ABCF->ab-angle a

Percentage Accurate: 18.5% → 54.2%

Time: 40.5s

Alternatives: 21

Speedup: 6.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.5% 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: 54.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)\right)}\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := -t\_1\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := t\_3 - {B\_m}^{2}\\ t_5 := 2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\\ t_6 := -0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot C\right)}}{t\_4}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\frac{\sqrt{F \cdot t\_1} \cdot t\_0}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot t\_6}}{t\_4}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{B\_m \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\sqrt{t\_6 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot \left(\frac{{B\_m}^{2}}{C} - 4 \cdot A\right)\right)\right)\right)}}{t\_4}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\frac{t\_0 \cdot \left(B\_m \cdot \sqrt{F}\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B\_m, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (* 2.0 (+ A (+ C (hypot (- A C) B_m))))))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (t_2 (- t_1))
        (t_3 (* (* 4.0 A) C))
        (t_4 (- t_3 (pow B_m 2.0)))
        (t_5 (* 2.0 (* (- (pow B_m 2.0) t_3) F)))
        (t_6 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
   (if (<= (pow B_m 2.0) 5e-128)
     (/ (sqrt (* t_5 (* 2.0 C))) t_4)
     (if (<= (pow B_m 2.0) 2e+31)
       (/ (* (sqrt (* F t_1)) t_0) t_2)
       (if (<= (pow B_m 2.0) 2e+46)
         (/ (sqrt (* t_5 t_6)) t_4)
         (if (<= (pow B_m 2.0) 2e+150)
           (/
            (*
             B_m
             (* (sqrt (+ A (+ C (hypot B_m (- A C))))) (- (sqrt (* 2.0 F)))))
            t_1)
           (if (<= (pow B_m 2.0) 5e+163)
             (/
              (sqrt
               (* t_6 (* 2.0 (* F (* C (- (/ (pow B_m 2.0) C) (* 4.0 A)))))))
              t_4)
             (if (<= (pow B_m 2.0) 2e+253)
               (/ (* t_0 (* B_m (sqrt F))) t_2)
               (*
                (* (sqrt (+ A (hypot B_m A))) (sqrt F))
                (/ (sqrt 2.0) (- B_m)))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((2.0 * (A + (C + hypot((A - C), B_m)))));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_2 = -t_1;
	double t_3 = (4.0 * A) * C;
	double t_4 = t_3 - pow(B_m, 2.0);
	double t_5 = 2.0 * ((pow(B_m, 2.0) - t_3) * F);
	double t_6 = (-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C);
	double tmp;
	if (pow(B_m, 2.0) <= 5e-128) {
		tmp = sqrt((t_5 * (2.0 * C))) / t_4;
	} else if (pow(B_m, 2.0) <= 2e+31) {
		tmp = (sqrt((F * t_1)) * t_0) / t_2;
	} else if (pow(B_m, 2.0) <= 2e+46) {
		tmp = sqrt((t_5 * t_6)) / t_4;
	} else if (pow(B_m, 2.0) <= 2e+150) {
		tmp = (B_m * (sqrt((A + (C + hypot(B_m, (A - C))))) * -sqrt((2.0 * F)))) / t_1;
	} else if (pow(B_m, 2.0) <= 5e+163) {
		tmp = sqrt((t_6 * (2.0 * (F * (C * ((pow(B_m, 2.0) / C) - (4.0 * A))))))) / t_4;
	} else if (pow(B_m, 2.0) <= 2e+253) {
		tmp = (t_0 * (B_m * sqrt(F))) / t_2;
	} else {
		tmp = (sqrt((A + hypot(B_m, A))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(2.0 * Float64(A + Float64(C + hypot(Float64(A - C), B_m)))))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(-t_1)
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64(t_3 - (B_m ^ 2.0))
	t_5 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_3) * F))
	t_6 = Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-128)
		tmp = Float64(sqrt(Float64(t_5 * Float64(2.0 * C))) / t_4);
	elseif ((B_m ^ 2.0) <= 2e+31)
		tmp = Float64(Float64(sqrt(Float64(F * t_1)) * t_0) / t_2);
	elseif ((B_m ^ 2.0) <= 2e+46)
		tmp = Float64(sqrt(Float64(t_5 * t_6)) / t_4);
	elseif ((B_m ^ 2.0) <= 2e+150)
		tmp = Float64(Float64(B_m * Float64(sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) * Float64(-sqrt(Float64(2.0 * F))))) / t_1);
	elseif ((B_m ^ 2.0) <= 5e+163)
		tmp = Float64(sqrt(Float64(t_6 * Float64(2.0 * Float64(F * Float64(C * Float64(Float64((B_m ^ 2.0) / C) - Float64(4.0 * A))))))) / t_4);
	elseif ((B_m ^ 2.0) <= 2e+253)
		tmp = Float64(Float64(t_0 * Float64(B_m * sqrt(F))) / t_2);
	else
		tmp = Float64(Float64(sqrt(Float64(A + hypot(B_m, A))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-t$95$1)}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-128], N[(N[Sqrt[N[(t$95$5 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+31], N[(N[(N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+46], N[(N[Sqrt[N[(t$95$5 * t$95$6), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+150], N[(N[(B$95$m * N[(N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+163], N[(N[Sqrt[N[(t$95$6 * N[(2.0 * N[(F * N[(C * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision] - N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+253], N[(N[(t$95$0 * N[(B$95$m * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (pow.f64 B 2) < 5.0000000000000001e-128

   1. Initial program 16.6%

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

   3. Taylor expanded in A around -inf 22.6%

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

   if 5.0000000000000001e-128 < (pow.f64 B 2) < 1.9999999999999999e31

   1. Initial program 44.4%

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

   3. Simplified 56.0%

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

      1. sqrt-prod 62.0%

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

      2. *-commutative 62.0%

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

      3. hypot-undefine 45.6%

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

      4. unpow2 45.6%

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

      5. unpow2 45.6%

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

      6. +-commutative 45.6%

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

      7. unpow2 45.6%

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

      8. unpow2 45.6%

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

      9. hypot-define 62.0%

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

   6. Applied egg-rr 62.0%

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

   if 1.9999999999999999e31 < (pow.f64 B 2) < 2e46

   1. Initial program 36.1%

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

   3. Taylor expanded in A around -inf 66.7%

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

   if 2e46 < (pow.f64 B 2) < 1.99999999999999996e150

   1. Initial program 36.2%

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

   3. Simplified 41.6%

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

   5. Taylor expanded in B around inf 41.3%

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

      1. pow1/2 41.3%

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

      2. unpow2 41.3%

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

      3. associate-*l* 41.2%

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

      4. unpow-prod-down 45.5%

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

      5. pow-prod-down 31.2%

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

      6. pow1/2 31.2%

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

      7. pow1/2 31.2%

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

      8. add-sqr-sqrt 32.2%

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

      9. associate-+r+ 32.5%

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

   7. Applied egg-rr 32.5%

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

      1. associate-*r* 32.5%

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

      2. unpow-prod-down 40.8%

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

      3. pow1/2 40.8%

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

      4. associate-+l+ 40.4%

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

   9. Applied egg-rr 40.4%

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

       1. unpow1/2 40.4%

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

   11. Simplified 40.4%

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

   if 1.99999999999999996e150 < (pow.f64 B 2) < 5e163

   1. Initial program 2.1%

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

   3. Taylor expanded in A around -inf 60.3%

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

   4. Taylor expanded in C around inf 60.6%

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

   if 5e163 < (pow.f64 B 2) < 1.9999999999999999e253

   1. Initial program 32.5%

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

   3. Simplified 38.8%

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

      1. sqrt-prod 44.8%

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

      2. *-commutative 44.8%

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

      3. hypot-undefine 32.6%

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

      4. unpow2 32.6%

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

      5. unpow2 32.6%

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

      6. +-commutative 32.6%

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

      7. unpow2 32.6%

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

      8. unpow2 32.6%

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

      9. hypot-define 44.8%

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

   6. Applied egg-rr 44.8%

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

   7. Taylor expanded in B around inf 23.9%

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

   if 1.9999999999999999e253 < (pow.f64 B 2)

