ABCF->ab-angle a

Percentage Accurate: 18.2% → 49.2%

Time: 28.6s

Alternatives: 17

Speedup: 4.8×

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: 18.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 49.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\\ t_1 := B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-72}:\\ \;\;\;\;\sqrt{\frac{t\_0}{t\_1}} \cdot \left(0 - {\left(2 \cdot F\right)}^{0.5}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{F \cdot t\_1}}{\frac{A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m}{\sqrt{2 \cdot t\_0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - \sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (+ A C) (hypot B_m (- A C))))
        (t_1 (+ (* B_m B_m) (* -4.0 (* A C))))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 5e-72)
     (* (sqrt (/ t_0 t_1)) (- 0.0 (pow (* 2.0 F) 0.5)))
     (if (<= t_3 INFINITY)
       (/
        (sqrt (* F t_1))
        (/ (- (* A (* 4.0 C)) (* B_m B_m)) (sqrt (* 2.0 t_0))))
       (* (/ (- 0.0 (sqrt 2.0)) B_m) (sqrt (* F (+ A (hypot B_m A)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (A + C) + hypot(B_m, (A - C));
	double t_1 = (B_m * B_m) + (-4.0 * (A * C));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= 5e-72) {
		tmp = sqrt((t_0 / t_1)) * (0.0 - pow((2.0 * F), 0.5));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((F * t_1)) / (((A * (4.0 * C)) - (B_m * B_m)) / sqrt((2.0 * t_0)));
	} else {
		tmp = ((0.0 - sqrt(2.0)) / B_m) * sqrt((F * (A + hypot(B_m, A))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (A + C) + Math.hypot(B_m, (A - C));
	double t_1 = (B_m * B_m) + (-4.0 * (A * C));
	double t_2 = (4.0 * A) * C;
	double t_3 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_2) * F)) * ((A + C) + Math.sqrt((Math.pow(B_m, 2.0) + Math.pow((A - C), 2.0)))))) / (t_2 - Math.pow(B_m, 2.0));
	double tmp;
	if (t_3 <= 5e-72) {
		tmp = Math.sqrt((t_0 / t_1)) * (0.0 - Math.pow((2.0 * F), 0.5));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((F * t_1)) / (((A * (4.0 * C)) - (B_m * B_m)) / Math.sqrt((2.0 * t_0)));
	} else {
		tmp = ((0.0 - Math.sqrt(2.0)) / B_m) * Math.sqrt((F * (A + Math.hypot(B_m, A))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (A + C) + math.hypot(B_m, (A - C))
	t_1 = (B_m * B_m) + (-4.0 * (A * C))
	t_2 = (4.0 * A) * C
	t_3 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_2) * F)) * ((A + C) + math.sqrt((math.pow(B_m, 2.0) + math.pow((A - C), 2.0)))))) / (t_2 - math.pow(B_m, 2.0))
	tmp = 0
	if t_3 <= 5e-72:
		tmp = math.sqrt((t_0 / t_1)) * (0.0 - math.pow((2.0 * F), 0.5))
	elif t_3 <= math.inf:
		tmp = math.sqrt((F * t_1)) / (((A * (4.0 * C)) - (B_m * B_m)) / math.sqrt((2.0 * t_0)))
	else:
		tmp = ((0.0 - math.sqrt(2.0)) / B_m) * math.sqrt((F * (A + math.hypot(B_m, A))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(A + C) + hypot(B_m, Float64(A - C)))
	t_1 = Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C)))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= 5e-72)
		tmp = Float64(sqrt(Float64(t_0 / t_1)) * Float64(0.0 - (Float64(2.0 * F) ^ 0.5)));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(F * t_1)) / Float64(Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m)) / sqrt(Float64(2.0 * t_0))));
	else
		tmp = Float64(Float64(Float64(0.0 - sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(A + hypot(B_m, A)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (A + C) + hypot(B_m, (A - C));
	t_1 = (B_m * B_m) + (-4.0 * (A * C));
	t_2 = (4.0 * A) * C;
	t_3 = sqrt(((2.0 * (((B_m ^ 2.0) - t_2) * F)) * ((A + C) + sqrt(((B_m ^ 2.0) + ((A - C) ^ 2.0)))))) / (t_2 - (B_m ^ 2.0));
	tmp = 0.0;
	if (t_3 <= 5e-72)
		tmp = sqrt((t_0 / t_1)) * (0.0 - ((2.0 * F) ^ 0.5));
	elseif (t_3 <= Inf)
		tmp = sqrt((F * t_1)) / (((A * (4.0 * C)) - (B_m * B_m)) / sqrt((2.0 * t_0)));
	else
		tmp = ((0.0 - sqrt(2.0)) / B_m) * sqrt((F * (A + hypot(B_m, A))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-72], N[(N[Sqrt[N[(t$95$0 / t$95$1), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 4.9999999999999996e-72

   1. Initial program 32.7%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \left(\sqrt{2}\right)\right)\right) \]

   5. Simplified 57.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\right)\right) \]

      2. sqrt-prod N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \left(\sqrt{F} \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left({2}^{\frac{1}{2}} \cdot \sqrt{F}\right) \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left({2}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}\right) \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      7. unpow-prod-down N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}}\right), \left({\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right)\right) \]

   7. Applied egg-rr 68.6%

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

   if 4.9999999999999996e-72 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 39.8%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 60.9%

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

   6. Applied egg-rr 72.5%

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

      1. associate-/l* N/A

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

      2. pow1/2 N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left({F}^{\frac{1}{2}} \cdot \frac{\sqrt{2 \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\right)}\right) \]

   8. Applied egg-rr 0.0%

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

   10. Applied egg-rr 72.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 19.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right)\right) \]

