

Fortran

NOTE: A and C 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 = 0.0d0
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	return 0.0;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	return 0.0

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	return 0.0
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp = code(A, B, C, F)
	tmp = 0.0;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := 0.0

Derivation

1. Initial program 21.3%

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

3. Simplified 29.0%

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

4. Taylor expanded in C around inf 13.7%

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

6. Applied egg-rr 2.1%

   \[\leadsto \color{blue}{0 - \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right) \cdot \left(F \cdot 0\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}} \]
7. Step-by-step derivation

   1. sub0-neg 2.1%

      \[\leadsto \color{blue}{-\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right) \cdot \left(F \cdot 0\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}} \]

   2. distribute-neg-frac 2.1%

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right) \cdot \left(F \cdot 0\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}} \]

   3. unpow1/2 2.1%

      \[\leadsto \frac{-\color{blue}{{\left(\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right) \cdot \left(F \cdot 0\right)\right)}^{0.5}}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

   4. mul0-rgt 2.1%

      \[\leadsto \frac{-{\left(\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right) \cdot \color{blue}{0}\right)}^{0.5}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

   5. metadata-eval 2.1%

      \[\leadsto \frac{-{\left(\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right) \cdot \color{blue}{\left(0 - 0\right)}\right)}^{0.5}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

   6. mul0-rgt 2.1%

      \[\leadsto \frac{-{\left(\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right) \cdot \left(\color{blue}{F \cdot 0} - 0\right)\right)}^{0.5}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

   7. mul0-rgt 2.1%

      \[\leadsto \frac{-{\left(\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right) \cdot \left(F \cdot 0 - \color{blue}{F \cdot 0}\right)\right)}^{0.5}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

   8. distribute-lft-out-- 2.1%

      \[\leadsto \frac{-{\color{blue}{\left(\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right) \cdot \left(F \cdot 0\right) - \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right) \cdot \left(F \cdot 0\right)\right)}}^{0.5}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

   9. +-inverses 3.7%

      \[\leadsto \frac{-{\color{blue}{0}}^{0.5}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

   10. metadata-eval 3.7%

       \[\leadsto \frac{-\color{blue}{0}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

   11. metadata-eval 3.7%

       \[\leadsto \frac{\color{blue}{0}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

   12. div0 3.8%

       \[\leadsto \color{blue}{0} \]

8. Simplified 3.8%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 3.8%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023301 
(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))))