   1. Initial program 4.2%

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

   3. Taylor expanded in C around 0 8.2%

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

      1. mul-1-neg 8.2%

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

      2. +-commutative 8.2%

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

      3. unpow2 8.2%

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

      4. unpow2 8.2%

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

      5. hypot-define 30.0%

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

   5. Simplified 30.0%

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

      1. pow1/2 30.0%

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

      2. *-commutative 30.0%

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

      3. unpow-prod-down 46.1%

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

      4. pow1/2 46.1%

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

      5. pow1/2 46.1%

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

   7. Applied egg-rr 46.1%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\frac{\sqrt{F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\sqrt{\left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right) \cdot \left(2 \cdot \left(F \cdot \left(C \cdot \left(\frac{{B}^{2}}{C} - 4 \cdot A\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)\right)} \cdot \left(B \cdot \sqrt{F}\right)}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{F \cdot t\_0}\\ t_2 := \frac{{B\_m}^{2}}{A}\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := t\_3 - {B\_m}^{2}\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_4}\\ t_6 := -t\_0\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot t\_2\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left({B\_m}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}{t\_4}\\ \mathbf{elif}\;t\_5 \leq -1 \cdot 10^{-219}:\\ \;\;\;\;\frac{t\_1 \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)\right)}}{t\_6}\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\sqrt{F \cdot \frac{\mathsf{fma}\left(-0.5, t\_2, 2 \cdot C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{t\_1 \cdot \sqrt{2 \cdot \left(2 \cdot C\right)}}{t\_6}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B\_m, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (sqrt (* F t_0)))
        (t_2 (/ (pow B_m 2.0) A))
        (t_3 (* (* 4.0 A) C))
        (t_4 (- t_3 (pow B_m 2.0)))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_3) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_4))
        (t_6 (- t_0)))
   (if (<= t_5 (- INFINITY))
     (/
      (*
       (sqrt (fma 2.0 C (* -0.5 t_2)))
       (sqrt (* 2.0 (* F (- (pow B_m 2.0) (* 4.0 (* A C)))))))
      t_4)
     (if (<= t_5 -1e-219)
       (/ (* t_1 (sqrt (* 2.0 (+ A (+ C (hypot (- A C) B_m)))))) t_6)
       (if (<= t_5 0.0)
         (*
          (sqrt
           (* F (/ (fma -0.5 t_2 (* 2.0 C)) (fma -4.0 (* A C) (pow B_m 2.0)))))
          (- (sqrt 2.0)))
         (if (<= t_5 INFINITY)
           (/ (* t_1 (sqrt (* 2.0 (* 2.0 C)))) t_6)
           (*
            (* (sqrt (+ A (hypot B_m A))) (sqrt F))
            (/ (sqrt 2.0) (- B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = sqrt((F * t_0));
	double t_2 = pow(B_m, 2.0) / A;
	double t_3 = (4.0 * A) * C;
	double t_4 = t_3 - pow(B_m, 2.0);
	double t_5 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_3) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_4;
	double t_6 = -t_0;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = (sqrt(fma(2.0, C, (-0.5 * t_2))) * sqrt((2.0 * (F * (pow(B_m, 2.0) - (4.0 * (A * C))))))) / t_4;
	} else if (t_5 <= -1e-219) {
		tmp = (t_1 * sqrt((2.0 * (A + (C + hypot((A - C), B_m)))))) / t_6;
	} else if (t_5 <= 0.0) {
		tmp = sqrt((F * (fma(-0.5, t_2, (2.0 * C)) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (t_1 * sqrt((2.0 * (2.0 * C)))) / t_6;
	} else {
		tmp = (sqrt((A + hypot(B_m, A))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = sqrt(Float64(F * t_0))
	t_2 = Float64((B_m ^ 2.0) / A)
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64(t_3 - (B_m ^ 2.0))
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_3) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_4)
	t_6 = Float64(-t_0)
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(fma(2.0, C, Float64(-0.5 * t_2))) * sqrt(Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - Float64(4.0 * Float64(A * C))))))) / t_4);
	elseif (t_5 <= -1e-219)
		tmp = Float64(Float64(t_1 * sqrt(Float64(2.0 * Float64(A + Float64(C + hypot(Float64(A - C), B_m)))))) / t_6);
	elseif (t_5 <= 0.0)
		tmp = Float64(sqrt(Float64(F * Float64(fma(-0.5, t_2, Float64(2.0 * C)) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(t_1 * sqrt(Float64(2.0 * Float64(2.0 * C)))) / t_6);
	else
		tmp = Float64(Float64(sqrt(Float64(A + hypot(B_m, A))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$6 = (-t$95$0)}, If[LessEqual[t$95$5, (-Infinity)], N[(N[(N[Sqrt[N[(2.0 * C + N[(-0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$5, -1e-219], N[(N[(t$95$1 * N[Sqrt[N[(2.0 * N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(N[Sqrt[N[(F * N[(N[(-0.5 * t$95$2 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(t$95$1 * N[Sqrt[N[(2.0 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision], N[(N[(N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 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))) < -inf.0

   1. Initial program 3.1%

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

   3. Taylor expanded in A around -inf 11.6%

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

      1. pow1/2 12.7%

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

      2. *-commutative 12.7%

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

      3. unpow-prod-down 19.5%

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

      4. pow1/2 19.2%

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

      5. +-commutative 19.2%

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

      6. fma-define 19.2%

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

      7. pow1/2 19.2%

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

      8. associate-*l* 19.2%

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

   5. Applied egg-rr 19.2%

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

   if -inf.0 < (/.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))) < -1e-219