   5. Simplified 19.8%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 5 \cdot 10^{-72}:\\ \;\;\;\;\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \left(0 - {\left(2 \cdot F\right)}^{0.5}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}{\frac{A \cdot \left(4 \cdot C\right) - B \cdot B}{\sqrt{2 \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - \sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 45.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 0 - \sqrt{2}\\ t_1 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_2 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B\_m \leq 2 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{F \cdot t\_2} \cdot \sqrt{2 \cdot \left(A + \left(C + t\_1\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{\left(A + C\right) + t\_1}{B\_m \cdot B\_m + t\_2}} \cdot \left(0 - {\left(2 \cdot F\right)}^{0.5}\right)\\ \mathbf{elif}\;B\_m \leq 7.2 \cdot 10^{+272}:\\ \;\;\;\;\frac{t\_0}{B\_m} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- 0.0 (sqrt 2.0)))
        (t_1 (hypot B_m (- A C)))
        (t_2 (* -4.0 (* A C))))
   (if (<= B_m 2e-153)
     (/
      (* (sqrt (* F t_2)) (sqrt (* 2.0 (+ A (+ C t_1)))))
      (- (* (* 4.0 A) C) (* B_m B_m)))
     (if (<= B_m 1.35e+154)
       (*
        (sqrt (/ (+ (+ A C) t_1) (+ (* B_m B_m) t_2)))
        (- 0.0 (pow (* 2.0 F) 0.5)))
       (if (<= B_m 7.2e+272)
         (* (/ t_0 B_m) (sqrt (* F (+ A (hypot B_m A)))))
         (* (sqrt (/ F B_m)) t_0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - sqrt(2.0);
	double t_1 = hypot(B_m, (A - C));
	double t_2 = -4.0 * (A * C);
	double tmp;
	if (B_m <= 2e-153) {
		tmp = (sqrt((F * t_2)) * sqrt((2.0 * (A + (C + t_1))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 1.35e+154) {
		tmp = sqrt((((A + C) + t_1) / ((B_m * B_m) + t_2))) * (0.0 - pow((2.0 * F), 0.5));
	} else if (B_m <= 7.2e+272) {
		tmp = (t_0 / B_m) * sqrt((F * (A + hypot(B_m, A))));
	} else {
		tmp = sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - Math.sqrt(2.0);
	double t_1 = Math.hypot(B_m, (A - C));
	double t_2 = -4.0 * (A * C);
	double tmp;
	if (B_m <= 2e-153) {
		tmp = (Math.sqrt((F * t_2)) * Math.sqrt((2.0 * (A + (C + t_1))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 1.35e+154) {
		tmp = Math.sqrt((((A + C) + t_1) / ((B_m * B_m) + t_2))) * (0.0 - Math.pow((2.0 * F), 0.5));
	} else if (B_m <= 7.2e+272) {
		tmp = (t_0 / B_m) * Math.sqrt((F * (A + Math.hypot(B_m, A))));
	} else {
		tmp = Math.sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = 0.0 - math.sqrt(2.0)
	t_1 = math.hypot(B_m, (A - C))
	t_2 = -4.0 * (A * C)
	tmp = 0
	if B_m <= 2e-153:
		tmp = (math.sqrt((F * t_2)) * math.sqrt((2.0 * (A + (C + t_1))))) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 1.35e+154:
		tmp = math.sqrt((((A + C) + t_1) / ((B_m * B_m) + t_2))) * (0.0 - math.pow((2.0 * F), 0.5))
	elif B_m <= 7.2e+272:
		tmp = (t_0 / B_m) * math.sqrt((F * (A + math.hypot(B_m, A))))
	else:
		tmp = math.sqrt((F / B_m)) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(0.0 - sqrt(2.0))
	t_1 = hypot(B_m, Float64(A - C))
	t_2 = Float64(-4.0 * Float64(A * C))
	tmp = 0.0
	if (B_m <= 2e-153)
		tmp = Float64(Float64(sqrt(Float64(F * t_2)) * sqrt(Float64(2.0 * Float64(A + Float64(C + t_1))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 1.35e+154)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + t_1) / Float64(Float64(B_m * B_m) + t_2))) * Float64(0.0 - (Float64(2.0 * F) ^ 0.5)));
	elseif (B_m <= 7.2e+272)
		tmp = Float64(Float64(t_0 / B_m) * sqrt(Float64(F * Float64(A + hypot(B_m, A)))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = 0.0 - sqrt(2.0);
	t_1 = hypot(B_m, (A - C));
	t_2 = -4.0 * (A * C);
	tmp = 0.0;
	if (B_m <= 2e-153)
		tmp = (sqrt((F * t_2)) * sqrt((2.0 * (A + (C + t_1))))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 1.35e+154)
		tmp = sqrt((((A + C) + t_1) / ((B_m * B_m) + t_2))) * (0.0 - ((2.0 * F) ^ 0.5));
	elseif (B_m <= 7.2e+272)
		tmp = (t_0 / B_m) * sqrt((F * (A + hypot(B_m, A))));
	else
		tmp = sqrt((F / B_m)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2e-153], N[(N[(N[Sqrt[N[(F * t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(A + N[(C + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.35e+154], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + t$95$1), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7.2e+272], N[(N[(t$95$0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 2.00000000000000008e-153

   1. Initial program 21.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 28.7%

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

   6. Applied egg-rr 38.6%

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

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-4 \cdot \left(A \cdot C\right)\right)}, F\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(A \cdot C\right)\right), F\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 25.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right), F\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 25.5%

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

   if 2.00000000000000008e-153 < B < 1.35000000000000003e154

   1. Initial program 29.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \left(\sqrt{2}\right)\right)\right) \]

   5. Simplified 54.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\right)\right) \]

      2. sqrt-prod N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \left(\sqrt{F} \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left({2}^{\frac{1}{2}} \cdot \sqrt{F}\right) \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left({2}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}\right) \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      7. unpow-prod-down N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}}\right), \left({\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right)\right) \]