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

      1. sqrt-prod 98.9%

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

      2. *-commutative 98.9%

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

      3. hypot-undefine 98.9%

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

      4. unpow2 98.9%

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

      5. unpow2 98.9%

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

      6. +-commutative 98.9%

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

      7. unpow2 98.9%

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

      8. unpow2 98.9%

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

      9. hypot-define 98.9%

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

   6. Applied egg-rr 98.9%

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

   if -1e-219 < (/.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))) < -0.0

   1. Initial program 3.4%

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

   3. Taylor expanded in A around -inf 27.2%

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

   4. Taylor expanded in F around 0 21.2%

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

      1. mul-1-neg 21.2%

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

      2. associate-/l* 21.0%

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

      3. fma-define 21.0%

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

      4. sub-neg 21.0%

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

      5. +-commutative 21.0%

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

      6. distribute-lft-neg-in 21.0%

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

      7. metadata-eval 21.0%

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

      8. fma-define 21.0%

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

   6. Simplified 21.0%

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

   if -0.0 < (/.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))) < +inf.0

   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 84.6%

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

      1. sqrt-prod 94.3%

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

      2. *-commutative 94.3%

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

      3. hypot-undefine 46.3%

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

      4. unpow2 46.3%

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

      5. unpow2 46.3%

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

      6. +-commutative 46.3%

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

      7. unpow2 46.3%

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

      8. unpow2 46.3%

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

      9. hypot-define 94.3%

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

   6. Applied egg-rr 94.3%

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

   7. Taylor expanded in A around -inf 56.7%

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

   if +inf.0 < (/.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 0.0%

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

   3. Taylor expanded in C around 0 1.8%

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

      1. mul-1-neg 1.8%

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

      2. +-commutative 1.8%

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

      3. unpow2 1.8%

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

      4. unpow2 1.8%

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

      5. hypot-define 19.0%

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

   5. Simplified 19.0%

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

      1. pow1/2 19.0%

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

      2. *-commutative 19.0%

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

      3. unpow-prod-down 30.2%

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

      4. pow1/2 30.2%

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

      5. pow1/2 30.2%

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

   7. Applied egg-rr 30.2%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{{B}^{2}}{A}\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\sqrt{F \cdot \frac{\mathsf{fma}\left(-0.5, \frac{{B}^{2}}{A}, 2 \cdot C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{2 \cdot \left(2 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := -0.5 \cdot \frac{{B\_m}^{2}}{A}\\ t_2 := t\_0 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{t\_1}\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+23}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt{\left(t\_1 + 2 \cdot C\right) \cdot \left(2 \cdot \left(F \cdot \left(C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right)\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+200}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B\_m, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (* -0.5 (/ (pow B_m 2.0) A)))
        (t_2 (- t_0 (pow B_m 2.0)))
        (t_3 (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt t_1))))))
   (if (<= (pow B_m 2.0) 5e-78)
     (/ (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C))) t_2)
     (if (<= (pow B_m 2.0) 2e+23)
       t_3
       (if (<= (pow B_m 2.0) 2e+46)
         (/ (sqrt (* (+ t_1 (* 2.0 C)) (* 2.0 (* F (* C (* A -4.0)))))) t_2)
         (if (<= (pow B_m 2.0) 2e+150)
           (* (sqrt 2.0) (* (sqrt F) (- (sqrt (/ 1.0 B_m)))))
           (if (<= (pow B_m 2.0) 1e+200)
             t_3
             (*
              (* (sqrt (+ A (hypot B_m A))) (sqrt F))
              (/ (sqrt 2.0) (- B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = -0.5 * (pow(B_m, 2.0) / A);
	double t_2 = t_0 - pow(B_m, 2.0);
	double t_3 = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt(t_1));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-78) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / t_2;
	} else if (pow(B_m, 2.0) <= 2e+23) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 2e+46) {
		tmp = sqrt(((t_1 + (2.0 * C)) * (2.0 * (F * (C * (A * -4.0)))))) / t_2;
	} else if (pow(B_m, 2.0) <= 2e+150) {
		tmp = sqrt(2.0) * (sqrt(F) * -sqrt((1.0 / B_m)));
	} else if (pow(B_m, 2.0) <= 1e+200) {
		tmp = t_3;
	} else {
		tmp = (sqrt((A + hypot(B_m, A))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = -0.5 * (Math.pow(B_m, 2.0) / A);
	double t_2 = t_0 - Math.pow(B_m, 2.0);
	double t_3 = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * -Math.sqrt(t_1));
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-78) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / t_2;
	} else if (Math.pow(B_m, 2.0) <= 2e+23) {
		tmp = t_3;
	} else if (Math.pow(B_m, 2.0) <= 2e+46) {
		tmp = Math.sqrt(((t_1 + (2.0 * C)) * (2.0 * (F * (C * (A * -4.0)))))) / t_2;
	} else if (Math.pow(B_m, 2.0) <= 2e+150) {
		tmp = Math.sqrt(2.0) * (Math.sqrt(F) * -Math.sqrt((1.0 / B_m)));
	} else if (Math.pow(B_m, 2.0) <= 1e+200) {
		tmp = t_3;
	} else {
		tmp = (Math.sqrt((A + Math.hypot(B_m, A))) * Math.sqrt(F)) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	t_1 = -0.5 * (math.pow(B_m, 2.0) / A)
	t_2 = t_0 - math.pow(B_m, 2.0)
	t_3 = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * -math.sqrt(t_1))
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e-78:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / t_2
	elif math.pow(B_m, 2.0) <= 2e+23:
		tmp = t_3
	elif math.pow(B_m, 2.0) <= 2e+46:
		tmp = math.sqrt(((t_1 + (2.0 * C)) * (2.0 * (F * (C * (A * -4.0)))))) / t_2
	elif math.pow(B_m, 2.0) <= 2e+150:
		tmp = math.sqrt(2.0) * (math.sqrt(F) * -math.sqrt((1.0 / B_m)))
	elif math.pow(B_m, 2.0) <= 1e+200:
		tmp = t_3
	else:
		tmp = (math.sqrt((A + math.hypot(B_m, A))) * math.sqrt(F)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(-0.5 * Float64((B_m ^ 2.0) / A))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	t_3 = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(t_1))))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-78)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / t_2);
	elseif ((B_m ^ 2.0) <= 2e+23)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 2e+46)
		tmp = Float64(sqrt(Float64(Float64(t_1 + Float64(2.0 * C)) * Float64(2.0 * Float64(F * Float64(C * Float64(A * -4.0)))))) / t_2);
	elseif ((B_m ^ 2.0) <= 2e+150)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(F) * Float64(-sqrt(Float64(1.0 / B_m)))));
	elseif ((B_m ^ 2.0) <= 1e+200)
		tmp = t_3;
	else
		tmp = Float64(Float64(sqrt(Float64(A + hypot(B_m, A))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	t_1 = -0.5 * ((B_m ^ 2.0) / A);
	t_2 = t_0 - (B_m ^ 2.0);
	t_3 = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt(t_1));
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e-78)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / t_2;
	elseif ((B_m ^ 2.0) <= 2e+23)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 2e+46)
		tmp = sqrt(((t_1 + (2.0 * C)) * (2.0 * (F * (C * (A * -4.0)))))) / t_2;
	elseif ((B_m ^ 2.0) <= 2e+150)
		tmp = sqrt(2.0) * (sqrt(F) * -sqrt((1.0 / B_m)));
	elseif ((B_m ^ 2.0) <= 1e+200)
		tmp = t_3;
	else
		tmp = (sqrt((A + hypot(B_m, A))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[t$95$1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-78], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+23], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+46], N[(N[Sqrt[N[(N[(t$95$1 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+150], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+200], t$95$3, N[(N[(N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B 2) < 4.9999999999999996e-78

   1. Initial program 20.2%

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

   3. Taylor expanded in A around -inf 23.6%

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

   if 4.9999999999999996e-78 < (pow.f64 B 2) < 1.9999999999999998e23 or 1.99999999999999996e150 < (pow.f64 B 2) < 9.9999999999999997e199

   1. Initial program 25.6%

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

   3. Taylor expanded in C around 0 5.6%

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

      1. mul-1-neg 5.6%

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

      2. +-commutative 5.6%

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

      3. unpow2 5.6%

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

      4. unpow2 5.6%

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

      5. hypot-define 6.2%

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

   5. Simplified 6.2%

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

      1. pow1/2 6.2%

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

      2. *-commutative 6.2%

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

      3. unpow-prod-down 7.7%

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

      4. pow1/2 7.7%

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

      5. pow1/2 7.7%

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

   7. Applied egg-rr 7.7%

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

   8. Taylor expanded in A around -inf 24.0%

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

   if 1.9999999999999998e23 < (pow.f64 B 2) < 2e46

   1. Initial program 42.2%

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

   3. Taylor expanded in A around -inf 41.9%

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

   4. Taylor expanded in B around 0 80.0%

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

      1. associate-*r* 80.0%

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

   6. Simplified 80.0%

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

   if 2e46 < (pow.f64 B 2) < 1.99999999999999996e150

   1. Initial program 36.2%

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

   3. Taylor expanded in B around inf 28.2%

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

      1. mul-1-neg 28.2%

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

   5. Simplified 28.2%

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

      1. pow1/2 28.2%

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

      2. div-inv 28.2%

         \[\leadsto -{\color{blue}{\left(F \cdot \frac{1}{B}\right)}}^{0.5} \cdot \sqrt{2} \]

      3. unpow-prod-down 32.0%

         \[\leadsto -\color{blue}{\left({F}^{0.5} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \cdot \sqrt{2} \]

      4. pow1/2 32.0%

         \[\leadsto -\left(\color{blue}{\sqrt{F}} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right) \cdot \sqrt{2} \]

   7. Applied egg-rr 32.0%

      \[\leadsto -\color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \cdot \sqrt{2} \]
   8. Step-by-step derivation

      1. unpow1/2 32.0%

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

   9. Simplified 32.0%

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

   if 9.9999999999999997e199 < (pow.f64 B 2)