   7. Applied egg-rr 59.2%

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

   if 1.35000000000000003e154 < B < 7.1999999999999995e272

   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

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 54.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right)\right) \]

   5. Simplified 54.5%

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

   if 7.1999999999999995e272 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left(\sqrt{2}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 77.0%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 77.0%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \left(0 - {\left(2 \cdot F\right)}^{0.5}\right)\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+272}:\\ \;\;\;\;\frac{0 - \sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 0 - \sqrt{2}\\ \mathbf{if}\;B\_m \leq 4.1 \cdot 10^{-176}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B\_m \cdot B\_m\right)}{C}}\right)\\ \mathbf{elif}\;B\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)}} \cdot \left(0 - {\left(2 \cdot F\right)}^{0.5}\right)\\ \mathbf{elif}\;B\_m \leq 2.55 \cdot 10^{+272}:\\ \;\;\;\;\frac{t\_0}{B\_m} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- 0.0 (sqrt 2.0))))
   (if (<= B_m 4.1e-176)
     (*
      0.25
      (*
       (/ 1.0 C)
       (sqrt (+ (* -16.0 (* C F)) (* 4.0 (/ (* F (* B_m B_m)) C))))))
     (if (<= B_m 1.35e+154)
       (*
        (sqrt
         (/ (+ (+ A C) (hypot B_m (- A C))) (+ (* B_m B_m) (* -4.0 (* A C)))))
        (- 0.0 (pow (* 2.0 F) 0.5)))
       (if (<= B_m 2.55e+272)
         (* (/ t_0 B_m) (sqrt (* F (+ A (hypot B_m A)))))
         (* (sqrt (/ F B_m)) t_0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - sqrt(2.0);
	double tmp;
	if (B_m <= 4.1e-176) {
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	} else if (B_m <= 1.35e+154) {
		tmp = sqrt((((A + C) + hypot(B_m, (A - C))) / ((B_m * B_m) + (-4.0 * (A * C))))) * (0.0 - pow((2.0 * F), 0.5));
	} else if (B_m <= 2.55e+272) {
		tmp = (t_0 / B_m) * sqrt((F * (A + hypot(B_m, A))));
	} else {
		tmp = sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - Math.sqrt(2.0);
	double tmp;
	if (B_m <= 4.1e-176) {
		tmp = 0.25 * ((1.0 / C) * Math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	} else if (B_m <= 1.35e+154) {
		tmp = Math.sqrt((((A + C) + Math.hypot(B_m, (A - C))) / ((B_m * B_m) + (-4.0 * (A * C))))) * (0.0 - Math.pow((2.0 * F), 0.5));
	} else if (B_m <= 2.55e+272) {
		tmp = (t_0 / B_m) * Math.sqrt((F * (A + Math.hypot(B_m, A))));
	} else {
		tmp = Math.sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = 0.0 - math.sqrt(2.0)
	tmp = 0
	if B_m <= 4.1e-176:
		tmp = 0.25 * ((1.0 / C) * math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))))
	elif B_m <= 1.35e+154:
		tmp = math.sqrt((((A + C) + math.hypot(B_m, (A - C))) / ((B_m * B_m) + (-4.0 * (A * C))))) * (0.0 - math.pow((2.0 * F), 0.5))
	elif B_m <= 2.55e+272:
		tmp = (t_0 / B_m) * math.sqrt((F * (A + math.hypot(B_m, A))))
	else:
		tmp = math.sqrt((F / B_m)) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(0.0 - sqrt(2.0))
	tmp = 0.0
	if (B_m <= 4.1e-176)
		tmp = Float64(0.25 * Float64(Float64(1.0 / C) * sqrt(Float64(Float64(-16.0 * Float64(C * F)) + Float64(4.0 * Float64(Float64(F * Float64(B_m * B_m)) / C))))));
	elseif (B_m <= 1.35e+154)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))))) * Float64(0.0 - (Float64(2.0 * F) ^ 0.5)));
	elseif (B_m <= 2.55e+272)
		tmp = Float64(Float64(t_0 / B_m) * sqrt(Float64(F * Float64(A + hypot(B_m, A)))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = 0.0 - sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 4.1e-176)
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	elseif (B_m <= 1.35e+154)
		tmp = sqrt((((A + C) + hypot(B_m, (A - C))) / ((B_m * B_m) + (-4.0 * (A * C))))) * (0.0 - ((2.0 * F) ^ 0.5));
	elseif (B_m <= 2.55e+272)
		tmp = (t_0 / B_m) * sqrt((F * (A + hypot(B_m, A))));
	else
		tmp = sqrt((F / B_m)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.1e-176], N[(0.25 * N[(N[(1.0 / C), $MachinePrecision] * N[Sqrt[N[(N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.35e+154], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.55e+272], N[(N[(t$95$0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 4.1000000000000003e-176

   1. Initial program 22.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 29.1%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 15.6%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(\frac{-2 \cdot \left(A \cdot C\right)}{A - C} + 2 \cdot A\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(0.5 \cdot \left(\frac{A \cdot C}{\left(A - C\right) \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)} + \frac{1}{A - C}\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \color{blue}{\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C}\right), \left(8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 10.4%

       \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \color{blue}{\left(2 \cdot \frac{\left(B \cdot B\right) \cdot \left(-0.5 \cdot \left(\left(B \cdot B\right) \cdot F\right) + 2 \cdot \left(\left(A \cdot A\right) \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around inf

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{1}{C}\right), \color{blue}{\left(\sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \left(\sqrt{\color{blue}{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}}\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\left(-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \left(\frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), C\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 20.0%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

   13. Simplified 20.0%

       \[\leadsto \color{blue}{0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{\left(B \cdot B\right) \cdot F}{C}}\right)} \]

   if 4.1000000000000003e-176 < B < 1.35000000000000003e154

   1. Initial program 26.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \left(\sqrt{2}\right)\right)\right) \]

   5. Simplified 51.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\right)\right) \]

      2. sqrt-prod N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \left(\sqrt{F} \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left({2}^{\frac{1}{2}} \cdot \sqrt{F}\right) \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left({2}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}\right) \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      7. unpow-prod-down N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}}\right), \left({\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right)\right) \]

   7. Applied egg-rr 55.6%

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

   if 1.35000000000000003e154 < B < 2.54999999999999993e272

   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

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 54.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right)\right) \]

   5. Simplified 54.5%

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

   if 2.54999999999999993e272 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left(\sqrt{2}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 77.0%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 77.0%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.1 \cdot 10^{-176}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B \cdot B\right)}{C}}\right)\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \left(0 - {\left(2 \cdot F\right)}^{0.5}\right)\\ \mathbf{elif}\;B \leq 2.55 \cdot 10^{+272}:\\ \;\;\;\;\frac{0 - \sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 0 - \sqrt{2}\\ \mathbf{if}\;B\_m \leq 9 \cdot 10^{-176}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B\_m \cdot B\_m\right)}{C}}\right)\\ \mathbf{elif}\;B\_m \leq 8 \cdot 10^{+150}:\\ \;\;\;\;{\left(2 \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)}\right)}^{0.5} \cdot \left(0 - \sqrt{F}\right)\\ \mathbf{elif}\;B\_m \leq 1.3 \cdot 10^{+273}:\\ \;\;\;\;\frac{t\_0}{B\_m} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- 0.0 (sqrt 2.0))))
   (if (<= B_m 9e-176)
     (*
      0.25
      (*
       (/ 1.0 C)
       (sqrt (+ (* -16.0 (* C F)) (* 4.0 (/ (* F (* B_m B_m)) C))))))
     (if (<= B_m 8e+150)
       (*
        (pow
         (*
          2.0
          (/ (+ (+ A C) (hypot B_m (- A C))) (+ (* B_m B_m) (* -4.0 (* A C)))))
         0.5)
        (- 0.0 (sqrt F)))
       (if (<= B_m 1.3e+273)
         (* (/ t_0 B_m) (sqrt (* F (+ A (hypot B_m A)))))
         (* (sqrt (/ F B_m)) t_0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - sqrt(2.0);
	double tmp;
	if (B_m <= 9e-176) {
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	} else if (B_m <= 8e+150) {
		tmp = pow((2.0 * (((A + C) + hypot(B_m, (A - C))) / ((B_m * B_m) + (-4.0 * (A * C))))), 0.5) * (0.0 - sqrt(F));
	} else if (B_m <= 1.3e+273) {
		tmp = (t_0 / B_m) * sqrt((F * (A + hypot(B_m, A))));
	} else {
		tmp = sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - Math.sqrt(2.0);
	double tmp;
	if (B_m <= 9e-176) {
		tmp = 0.25 * ((1.0 / C) * Math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	} else if (B_m <= 8e+150) {
		tmp = Math.pow((2.0 * (((A + C) + Math.hypot(B_m, (A - C))) / ((B_m * B_m) + (-4.0 * (A * C))))), 0.5) * (0.0 - Math.sqrt(F));
	} else if (B_m <= 1.3e+273) {
		tmp = (t_0 / B_m) * Math.sqrt((F * (A + Math.hypot(B_m, A))));
	} else {
		tmp = Math.sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = 0.0 - math.sqrt(2.0)
	tmp = 0
	if B_m <= 9e-176:
		tmp = 0.25 * ((1.0 / C) * math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))))
	elif B_m <= 8e+150:
		tmp = math.pow((2.0 * (((A + C) + math.hypot(B_m, (A - C))) / ((B_m * B_m) + (-4.0 * (A * C))))), 0.5) * (0.0 - math.sqrt(F))
	elif B_m <= 1.3e+273:
		tmp = (t_0 / B_m) * math.sqrt((F * (A + math.hypot(B_m, A))))
	else:
		tmp = math.sqrt((F / B_m)) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(0.0 - sqrt(2.0))
	tmp = 0.0
	if (B_m <= 9e-176)
		tmp = Float64(0.25 * Float64(Float64(1.0 / C) * sqrt(Float64(Float64(-16.0 * Float64(C * F)) + Float64(4.0 * Float64(Float64(F * Float64(B_m * B_m)) / C))))));
	elseif (B_m <= 8e+150)
		tmp = Float64((Float64(2.0 * Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))))) ^ 0.5) * Float64(0.0 - sqrt(F)));
	elseif (B_m <= 1.3e+273)
		tmp = Float64(Float64(t_0 / B_m) * sqrt(Float64(F * Float64(A + hypot(B_m, A)))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = 0.0 - sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 9e-176)
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	elseif (B_m <= 8e+150)
		tmp = ((2.0 * (((A + C) + hypot(B_m, (A - C))) / ((B_m * B_m) + (-4.0 * (A * C))))) ^ 0.5) * (0.0 - sqrt(F));
	elseif (B_m <= 1.3e+273)
		tmp = (t_0 / B_m) * sqrt((F * (A + hypot(B_m, A))));
	else
		tmp = sqrt((F / B_m)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 9e-176], N[(0.25 * N[(N[(1.0 / C), $MachinePrecision] * N[Sqrt[N[(N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 8e+150], N[(N[Power[N[(2.0 * N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[(0.0 - N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.3e+273], N[(N[(t$95$0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 9e-176

   1. Initial program 22.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 29.1%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 15.6%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(\frac{-2 \cdot \left(A \cdot C\right)}{A - C} + 2 \cdot A\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(0.5 \cdot \left(\frac{A \cdot C}{\left(A - C\right) \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)} + \frac{1}{A - C}\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \color{blue}{\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C}\right), \left(8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 10.4%