   1. Initial program 8.4%

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

   3. Taylor expanded in C around 0 10.3%

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

      1. mul-1-neg 10.3%

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

      2. +-commutative 10.3%

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

      3. unpow2 10.3%

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

      4. unpow2 10.3%

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

      5. hypot-define 29.9%

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

   5. Simplified 29.9%

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

      1. pow1/2 29.9%

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

      2. *-commutative 29.9%

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

      3. unpow-prod-down 44.3%

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

      4. pow1/2 44.3%

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

      5. pow1/2 44.3%

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

   7. Applied egg-rr 44.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B}^{2}}{A}}\right)\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt{\left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right) \cdot \left(2 \cdot \left(F \cdot \left(C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{\frac{1}{B}}\right)\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{+200}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B}^{2}}{A}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ t_2 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{B\_m \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)\right)} \cdot \left(B\_m \cdot \sqrt{F}\right)}{-t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B\_m, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_0) F))
            (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
          (- t_0 (pow B_m 2.0))))
        (t_2 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 500.0)
     t_1
     (if (<= (pow B_m 2.0) 2e+150)
       (/
        (* B_m (* (sqrt (+ A (+ C (hypot B_m (- A C))))) (- (sqrt (* 2.0 F)))))
        t_2)
       (if (<= (pow B_m 2.0) 5e+163)
         t_1
         (if (<= (pow B_m 2.0) 2e+253)
           (/
            (* (sqrt (* 2.0 (+ A (+ C (hypot (- A C) B_m))))) (* B_m (sqrt F)))
            (- t_2))
           (*
            (* (sqrt (+ A (hypot B_m A))) (sqrt F))
            (/ (sqrt 2.0) (- B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / (t_0 - pow(B_m, 2.0));
	double t_2 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 500.0) {
		tmp = t_1;
	} else if (pow(B_m, 2.0) <= 2e+150) {
		tmp = (B_m * (sqrt((A + (C + hypot(B_m, (A - C))))) * -sqrt((2.0 * F)))) / t_2;
	} else if (pow(B_m, 2.0) <= 5e+163) {
		tmp = t_1;
	} else if (pow(B_m, 2.0) <= 2e+253) {
		tmp = (sqrt((2.0 * (A + (C + hypot((A - C), B_m))))) * (B_m * sqrt(F))) / -t_2;
	} else {
		tmp = (sqrt((A + hypot(B_m, A))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / Float64(t_0 - (B_m ^ 2.0)))
	t_2 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 500.0)
		tmp = t_1;
	elseif ((B_m ^ 2.0) <= 2e+150)
		tmp = Float64(Float64(B_m * Float64(sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) * Float64(-sqrt(Float64(2.0 * F))))) / t_2);
	elseif ((B_m ^ 2.0) <= 5e+163)
		tmp = t_1;
	elseif ((B_m ^ 2.0) <= 2e+253)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(A + Float64(C + hypot(Float64(A - C), B_m))))) * Float64(B_m * sqrt(F))) / Float64(-t_2));
	else
		tmp = Float64(Float64(sqrt(Float64(A + hypot(B_m, A))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 500.0], t$95$1, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+150], N[(N[(B$95$m * N[(N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+163], t$95$1, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+253], N[(N[(N[Sqrt[N[(2.0 * N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(B$95$m * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-t$95$2)), $MachinePrecision], N[(N[(N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B 2) < 500 or 1.99999999999999996e150 < (pow.f64 B 2) < 5e163

   1. Initial program 20.1%

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

   3. Taylor expanded in A around -inf 26.5%

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

   if 500 < (pow.f64 B 2) < 1.99999999999999996e150

   1. Initial program 40.9%

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

   3. Simplified 47.8%

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

   5. Taylor expanded in B around inf 44.6%

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

      1. pow1/2 44.6%

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

      2. unpow2 44.6%

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

      3. associate-*l* 44.6%

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

      4. unpow-prod-down 47.6%

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

      5. pow-prod-down 28.1%

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

      6. pow1/2 28.1%

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

      7. pow1/2 28.1%

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

      8. add-sqr-sqrt 29.0%

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

      9. associate-+r+ 29.2%

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

   7. Applied egg-rr 29.2%

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

      1. associate-*r* 29.2%

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

      2. unpow-prod-down 35.0%

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

      3. pow1/2 35.0%

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

      4. associate-+l+ 34.8%

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

   9. Applied egg-rr 34.8%

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

       1. unpow1/2 34.8%

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

   11. Simplified 34.8%

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

   if 5e163 < (pow.f64 B 2) < 1.9999999999999999e253

   1. Initial program 32.5%

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

   3. Simplified 38.8%

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

      1. sqrt-prod 44.8%

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

      2. *-commutative 44.8%

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

      3. hypot-undefine 32.6%

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

      4. unpow2 32.6%

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

      5. unpow2 32.6%

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

      6. +-commutative 32.6%

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

      7. unpow2 32.6%

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

      8. unpow2 32.6%

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

      9. hypot-define 44.8%

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

   6. Applied egg-rr 44.8%

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

   7. Taylor expanded in B around inf 23.9%

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

   if 1.9999999999999999e253 < (pow.f64 B 2)

   1. Initial program 4.2%

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

   3. Taylor expanded in C around 0 8.2%

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

      1. mul-1-neg 8.2%

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

      2. +-commutative 8.2%

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

      3. unpow2 8.2%

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

      4. unpow2 8.2%

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

      5. hypot-define 30.0%

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

   5. Simplified 30.0%

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

      1. pow1/2 30.0%

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

      2. *-commutative 30.0%

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

      3. unpow-prod-down 46.1%

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

      4. pow1/2 46.1%

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

      5. pow1/2 46.1%

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

   7. Applied egg-rr 46.1%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 500:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)\right)} \cdot \left(B \cdot \sqrt{F}\right)}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := -0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\\ t_2 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := t\_0 - {B\_m}^{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 500:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot t\_1}}{t\_3}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{B\_m \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot \left(\frac{{B\_m}^{2}}{C} - 4 \cdot A\right)\right)\right)\right)}}{t\_3}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)\right)} \cdot \left(B\_m \cdot \sqrt{F}\right)}{-t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B\_m, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C)))
        (t_2 (fma B_m B_m (* A (* C -4.0))))
        (t_3 (- t_0 (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 500.0)
     (/ (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) t_1)) t_3)
     (if (<= (pow B_m 2.0) 2e+150)
       (/
        (* B_m (* (sqrt (+ A (+ C (hypot B_m (- A C))))) (- (sqrt (* 2.0 F)))))
        t_2)
       (if (<= (pow B_m 2.0) 5e+163)
         (/
          (sqrt (* t_1 (* 2.0 (* F (* C (- (/ (pow B_m 2.0) C) (* 4.0 A)))))))
          t_3)
         (if (<= (pow B_m 2.0) 2e+253)
           (/
            (* (sqrt (* 2.0 (+ A (+ C (hypot (- A C) B_m))))) (* B_m (sqrt F)))
            (- t_2))
           (*
            (* (sqrt (+ A (hypot B_m A))) (sqrt F))
            (/ (sqrt 2.0) (- B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = (-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C);
	double t_2 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_3 = t_0 - pow(B_m, 2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 500.0) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * t_1)) / t_3;
	} else if (pow(B_m, 2.0) <= 2e+150) {
		tmp = (B_m * (sqrt((A + (C + hypot(B_m, (A - C))))) * -sqrt((2.0 * F)))) / t_2;
	} else if (pow(B_m, 2.0) <= 5e+163) {
		tmp = sqrt((t_1 * (2.0 * (F * (C * ((pow(B_m, 2.0) / C) - (4.0 * A))))))) / t_3;
	} else if (pow(B_m, 2.0) <= 2e+253) {
		tmp = (sqrt((2.0 * (A + (C + hypot((A - C), B_m))))) * (B_m * sqrt(F))) / -t_2;
	} else {
		tmp = (sqrt((A + hypot(B_m, A))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C))
	t_2 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_3 = Float64(t_0 - (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 500.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * t_1)) / t_3);
	elseif ((B_m ^ 2.0) <= 2e+150)
		tmp = Float64(Float64(B_m * Float64(sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) * Float64(-sqrt(Float64(2.0 * F))))) / t_2);
	elseif ((B_m ^ 2.0) <= 5e+163)
		tmp = Float64(sqrt(Float64(t_1 * Float64(2.0 * Float64(F * Float64(C * Float64(Float64((B_m ^ 2.0) / C) - Float64(4.0 * A))))))) / t_3);
	elseif ((B_m ^ 2.0) <= 2e+253)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(A + Float64(C + hypot(Float64(A - C), B_m))))) * Float64(B_m * sqrt(F))) / Float64(-t_2));
	else
		tmp = Float64(Float64(sqrt(Float64(A + hypot(B_m, A))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 500.0], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+150], N[(N[(B$95$m * N[(N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+163], N[(N[Sqrt[N[(t$95$1 * N[(2.0 * N[(F * N[(C * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision] - N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+253], N[(N[(N[Sqrt[N[(2.0 * N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(B$95$m * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-t$95$2)), $MachinePrecision], N[(N[(N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B 2) < 500