       \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \color{blue}{\left(2 \cdot \frac{\left(B \cdot B\right) \cdot \left(-0.5 \cdot \left(\left(B \cdot B\right) \cdot F\right) + 2 \cdot \left(\left(A \cdot A\right) \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around inf

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{1}{C}\right), \color{blue}{\left(\sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \left(\sqrt{\color{blue}{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}}\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\left(-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \left(\frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), C\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 20.0%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

   13. Simplified 20.0%

       \[\leadsto \color{blue}{0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{\left(B \cdot B\right) \cdot F}{C}}\right)} \]

   if 9e-176 < B < 7.99999999999999985e150

   1. Initial program 27.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \left(\sqrt{2}\right)\right)\right) \]

   5. Simplified 50.9%

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

      1. sqrt-unprod N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\left(F \cdot \frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right) \cdot 2}\right)\right) \]

      2. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({\left(\left(F \cdot \frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right) \cdot 2\right)}^{\frac{1}{2}}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({\left(F \cdot \left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \cdot 2\right)\right)}^{\frac{1}{2}}\right)\right) \]

      4. unpow-prod-down N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({F}^{\frac{1}{2}} \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \cdot 2\right)}^{\frac{1}{2}}\right)\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{F} \cdot {\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \cdot 2\right)}^{\frac{1}{2}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{F}\right), \left({\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \cdot 2\right)}^{\frac{1}{2}}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \left({\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \cdot 2\right)}^{\frac{1}{2}}\right)\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(F\right), \mathsf{pow.f64}\left(\left(\frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \cdot 2\right), \frac{1}{2}\right)\right)\right) \]

   7. Applied egg-rr 55.0%

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

   if 7.99999999999999985e150 < B < 1.29999999999999997e273

   1. Initial program 0.1%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 55.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right)\right) \]

   5. Simplified 55.9%

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

   if 1.29999999999999997e273 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left(\sqrt{2}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 77.0%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 77.0%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-176}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B \cdot B\right)}{C}}\right)\\ \mathbf{elif}\;B \leq 8 \cdot 10^{+150}:\\ \;\;\;\;{\left(2 \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{0.5} \cdot \left(0 - \sqrt{F}\right)\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+273}:\\ \;\;\;\;\frac{0 - \sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 0 - \sqrt{2}\\ \mathbf{if}\;B\_m \leq 1.5 \cdot 10^{-146}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B\_m \cdot B\_m\right)}{C}}\right)\\ \mathbf{elif}\;B\_m \leq 3.7 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)}} \cdot t\_0\\ \mathbf{elif}\;B\_m \leq 5.1 \cdot 10^{+272}:\\ \;\;\;\;\frac{t\_0}{B\_m} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- 0.0 (sqrt 2.0))))
   (if (<= B_m 1.5e-146)
     (*
      0.25
      (*
       (/ 1.0 C)
       (sqrt (+ (* -16.0 (* C F)) (* 4.0 (/ (* F (* B_m B_m)) C))))))
     (if (<= B_m 3.7e+146)
       (*
        (sqrt
         (*
          F
          (/
           (+ (+ A C) (hypot B_m (- A C)))
           (+ (* B_m B_m) (* -4.0 (* A C))))))
        t_0)
       (if (<= B_m 5.1e+272)
         (* (/ t_0 B_m) (sqrt (* F (+ A (hypot B_m A)))))
         (* (sqrt (/ F B_m)) t_0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - sqrt(2.0);
	double tmp;
	if (B_m <= 1.5e-146) {
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	} else if (B_m <= 3.7e+146) {
		tmp = sqrt((F * (((A + C) + hypot(B_m, (A - C))) / ((B_m * B_m) + (-4.0 * (A * C)))))) * t_0;
	} else if (B_m <= 5.1e+272) {
		tmp = (t_0 / B_m) * sqrt((F * (A + hypot(B_m, A))));
	} else {
		tmp = sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - Math.sqrt(2.0);
	double tmp;
	if (B_m <= 1.5e-146) {
		tmp = 0.25 * ((1.0 / C) * Math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	} else if (B_m <= 3.7e+146) {
		tmp = Math.sqrt((F * (((A + C) + Math.hypot(B_m, (A - C))) / ((B_m * B_m) + (-4.0 * (A * C)))))) * t_0;
	} else if (B_m <= 5.1e+272) {
		tmp = (t_0 / B_m) * Math.sqrt((F * (A + Math.hypot(B_m, A))));
	} else {
		tmp = Math.sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = 0.0 - math.sqrt(2.0)
	tmp = 0
	if B_m <= 1.5e-146:
		tmp = 0.25 * ((1.0 / C) * math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))))
	elif B_m <= 3.7e+146:
		tmp = math.sqrt((F * (((A + C) + math.hypot(B_m, (A - C))) / ((B_m * B_m) + (-4.0 * (A * C)))))) * t_0
	elif B_m <= 5.1e+272:
		tmp = (t_0 / B_m) * math.sqrt((F * (A + math.hypot(B_m, A))))
	else:
		tmp = math.sqrt((F / B_m)) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(0.0 - sqrt(2.0))
	tmp = 0.0
	if (B_m <= 1.5e-146)
		tmp = Float64(0.25 * Float64(Float64(1.0 / C) * sqrt(Float64(Float64(-16.0 * Float64(C * F)) + Float64(4.0 * Float64(Float64(F * Float64(B_m * B_m)) / C))))));
	elseif (B_m <= 3.7e+146)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C)))))) * t_0);
	elseif (B_m <= 5.1e+272)
		tmp = Float64(Float64(t_0 / B_m) * sqrt(Float64(F * Float64(A + hypot(B_m, A)))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = 0.0 - sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 1.5e-146)
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	elseif (B_m <= 3.7e+146)
		tmp = sqrt((F * (((A + C) + hypot(B_m, (A - C))) / ((B_m * B_m) + (-4.0 * (A * C)))))) * t_0;
	elseif (B_m <= 5.1e+272)
		tmp = (t_0 / B_m) * sqrt((F * (A + hypot(B_m, A))));
	else
		tmp = sqrt((F / B_m)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.5e-146], N[(0.25 * N[(N[(1.0 / C), $MachinePrecision] * N[Sqrt[N[(N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 3.7e+146], N[(N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 5.1e+272], N[(N[(t$95$0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.50000000000000009e-146