   1. Initial program 20.8%

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

   3. Taylor expanded in A around -inf 25.1%

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

   if 500 < (pow.f64 B 2) < 1.99999999999999996e150

   1. Initial program 40.9%

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

   3. Simplified 47.8%

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

   5. Taylor expanded in B around inf 44.6%

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

      1. pow1/2 44.6%

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

      2. unpow2 44.6%

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

      3. associate-*l* 44.6%

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

      4. unpow-prod-down 47.6%

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

      5. pow-prod-down 28.1%

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

      6. pow1/2 28.1%

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

      7. pow1/2 28.1%

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

      8. add-sqr-sqrt 29.0%

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

      9. associate-+r+ 29.2%

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

   7. Applied egg-rr 29.2%

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

      1. associate-*r* 29.2%

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

      2. unpow-prod-down 35.0%

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

      3. pow1/2 35.0%

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

      4. associate-+l+ 34.8%

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

   9. Applied egg-rr 34.8%

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

       1. unpow1/2 34.8%

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

   11. Simplified 34.8%

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

   if 1.99999999999999996e150 < (pow.f64 B 2) < 5e163

   1. Initial program 2.1%

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

   3. Taylor expanded in A around -inf 60.3%

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

   4. Taylor expanded in C around inf 60.6%

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

   if 5e163 < (pow.f64 B 2) < 1.9999999999999999e253

   1. Initial program 32.5%

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

   3. Simplified 38.8%

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

      1. sqrt-prod 44.8%

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

      2. *-commutative 44.8%

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

      3. hypot-undefine 32.6%

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

      4. unpow2 32.6%

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

      5. unpow2 32.6%

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

      6. +-commutative 32.6%

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

      7. unpow2 32.6%

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

      8. unpow2 32.6%

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

      9. hypot-define 44.8%

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

   6. Applied egg-rr 44.8%

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

   7. Taylor expanded in B around inf 23.9%

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

   if 1.9999999999999999e253 < (pow.f64 B 2)

   1. Initial program 4.2%

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

   3. Taylor expanded in C around 0 8.2%

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

      1. mul-1-neg 8.2%

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

      2. +-commutative 8.2%

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

      3. unpow2 8.2%

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

      4. unpow2 8.2%

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

      5. hypot-define 30.0%

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

   5. Simplified 30.0%

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

      1. pow1/2 30.0%

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

      2. *-commutative 30.0%

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

      3. unpow-prod-down 46.1%

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

      4. pow1/2 46.1%

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

      5. pow1/2 46.1%

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

   7. Applied egg-rr 46.1%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 500:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\sqrt{\left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right) \cdot \left(2 \cdot \left(F \cdot \left(C \cdot \left(\frac{{B}^{2}}{C} - 4 \cdot A\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)\right)} \cdot \left(B \cdot \sqrt{F}\right)}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := -0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\\ t_2 := t\_0 - {B\_m}^{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 500:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot t\_1}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot \left(\frac{{B\_m}^{2}}{C} - 4 \cdot A\right)\right)\right)\right)}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)\right)} \cdot \left(B\_m \cdot \sqrt{F}\right)}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B\_m, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C)))
        (t_2 (- t_0 (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 500.0)
     (/ (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) t_1)) t_2)
     (if (<= (pow B_m 2.0) 2e+150)
       (*
        (sqrt
         (*
          F
          (/
           (+ A (+ C (hypot B_m (- A C))))
           (fma -4.0 (* A C) (pow B_m 2.0)))))
        (- (sqrt 2.0)))
       (if (<= (pow B_m 2.0) 5e+163)
         (/
          (sqrt (* t_1 (* 2.0 (* F (* C (- (/ (pow B_m 2.0) C) (* 4.0 A)))))))
          t_2)
         (if (<= (pow B_m 2.0) 2e+253)
           (/
            (* (sqrt (* 2.0 (+ A (+ C (hypot (- A C) B_m))))) (* B_m (sqrt F)))
            (- (fma B_m B_m (* A (* C -4.0)))))
           (*
            (* (sqrt (+ A (hypot B_m A))) (sqrt F))
            (/ (sqrt 2.0) (- B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = (-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C);
	double t_2 = t_0 - pow(B_m, 2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 500.0) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * t_1)) / t_2;
	} else if (pow(B_m, 2.0) <= 2e+150) {
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 5e+163) {
		tmp = sqrt((t_1 * (2.0 * (F * (C * ((pow(B_m, 2.0) / C) - (4.0 * A))))))) / t_2;
	} else if (pow(B_m, 2.0) <= 2e+253) {
		tmp = (sqrt((2.0 * (A + (C + hypot((A - C), B_m))))) * (B_m * sqrt(F))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = (sqrt((A + hypot(B_m, A))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 500.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * t_1)) / t_2);
	elseif ((B_m ^ 2.0) <= 2e+150)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 5e+163)
		tmp = Float64(sqrt(Float64(t_1 * Float64(2.0 * Float64(F * Float64(C * Float64(Float64((B_m ^ 2.0) / C) - Float64(4.0 * A))))))) / t_2);
	elseif ((B_m ^ 2.0) <= 2e+253)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(A + Float64(C + hypot(Float64(A - C), B_m))))) * Float64(B_m * sqrt(F))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(Float64(sqrt(Float64(A + hypot(B_m, A))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 500.0], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+150], N[(N[Sqrt[N[(F * N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+163], N[(N[Sqrt[N[(t$95$1 * N[(2.0 * N[(F * N[(C * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision] - N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+253], N[(N[(N[Sqrt[N[(2.0 * N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(B$95$m * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B 2) < 500

   1. Initial program 20.8%

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

   3. Taylor expanded in A around -inf 25.1%

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

   if 500 < (pow.f64 B 2) < 1.99999999999999996e150

   1. Initial program 40.9%

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

   3. Taylor expanded in F around 0 35.4%

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

      1. mul-1-neg 35.4%

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

   5. Simplified 59.7%

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

   if 1.99999999999999996e150 < (pow.f64 B 2) < 5e163

   1. Initial program 2.1%

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

   3. Taylor expanded in A around -inf 60.3%

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

   4. Taylor expanded in C around inf 60.6%

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

   if 5e163 < (pow.f64 B 2) < 1.9999999999999999e253

   1. Initial program 32.5%

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

   3. Simplified 38.8%

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

      1. sqrt-prod 44.8%

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

      2. *-commutative 44.8%

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

      3. hypot-undefine 32.6%

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

      4. unpow2 32.6%

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

      5. unpow2 32.6%

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

      6. +-commutative 32.6%

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

      7. unpow2 32.6%

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

      8. unpow2 32.6%

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

      9. hypot-define 44.8%

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

   6. Applied egg-rr 44.8%

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

   7. Taylor expanded in B around inf 23.9%

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

   if 1.9999999999999999e253 < (pow.f64 B 2)

   1. Initial program 4.2%

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

   3. Taylor expanded in C around 0 8.2%

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

      1. mul-1-neg 8.2%

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

      2. +-commutative 8.2%

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

      3. unpow2 8.2%

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

      4. unpow2 8.2%

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

      5. hypot-define 30.0%

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

   5. Simplified 30.0%

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

      1. pow1/2 30.0%

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

      2. *-commutative 30.0%

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

      3. unpow-prod-down 46.1%

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

      4. pow1/2 46.1%

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

      5. pow1/2 46.1%

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

   7. Applied egg-rr 46.1%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 500:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\sqrt{\left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right) \cdot \left(2 \cdot \left(F \cdot \left(C \cdot \left(\frac{{B}^{2}}{C} - 4 \cdot A\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)\right)} \cdot \left(B \cdot \sqrt{F}\right)}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{2}}{B\_m}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-11}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{B\_m \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+200}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{F} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B\_m}^{2}}{A}}\right)\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+271}:\\ \;\;\;\;t\_1 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B\_m, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)) (t_1 (/ (sqrt 2.0) B_m)))
   (if (<= (pow B_m 2.0) 1e-11)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 2e+150)
       (/
        (* B_m (* (sqrt (+ A (+ C (hypot B_m (- A C))))) (- (sqrt (* 2.0 F)))))
        (fma B_m B_m (* A (* C -4.0))))
       (if (<= (pow B_m 2.0) 1e+200)
         (* t_1 (* (sqrt F) (- (sqrt (* -0.5 (/ (pow B_m 2.0) A))))))
         (if (<= (pow B_m 2.0) 2e+271)
           (* t_1 (- (sqrt (* F (+ C (hypot B_m C))))))
           (*
            (* (sqrt (+ A (hypot B_m A))) (sqrt F))
            (/ (sqrt 2.0) (- B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = sqrt(2.0) / B_m;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-11) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+150) {
		tmp = (B_m * (sqrt((A + (C + hypot(B_m, (A - C))))) * -sqrt((2.0 * F)))) / fma(B_m, B_m, (A * (C * -4.0)));
	} else if (pow(B_m, 2.0) <= 1e+200) {
		tmp = t_1 * (sqrt(F) * -sqrt((-0.5 * (pow(B_m, 2.0) / A))));
	} else if (pow(B_m, 2.0) <= 2e+271) {
		tmp = t_1 * -sqrt((F * (C + hypot(B_m, C))));
	} else {
		tmp = (sqrt((A + hypot(B_m, A))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(2.0) / B_m)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-11)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+150)
		tmp = Float64(Float64(B_m * Float64(sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) * Float64(-sqrt(Float64(2.0 * F))))) / fma(B_m, B_m, Float64(A * Float64(C * -4.0))));
	elseif ((B_m ^ 2.0) <= 1e+200)
		tmp = Float64(t_1 * Float64(sqrt(F) * Float64(-sqrt(Float64(-0.5 * Float64((B_m ^ 2.0) / A))))));
	elseif ((B_m ^ 2.0) <= 2e+271)
		tmp = Float64(t_1 * Float64(-sqrt(Float64(F * Float64(C + hypot(B_m, C))))));
	else
		tmp = Float64(Float64(sqrt(Float64(A + hypot(B_m, A))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-11], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+150], N[(N[(B$95$m * N[(N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+200], N[(t$95$1 * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+271], N[(t$95$1 * (-N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B 2) < 9.99999999999999939e-12