   1. Initial program 21.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 28.7%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 15.2%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(\frac{-2 \cdot \left(A \cdot C\right)}{A - C} + 2 \cdot A\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(0.5 \cdot \left(\frac{A \cdot C}{\left(A - C\right) \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)} + \frac{1}{A - C}\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \color{blue}{\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C}\right), \left(8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 10.1%

       \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \color{blue}{\left(2 \cdot \frac{\left(B \cdot B\right) \cdot \left(-0.5 \cdot \left(\left(B \cdot B\right) \cdot F\right) + 2 \cdot \left(\left(A \cdot A\right) \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around inf

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{1}{C}\right), \color{blue}{\left(\sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \left(\sqrt{\color{blue}{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}}\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\left(-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \left(\frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), C\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 19.3%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

   13. Simplified 19.3%

       \[\leadsto \color{blue}{0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{\left(B \cdot B\right) \cdot F}{C}}\right)} \]

   if 1.50000000000000009e-146 < B < 3.70000000000000004e146

   1. Initial program 29.4%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \left(\sqrt{2}\right)\right)\right) \]

   5. Simplified 54.0%

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

   if 3.70000000000000004e146 < B < 5.09999999999999985e272

   1. Initial program 0.1%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 55.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right)\right) \]

   5. Simplified 55.9%

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

   if 5.09999999999999985e272 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left(\sqrt{2}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 77.0%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 77.0%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.5 \cdot 10^{-146}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B \cdot B\right)}{C}}\right)\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{+272}:\\ \;\;\;\;\frac{0 - \sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 0 - \sqrt{2}\\ \mathbf{if}\;B\_m \leq 8 \cdot 10^{-147}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B\_m \cdot B\_m\right)}{C}}\right)\\ \mathbf{elif}\;B\_m \leq 7.1 \cdot 10^{+103}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)}}\\ \mathbf{elif}\;B\_m \leq 2.4 \cdot 10^{+273}:\\ \;\;\;\;\frac{t\_0}{B\_m} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- 0.0 (sqrt 2.0))))
   (if (<= B_m 8e-147)
     (*
      0.25
      (*
       (/ 1.0 C)
       (sqrt (+ (* -16.0 (* C F)) (* 4.0 (/ (* F (* B_m B_m)) C))))))
     (if (<= B_m 7.1e+103)
       (-
        0.0
        (sqrt
         (/
          (* 2.0 (* F (+ (+ A C) (hypot B_m (- A C)))))
          (+ (* B_m B_m) (* -4.0 (* A C))))))
       (if (<= B_m 2.4e+273)
         (* (/ t_0 B_m) (sqrt (* F (+ A (hypot B_m A)))))
         (* (sqrt (/ F B_m)) t_0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - sqrt(2.0);
	double tmp;
	if (B_m <= 8e-147) {
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	} else if (B_m <= 7.1e+103) {
		tmp = 0.0 - sqrt(((2.0 * (F * ((A + C) + hypot(B_m, (A - C))))) / ((B_m * B_m) + (-4.0 * (A * C)))));
	} else if (B_m <= 2.4e+273) {
		tmp = (t_0 / B_m) * sqrt((F * (A + hypot(B_m, A))));
	} else {
		tmp = sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - Math.sqrt(2.0);
	double tmp;
	if (B_m <= 8e-147) {
		tmp = 0.25 * ((1.0 / C) * Math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	} else if (B_m <= 7.1e+103) {
		tmp = 0.0 - Math.sqrt(((2.0 * (F * ((A + C) + Math.hypot(B_m, (A - C))))) / ((B_m * B_m) + (-4.0 * (A * C)))));
	} else if (B_m <= 2.4e+273) {
		tmp = (t_0 / B_m) * Math.sqrt((F * (A + Math.hypot(B_m, A))));
	} else {
		tmp = Math.sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = 0.0 - math.sqrt(2.0)
	tmp = 0
	if B_m <= 8e-147:
		tmp = 0.25 * ((1.0 / C) * math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))))
	elif B_m <= 7.1e+103:
		tmp = 0.0 - math.sqrt(((2.0 * (F * ((A + C) + math.hypot(B_m, (A - C))))) / ((B_m * B_m) + (-4.0 * (A * C)))))
	elif B_m <= 2.4e+273:
		tmp = (t_0 / B_m) * math.sqrt((F * (A + math.hypot(B_m, A))))
	else:
		tmp = math.sqrt((F / B_m)) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(0.0 - sqrt(2.0))
	tmp = 0.0
	if (B_m <= 8e-147)
		tmp = Float64(0.25 * Float64(Float64(1.0 / C) * sqrt(Float64(Float64(-16.0 * Float64(C * F)) + Float64(4.0 * Float64(Float64(F * Float64(B_m * B_m)) / C))))));
	elseif (B_m <= 7.1e+103)
		tmp = Float64(0.0 - sqrt(Float64(Float64(2.0 * Float64(F * Float64(Float64(A + C) + hypot(B_m, Float64(A - C))))) / Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))))));
	elseif (B_m <= 2.4e+273)
		tmp = Float64(Float64(t_0 / B_m) * sqrt(Float64(F * Float64(A + hypot(B_m, A)))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = 0.0 - sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 8e-147)
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	elseif (B_m <= 7.1e+103)
		tmp = 0.0 - sqrt(((2.0 * (F * ((A + C) + hypot(B_m, (A - C))))) / ((B_m * B_m) + (-4.0 * (A * C)))));
	elseif (B_m <= 2.4e+273)
		tmp = (t_0 / B_m) * sqrt((F * (A + hypot(B_m, A))));
	else
		tmp = sqrt((F / B_m)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 8e-147], N[(0.25 * N[(N[(1.0 / C), $MachinePrecision] * N[Sqrt[N[(N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7.1e+103], N[(0.0 - N[Sqrt[N[(N[(2.0 * N[(F * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.4e+273], N[(N[(t$95$0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 7.9999999999999998e-147