   1. Initial program 20.4%

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

   3. Taylor expanded in A around -inf 23.7%

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

   if 9.99999999999999939e-12 < (pow.f64 B 2) < 1.99999999999999996e150

   1. Initial program 40.4%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in B around inf 43.7%

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

      1. pow1/2 43.7%

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

      2. unpow2 43.7%

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

      3. associate-*l* 41.1%

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

      4. unpow-prod-down 43.8%

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

      5. pow-prod-down 25.7%

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

      6. pow1/2 25.7%

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

      7. pow1/2 25.7%

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

      8. add-sqr-sqrt 26.7%

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

      9. associate-+r+ 26.9%

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

   7. Applied egg-rr 26.9%

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

      1. associate-*r* 26.9%

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

      2. unpow-prod-down 32.3%

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

      3. pow1/2 32.3%

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

      4. associate-+l+ 32.0%

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

   9. Applied egg-rr 32.0%

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

       1. unpow1/2 32.0%

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

   11. Simplified 32.0%

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

   if 1.99999999999999996e150 < (pow.f64 B 2) < 9.9999999999999997e199

   1. Initial program 10.0%

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

   3. Taylor expanded in C around 0 2.9%

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

      1. mul-1-neg 2.9%

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

      2. +-commutative 2.9%

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

      3. unpow2 2.9%

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

      4. unpow2 2.9%

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

      5. hypot-define 4.0%

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

   5. Simplified 4.0%

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

      1. pow1/2 4.0%

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

      2. *-commutative 4.0%

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

      3. unpow-prod-down 7.4%

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

      4. pow1/2 7.4%

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

      5. pow1/2 7.4%

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

   7. Applied egg-rr 7.4%

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

   8. Taylor expanded in A around -inf 48.2%

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

   if 9.9999999999999997e199 < (pow.f64 B 2) < 1.99999999999999991e271

   1. Initial program 47.5%

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

   3. Taylor expanded in A around 0 39.7%

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

      1. mul-1-neg 39.7%

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

      2. unpow2 39.7%

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

      3. unpow2 39.7%

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

      4. hypot-define 40.5%

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

   5. Simplified 40.5%

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

   if 1.99999999999999991e271 < (pow.f64 B 2)

   1. Initial program 1.6%

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

   3. Taylor expanded in C around 0 5.9%

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

      1. mul-1-neg 5.9%

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

      2. +-commutative 5.9%

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

      3. unpow2 5.9%

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

      4. unpow2 5.9%

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

      5. hypot-define 28.9%

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

   5. Simplified 28.9%

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

      1. pow1/2 28.9%

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

      2. *-commutative 28.9%

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

      3. unpow-prod-down 45.8%

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

      4. pow1/2 45.8%

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

      5. pow1/2 45.8%

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

   7. Applied egg-rr 45.8%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-11}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+200}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B}^{2}}{A}}\right)\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, C - A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 500:\\ \;\;\;\;\sqrt{-0.5 \cdot \left({B\_m}^{2} \cdot \frac{F}{A}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-225)
   (/
    (sqrt (* (* -8.0 (* A (* C F))) (+ A (+ C (hypot B_m (- C A))))))
    (- (* 4.0 (* A C)) (pow B_m 2.0)))
   (if (<= (pow B_m 2.0) 500.0)
     (* (sqrt (* -0.5 (* (pow B_m 2.0) (/ F A)))) (/ (sqrt 2.0) (- B_m)))
     (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-225) {
		tmp = sqrt(((-8.0 * (A * (C * F))) * (A + (C + hypot(B_m, (C - A)))))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 500.0) {
		tmp = sqrt((-0.5 * (pow(B_m, 2.0) * (F / A)))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-225) {
		tmp = Math.sqrt(((-8.0 * (A * (C * F))) * (A + (C + Math.hypot(B_m, (C - A)))))) / ((4.0 * (A * C)) - Math.pow(B_m, 2.0));
	} else if (Math.pow(B_m, 2.0) <= 500.0) {
		tmp = Math.sqrt((-0.5 * (Math.pow(B_m, 2.0) * (F / A)))) * (Math.sqrt(2.0) / -B_m);
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-225:
		tmp = math.sqrt(((-8.0 * (A * (C * F))) * (A + (C + math.hypot(B_m, (C - A)))))) / ((4.0 * (A * C)) - math.pow(B_m, 2.0))
	elif math.pow(B_m, 2.0) <= 500.0:
		tmp = math.sqrt((-0.5 * (math.pow(B_m, 2.0) * (F / A)))) * (math.sqrt(2.0) / -B_m)
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-225)
		tmp = Float64(sqrt(Float64(Float64(-8.0 * Float64(A * Float64(C * F))) * Float64(A + Float64(C + hypot(B_m, Float64(C - A)))))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 500.0)
		tmp = Float64(sqrt(Float64(-0.5 * Float64((B_m ^ 2.0) * Float64(F / A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-225)
		tmp = sqrt(((-8.0 * (A * (C * F))) * (A + (C + hypot(B_m, (C - A)))))) / ((4.0 * (A * C)) - (B_m ^ 2.0));
	elseif ((B_m ^ 2.0) <= 500.0)
		tmp = sqrt((-0.5 * ((B_m ^ 2.0) * (F / A)))) * (sqrt(2.0) / -B_m);
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-225], N[(N[Sqrt[N[(N[(-8.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(C - A), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 500.0], N[(N[Sqrt[N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 1.9999999999999999e-225

   1. Initial program 14.1%

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

   3. Simplified 27.5%

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

   5. Taylor expanded in A around inf 23.3%

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

   if 1.9999999999999999e-225 < (pow.f64 B 2) < 500

   1. Initial program 33.2%

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

   3. Taylor expanded in C around 0 7.9%

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

      1. mul-1-neg 7.9%

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

      2. +-commutative 7.9%

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

      3. unpow2 7.9%

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

      4. unpow2 7.9%

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

      5. hypot-define 8.2%

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

   5. Simplified 8.2%

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

   6. Taylor expanded in A around -inf 9.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{{B}^{2} \cdot F}{A}}} \]
   7. Step-by-step derivation

      1. associate-/l* 8.9%

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

   8. Simplified 8.9%

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

   if 500 < (pow.f64 B 2)

   1. Initial program 16.6%

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

   3. Taylor expanded in B around inf 27.7%

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

      1. mul-1-neg 27.7%

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

   5. Simplified 27.7%

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

      1. sqrt-div 33.7%

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

   7. Applied egg-rr 33.7%

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

      1. associate-*l/ 33.7%

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

      2. pow1/2 33.7%

         \[\leadsto -\frac{\color{blue}{{F}^{0.5}} \cdot \sqrt{2}}{\sqrt{B}} \]

      3. pow1/2 33.7%

         \[\leadsto -\frac{{F}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{\sqrt{B}} \]

      4. pow-prod-down 33.8%

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

   9. Applied egg-rr 33.8%

      \[\leadsto -\color{blue}{\frac{{\left(F \cdot 2\right)}^{0.5}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. unpow1/2 33.8%

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

   11. Simplified 33.8%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, C - A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 500:\\ \;\;\;\;\sqrt{-0.5 \cdot \left({B}^{2} \cdot \frac{F}{A}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B\_m, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 2e+46)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (* (* (sqrt (+ A (hypot B_m A))) (sqrt F)) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 2e+46) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = (sqrt((A + hypot(B_m, A))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e+46) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = (Math.sqrt((A + Math.hypot(B_m, A))) * Math.sqrt(F)) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e+46:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = (math.sqrt((A + math.hypot(B_m, A))) * math.sqrt(F)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+46)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(A + hypot(B_m, A))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e+46)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = (sqrt((A + hypot(B_m, A))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+46], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 2e46

   1. Initial program 23.1%

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

   3. Taylor expanded in A around -inf 25.2%

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

   if 2e46 < (pow.f64 B 2)

   1. Initial program 13.8%

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

   3. Taylor expanded in C around 0 13.6%

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

      1. mul-1-neg 13.6%

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

      2. +-commutative 13.6%

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

      3. unpow2 13.6%

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

      4. unpow2 13.6%

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

      5. hypot-define 27.9%

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

   5. Simplified 27.9%

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

      1. pow1/2 27.9%

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

      2. *-commutative 27.9%

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

      3. unpow-prod-down 39.3%

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

      4. pow1/2 39.3%

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

      5. pow1/2 39.3%

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

   7. Applied egg-rr 39.3%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 2e+46)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 2e+46) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    if ((b_m ** 2.0d0) <= 2d+46) then
        tmp = sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * (2.0d0 * c))) / (t_0 - (b_m ** 2.0d0))
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e+46) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e+46:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+46)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e+46)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+46], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 2e46

   1. Initial program 23.1%

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

   3. Taylor expanded in A around -inf 25.2%

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

   if 2e46 < (pow.f64 B 2)

   1. Initial program 13.8%

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

   3. Taylor expanded in B around inf 29.0%

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

      1. mul-1-neg 29.0%

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

   5. Simplified 29.0%

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

      1. sqrt-div 35.5%

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

   7. Applied egg-rr 35.5%

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

      1. associate-*l/ 35.5%

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

      2. pow1/2 35.5%

         \[\leadsto -\frac{\color{blue}{{F}^{0.5}} \cdot \sqrt{2}}{\sqrt{B}} \]

      3. pow1/2 35.5%

         \[\leadsto -\frac{{F}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{\sqrt{B}} \]

      4. pow-prod-down 35.6%

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

   9. Applied egg-rr 35.6%

      \[\leadsto -\color{blue}{\frac{{\left(F \cdot 2\right)}^{0.5}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. unpow1/2 35.6%

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

   11. Simplified 35.6%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2900:\\ \;\;\;\;\sqrt{-0.5 \cdot \left({B\_m}^{2} \cdot \frac{F}{A}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2900.0)
   (* (sqrt (* -0.5 (* (pow B_m 2.0) (/ F A)))) (/ (sqrt 2.0) (- B_m)))
   (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2900.0) {
		tmp = sqrt((-0.5 * (pow(B_m, 2.0) * (F / A)))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 2900.0d0) then
        tmp = sqrt(((-0.5d0) * ((b_m ** 2.0d0) * (f / a)))) * (sqrt(2.0d0) / -b_m)
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2900.0) {
		tmp = Math.sqrt((-0.5 * (Math.pow(B_m, 2.0) * (F / A)))) * (Math.sqrt(2.0) / -B_m);
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 2900.0:
		tmp = math.sqrt((-0.5 * (math.pow(B_m, 2.0) * (F / A)))) * (math.sqrt(2.0) / -B_m)
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2900.0)
		tmp = Float64(sqrt(Float64(-0.5 * Float64((B_m ^ 2.0) * Float64(F / A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 2900.0)
		tmp = sqrt((-0.5 * ((B_m ^ 2.0) * (F / A)))) * (sqrt(2.0) / -B_m);
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2900.0], N[(N[Sqrt[N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2900

   1. Initial program 18.7%

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

   3. Taylor expanded in C around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. +-commutative 3.5%

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

      3. unpow2 3.5%

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

      4. unpow2 3.5%

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

      5. hypot-define 4.2%

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

   5. Simplified 4.2%

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

   6. Taylor expanded in A around -inf 5.5%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{{B}^{2} \cdot F}{A}}} \]
   7. Step-by-step derivation

      1. associate-/l* 4.8%

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

   8. Simplified 4.8%

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

   if 2900 < B

   1. Initial program 18.5%

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

   3. Taylor expanded in B around inf 49.2%

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

      1. mul-1-neg 49.2%

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

   5. Simplified 49.2%

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

      1. sqrt-div 61.5%

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

   7. Applied egg-rr 61.5%

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

      1. associate-*l/ 61.4%

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

      2. pow1/2 61.4%

         \[\leadsto -\frac{\color{blue}{{F}^{0.5}} \cdot \sqrt{2}}{\sqrt{B}} \]

      3. pow1/2 61.4%

         \[\leadsto -\frac{{F}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{\sqrt{B}} \]

      4. pow-prod-down 61.6%

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

   9. Applied egg-rr 61.6%

      \[\leadsto -\color{blue}{\frac{{\left(F \cdot 2\right)}^{0.5}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. unpow1/2 61.6%

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

   11. Simplified 61.6%

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

4. Final simplification 21.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2900:\\ \;\;\;\;\sqrt{-0.5 \cdot \left({B}^{2} \cdot \frac{F}{A}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 700:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{{B\_m}^{2} \cdot F}{A}} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 700.0)
   (* (sqrt (* -0.5 (/ (* (pow B_m 2.0) F) A))) (/ (sqrt 2.0) (- B_m)))
   (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 700.0) {
		tmp = sqrt((-0.5 * ((pow(B_m, 2.0) * F) / A))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 700.0d0) then
        tmp = sqrt(((-0.5d0) * (((b_m ** 2.0d0) * f) / a))) * (sqrt(2.0d0) / -b_m)
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 700.0) {
		tmp = Math.sqrt((-0.5 * ((Math.pow(B_m, 2.0) * F) / A))) * (Math.sqrt(2.0) / -B_m);
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 700.0:
		tmp = math.sqrt((-0.5 * ((math.pow(B_m, 2.0) * F) / A))) * (math.sqrt(2.0) / -B_m)
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 700.0)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(Float64((B_m ^ 2.0) * F) / A))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 700.0)
		tmp = sqrt((-0.5 * (((B_m ^ 2.0) * F) / A))) * (sqrt(2.0) / -B_m);
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 700.0], N[(N[Sqrt[N[(-0.5 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 700

   1. Initial program 18.7%

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

   3. Taylor expanded in C around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. +-commutative 3.5%

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

      3. unpow2 3.5%

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

      4. unpow2 3.5%

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

      5. hypot-define 4.2%

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

   5. Simplified 4.2%

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

   6. Taylor expanded in A around -inf 5.5%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{{B}^{2} \cdot F}{A}}} \]

   if 700 < B

   1. Initial program 18.5%

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

   3. Taylor expanded in B around inf 49.2%

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

      1. mul-1-neg 49.2%

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

   5. Simplified 49.2%

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

      1. sqrt-div 61.5%

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

   7. Applied egg-rr 61.5%

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

      1. associate-*l/ 61.4%

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

      2. pow1/2 61.4%

         \[\leadsto -\frac{\color{blue}{{F}^{0.5}} \cdot \sqrt{2}}{\sqrt{B}} \]

      3. pow1/2 61.4%

         \[\leadsto -\frac{{F}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{\sqrt{B}} \]

      4. pow-prod-down 61.6%

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

   9. Applied egg-rr 61.6%

      \[\leadsto -\color{blue}{\frac{{\left(F \cdot 2\right)}^{0.5}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. unpow1/2 61.6%

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

   11. Simplified 61.6%

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

4. Final simplification 21.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 700:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{{B}^{2} \cdot F}{A}} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 8 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 8e+26)
   (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (+ C (hypot B_m C))))))
   (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 8e+26) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (C + hypot(B_m, C))));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 8e+26) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((F * (C + Math.hypot(B_m, C))));
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 8e+26:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((F * (C + math.hypot(B_m, C))))
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 8e+26)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(C + hypot(B_m, C))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 8e+26)
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (C + hypot(B_m, C))));
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 8e+26], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 8.00000000000000038e26

   1. Initial program 24.2%

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

   3. Taylor expanded in A around 0 10.3%

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

      1. mul-1-neg 10.3%

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

      2. unpow2 10.3%

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

      3. unpow2 10.3%

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

      4. hypot-define 20.3%

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

   5. Simplified 20.3%

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

   if 8.00000000000000038e26 < F

   1. Initial program 9.5%

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

   3. Taylor expanded in B around inf 20.0%

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

      1. mul-1-neg 20.0%

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

   5. Simplified 20.0%

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

      1. sqrt-div 21.1%

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

   7. Applied egg-rr 21.1%

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

      1. associate-*l/ 21.1%

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

      2. pow1/2 21.1%

         \[\leadsto -\frac{\color{blue}{{F}^{0.5}} \cdot \sqrt{2}}{\sqrt{B}} \]