   1. Initial program 21.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 28.7%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 15.2%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(\frac{-2 \cdot \left(A \cdot C\right)}{A - C} + 2 \cdot A\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(0.5 \cdot \left(\frac{A \cdot C}{\left(A - C\right) \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)} + \frac{1}{A - C}\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \color{blue}{\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C}\right), \left(8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 10.1%

       \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \color{blue}{\left(2 \cdot \frac{\left(B \cdot B\right) \cdot \left(-0.5 \cdot \left(\left(B \cdot B\right) \cdot F\right) + 2 \cdot \left(\left(A \cdot A\right) \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around inf

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{1}{C}\right), \color{blue}{\left(\sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \left(\sqrt{\color{blue}{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}}\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\left(-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \left(\frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), C\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 19.3%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

   13. Simplified 19.3%

       \[\leadsto \color{blue}{0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{\left(B \cdot B\right) \cdot F}{C}}\right)} \]

   if 7.9999999999999998e-147 < B < 7.1000000000000002e103

   1. Initial program 32.7%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \left(\sqrt{2}\right)\right)\right) \]

   5. Simplified 50.6%

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

      1. sqrt-unprod N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\left(F \cdot \frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right) \cdot 2}\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(F \cdot \frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right) \cdot 2\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F \cdot \left(\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \cdot 2\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(F \cdot \left(\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right) \cdot 2}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(F \cdot \left(\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right) \cdot 2\right), \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 45.2%

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

   if 7.1000000000000002e103 < B < 2.4000000000000002e273

   1. Initial program 3.2%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 52.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right)\right) \]

   5. Simplified 52.8%

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

   if 2.4000000000000002e273 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left(\sqrt{2}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 77.0%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 77.0%

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

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8 \cdot 10^{-147}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B \cdot B\right)}{C}}\right)\\ \mathbf{elif}\;B \leq 7.1 \cdot 10^{+103}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{+273}:\\ \;\;\;\;\frac{0 - \sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 9.6 \cdot 10^{-147}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B\_m \cdot B\_m\right)}{C}}\right)\\ \mathbf{elif}\;B\_m \leq 4.1 \cdot 10^{+104}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 9.6e-147)
   (*
    0.25
    (* (/ 1.0 C) (sqrt (+ (* -16.0 (* C F)) (* 4.0 (/ (* F (* B_m B_m)) C))))))
   (if (<= B_m 4.1e+104)
     (-
      0.0
      (sqrt
       (/
        (* 2.0 (* F (+ (+ A C) (hypot B_m (- A C)))))
        (+ (* B_m B_m) (* -4.0 (* A C))))))
     (* (sqrt (/ F B_m)) (- 0.0 (sqrt 2.0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9.6e-147) {
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	} else if (B_m <= 4.1e+104) {
		tmp = 0.0 - sqrt(((2.0 * (F * ((A + C) + hypot(B_m, (A - C))))) / ((B_m * B_m) + (-4.0 * (A * C)))));
	} else {
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9.6e-147) {
		tmp = 0.25 * ((1.0 / C) * Math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	} else if (B_m <= 4.1e+104) {
		tmp = 0.0 - Math.sqrt(((2.0 * (F * ((A + C) + Math.hypot(B_m, (A - C))))) / ((B_m * B_m) + (-4.0 * (A * C)))));
	} else {
		tmp = Math.sqrt((F / B_m)) * (0.0 - Math.sqrt(2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 9.6e-147:
		tmp = 0.25 * ((1.0 / C) * math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))))
	elif B_m <= 4.1e+104:
		tmp = 0.0 - math.sqrt(((2.0 * (F * ((A + C) + math.hypot(B_m, (A - C))))) / ((B_m * B_m) + (-4.0 * (A * C)))))
	else:
		tmp = math.sqrt((F / B_m)) * (0.0 - math.sqrt(2.0))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 9.6e-147)
		tmp = Float64(0.25 * Float64(Float64(1.0 / C) * sqrt(Float64(Float64(-16.0 * Float64(C * F)) + Float64(4.0 * Float64(Float64(F * Float64(B_m * B_m)) / C))))));
	elseif (B_m <= 4.1e+104)
		tmp = Float64(0.0 - sqrt(Float64(Float64(2.0 * Float64(F * Float64(Float64(A + C) + hypot(B_m, Float64(A - C))))) / Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))))));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(0.0 - sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 9.6e-147)
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	elseif (B_m <= 4.1e+104)
		tmp = 0.0 - sqrt(((2.0 * (F * ((A + C) + hypot(B_m, (A - C))))) / ((B_m * B_m) + (-4.0 * (A * C)))));
	else
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 9.6e-147], N[(0.25 * N[(N[(1.0 / C), $MachinePrecision] * N[Sqrt[N[(N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.1e+104], N[(0.0 - N[Sqrt[N[(N[(2.0 * N[(F * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 9.59999999999999994e-147

   1. Initial program 21.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 28.7%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 15.2%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(\frac{-2 \cdot \left(A \cdot C\right)}{A - C} + 2 \cdot A\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(0.5 \cdot \left(\frac{A \cdot C}{\left(A - C\right) \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)} + \frac{1}{A - C}\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \color{blue}{\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C}\right), \left(8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 10.1%