      3. pow1/2 21.1%

         \[\leadsto -\frac{{F}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{\sqrt{B}} \]

      4. pow-prod-down 21.1%

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

   9. Applied egg-rr 21.1%

      \[\leadsto -\color{blue}{\frac{{\left(F \cdot 2\right)}^{0.5}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. unpow1/2 21.1%

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

   11. Simplified 21.1%

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

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 8 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in B around inf 15.8%

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

   1. mul-1-neg 15.8%

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

5. Simplified 15.8%

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

   1. pow1 15.8%

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

   2. sqrt-unprod 15.9%

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

7. Applied egg-rr 15.9%

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

   1. unpow1 15.9%

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

9. Simplified 15.9%

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

    1. add-sqr-sqrt 15.9%

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

    2. pow1/2 15.9%

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

    3. pow1/2 16.0%

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

    4. pow-prod-down 18.7%

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

    5. pow2 18.7%

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

    6. associate-*l/ 18.7%

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

11. Applied egg-rr 18.7%

    \[\leadsto -\sqrt{\color{blue}{{\left({\left(\frac{F \cdot 2}{B}\right)}^{2}\right)}^{0.5}}} \]
12. Step-by-step derivation

    1. unpow1/2 18.7%

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

    2. unpow2 18.7%

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

    3. rem-sqrt-square 28.8%

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

    4. associate-/l* 28.8%

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

13. Simplified 28.8%

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

14. Final simplification 28.8%

    \[\leadsto -\sqrt{\left|F \cdot \frac{2}{B}\right|} \]
15. Add Preprocessing

Alternative 15: 35.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in B around inf 15.8%

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

   1. mul-1-neg 15.8%

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

5. Simplified 15.8%

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

   1. sqrt-div 19.4%

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

7. Applied egg-rr 19.4%

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

   1. sqrt-undiv 15.8%

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

   2. sqrt-prod 15.9%

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

   3. pow1 15.9%

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

   4. associate-*l/ 15.9%

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

9. Applied egg-rr 15.9%

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

    1. unpow1 15.9%

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

    2. associate-/l* 15.9%

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

11. Simplified 15.9%

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

    1. pow1/2 16.0%

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

    2. *-commutative 16.0%

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

    3. unpow-prod-down 19.4%

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

    4. pow1/2 19.4%

       \[\leadsto -\color{blue}{\sqrt{\frac{2}{B}}} \cdot {F}^{0.5} \]

    5. pow1/2 19.4%

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

13. Applied egg-rr 19.4%

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

14. Final simplification 19.4%

    \[\leadsto \sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right) \]
15. Add Preprocessing

Alternative 16: 35.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in B around inf 15.8%

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

   1. mul-1-neg 15.8%

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

5. Simplified 15.8%

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

   1. sqrt-div 19.4%

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

7. Applied egg-rr 19.4%

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

   1. associate-*l/ 19.4%

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

   2. pow1/2 19.4%

      \[\leadsto -\frac{\color{blue}{{F}^{0.5}} \cdot \sqrt{2}}{\sqrt{B}} \]

   3. pow1/2 19.4%

      \[\leadsto -\frac{{F}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{\sqrt{B}} \]

   4. pow-prod-down 19.4%

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

9. Applied egg-rr 19.4%

   \[\leadsto -\color{blue}{\frac{{\left(F \cdot 2\right)}^{0.5}}{\sqrt{B}}} \]
10. Step-by-step derivation

    1. unpow1/2 19.4%

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

11. Simplified 19.4%

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

12. Final simplification 19.4%

    \[\leadsto \frac{\sqrt{2 \cdot F}}{-\sqrt{B}} \]
13. Add Preprocessing

Alternative 17: 27.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -{\left(2 \cdot \frac{F}{B\_m}\right)}^{0.5} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (pow (* 2.0 (/ F B_m)) 0.5)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -pow((2.0 * (F / B_m)), 0.5);
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -((2.0d0 * (f / b_m)) ** 0.5d0)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.pow((2.0 * (F / B_m)), 0.5);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.pow((2.0 * (F / B_m)), 0.5)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-(Float64(2.0 * Float64(F / B_m)) ^ 0.5))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -((2.0 * (F / B_m)) ^ 0.5);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Power[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision])

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in B around inf 15.8%

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

   1. mul-1-neg 15.8%

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

5. Simplified 15.8%

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

   1. sqrt-unprod 15.9%

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

   2. pow1/2 16.0%

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

7. Applied egg-rr 16.0%

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

8. Final simplification 16.0%

   \[\leadsto -{\left(2 \cdot \frac{F}{B}\right)}^{0.5} \]
9. Add Preprocessing

Alternative 18: 27.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -{\left(\frac{2 \cdot F}{B\_m}\right)}^{0.5} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (pow (/ (* 2.0 F) B_m) 0.5)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -pow(((2.0 * F) / B_m), 0.5);
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -(((2.0d0 * f) / b_m) ** 0.5d0)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.pow(((2.0 * F) / B_m), 0.5);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.pow(((2.0 * F) / B_m), 0.5)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-(Float64(Float64(2.0 * F) / B_m) ^ 0.5))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -(((2.0 * F) / B_m) ^ 0.5);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Power[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision], 0.5], $MachinePrecision])

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in B around inf 15.8%

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

   1. mul-1-neg 15.8%

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

5. Simplified 15.8%

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

   1. sqrt-div 19.4%

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

7. Applied egg-rr 19.4%

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

   1. sqrt-undiv 15.8%

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

   2. sqrt-prod 15.9%

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

   3. pow1/2 16.0%

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

   4. associate-*l/ 16.0%

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

9. Applied egg-rr 16.0%

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

10. Final simplification 16.0%

    \[\leadsto -{\left(\frac{2 \cdot F}{B}\right)}^{0.5} \]
11. Add Preprocessing

Alternative 19: 0.8% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in B around inf 15.8%

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

   1. mul-1-neg 15.8%

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

5. Simplified 15.8%

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

   1. pow1 15.8%

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

   2. sqrt-unprod 15.9%

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

7. Applied egg-rr 15.9%

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

   1. unpow1 15.9%

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

9. Simplified 15.9%

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

    1. add-sqr-sqrt 15.9%

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

    2. pow1/2 15.9%

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

    3. pow1/2 16.0%

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

    4. pow-prod-down 18.7%

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

    5. pow2 18.7%

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

    6. associate-*l/ 18.7%

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

11. Applied egg-rr 18.7%

    \[\leadsto -\sqrt{\color{blue}{{\left({\left(\frac{F \cdot 2}{B}\right)}^{2}\right)}^{0.5}}} \]
12. Step-by-step derivation

    1. unpow1/2 18.7%

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

    2. associate-/l* 18.6%

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

13. Simplified 18.6%

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

14. Taylor expanded in F around -inf 13.7%

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

15. Final simplification 13.7%

    \[\leadsto -\sqrt{\frac{F}{B} \cdot -2} \]
16. Add Preprocessing

Alternative 20: 27.2% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in B around inf 15.8%

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

   1. mul-1-neg 15.8%

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

5. Simplified 15.8%

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

   1. sqrt-div 19.4%

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

7. Applied egg-rr 19.4%

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

   1. sqrt-undiv 15.8%

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

   2. sqrt-prod 15.9%

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

   3. pow1 15.9%

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

   4. associate-*l/ 15.9%

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

9. Applied egg-rr 15.9%

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

    1. unpow1 15.9%

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

    2. associate-/l* 15.9%

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

11. Simplified 15.9%

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

12. Final simplification 15.9%

    \[\leadsto -\sqrt{F \cdot \frac{2}{B}} \]
13. Add Preprocessing

Alternative 21: 27.2% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in B around inf 15.8%

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

   1. mul-1-neg 15.8%

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

5. Simplified 15.8%

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

   1. pow1 15.8%

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

   2. sqrt-unprod 15.9%

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

7. Applied egg-rr 15.9%

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

   1. unpow1 15.9%

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

9. Simplified 15.9%

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

10. Final simplification 15.9%

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

Reproduce

herbie shell --seed 2024054 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))