       \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \color{blue}{\left(2 \cdot \frac{\left(B \cdot B\right) \cdot \left(-0.5 \cdot \left(\left(B \cdot B\right) \cdot F\right) + 2 \cdot \left(\left(A \cdot A\right) \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around inf

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{1}{C}\right), \color{blue}{\left(\sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \left(\sqrt{\color{blue}{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}}\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\left(-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \left(\frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), C\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 19.3%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

   13. Simplified 19.3%

       \[\leadsto \color{blue}{0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{\left(B \cdot B\right) \cdot F}{C}}\right)} \]

   if 9.59999999999999994e-147 < B < 4.09999999999999985e104

   1. Initial program 32.7%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \left(\sqrt{2}\right)\right)\right) \]

   5. Simplified 50.6%

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

      1. sqrt-unprod N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\left(F \cdot \frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right) \cdot 2}\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(F \cdot \frac{\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right) \cdot 2\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F \cdot \left(\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \cdot 2\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(F \cdot \left(\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right) \cdot 2}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(F \cdot \left(\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right) \cdot 2\right), \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 45.2%

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

   if 4.09999999999999985e104 < B

   1. Initial program 2.7%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left(\sqrt{2}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 59.8%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 59.8%

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

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.6 \cdot 10^{-147}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B \cdot B\right)}{C}}\right)\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{+104}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 0 - \sqrt{2}\\ \mathbf{if}\;C \leq -1.25 \cdot 10^{-62}:\\ \;\;\;\;0 - \sqrt{2} \cdot \sqrt{F \cdot \frac{-0.5}{C}}\\ \mathbf{elif}\;C \leq 7.8 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + 1}{B\_m}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- 0.0 (sqrt 2.0))))
   (if (<= C -1.25e-62)
     (- 0.0 (* (sqrt 2.0) (sqrt (* F (/ -0.5 C)))))
     (if (<= C 7.8e-11)
       (* (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) 1.0) B_m))) t_0)
       (* (sqrt (* F (/ -0.5 A))) t_0)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - sqrt(2.0);
	double tmp;
	if (C <= -1.25e-62) {
		tmp = 0.0 - (sqrt(2.0) * sqrt((F * (-0.5 / C))));
	} else if (C <= 7.8e-11) {
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + 1.0) / B_m))) * t_0;
	} else {
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
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 = 0.0d0 - sqrt(2.0d0)
    if (c <= (-1.25d-62)) then
        tmp = 0.0d0 - (sqrt(2.0d0) * sqrt((f * ((-0.5d0) / c))))
    else if (c <= 7.8d-11) then
        tmp = sqrt((f * ((((a / b_m) + (c / b_m)) + 1.0d0) / b_m))) * t_0
    else
        tmp = sqrt((f * ((-0.5d0) / a))) * t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - Math.sqrt(2.0);
	double tmp;
	if (C <= -1.25e-62) {
		tmp = 0.0 - (Math.sqrt(2.0) * Math.sqrt((F * (-0.5 / C))));
	} else if (C <= 7.8e-11) {
		tmp = Math.sqrt((F * ((((A / B_m) + (C / B_m)) + 1.0) / B_m))) * t_0;
	} else {
		tmp = Math.sqrt((F * (-0.5 / A))) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = 0.0 - math.sqrt(2.0)
	tmp = 0
	if C <= -1.25e-62:
		tmp = 0.0 - (math.sqrt(2.0) * math.sqrt((F * (-0.5 / C))))
	elif C <= 7.8e-11:
		tmp = math.sqrt((F * ((((A / B_m) + (C / B_m)) + 1.0) / B_m))) * t_0
	else:
		tmp = math.sqrt((F * (-0.5 / A))) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(0.0 - sqrt(2.0))
	tmp = 0.0
	if (C <= -1.25e-62)
		tmp = Float64(0.0 - Float64(sqrt(2.0) * sqrt(Float64(F * Float64(-0.5 / C)))));
	elseif (C <= 7.8e-11)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + 1.0) / B_m))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = 0.0 - sqrt(2.0);
	tmp = 0.0;
	if (C <= -1.25e-62)
		tmp = 0.0 - (sqrt(2.0) * sqrt((F * (-0.5 / C))));
	elseif (C <= 7.8e-11)
		tmp = sqrt((F * ((((A / B_m) + (C / B_m)) + 1.0) / B_m))) * t_0;
	else
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -1.25e-62], N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 7.8e-11], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.25e-62

   1. Initial program 11.5%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \left(\sqrt{2}\right)\right)\right) \]

   5. Simplified 19.2%

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

   6. Taylor expanded in C around -inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\frac{-1}{2}}{C}\right)}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 47.5%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\frac{-1}{2}, C\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   8. Simplified 47.5%

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

   if -1.25e-62 < C < 7.80000000000000021e-11

   1. Initial program 25.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \left(\sqrt{2}\right)\right)\right) \]

   5. Simplified 36.7%

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

   6. Taylor expanded in B around inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{1 + \left(\frac{A}{B} + \frac{C}{B}\right)}{B}\right)}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(1 + \left(\frac{A}{B} + \frac{C}{B}\right)\right), B\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{A}{B} + \frac{C}{B}\right)\right), B\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{A}{B}\right), \left(\frac{C}{B}\right)\right)\right), B\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(A, B\right), \left(\frac{C}{B}\right)\right)\right), B\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      5. /-lowering-/.f64 28.1%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(A, B\right), \mathsf{/.f64}\left(C, B\right)\right)\right), B\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   8. Simplified 28.1%

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

   if 7.80000000000000021e-11 < C

   1. Initial program 19.8%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \left(\sqrt{2}\right)\right)\right) \]

   5. Simplified 40.0%

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

   6. Taylor expanded in C around inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\frac{-1}{2}}{A}\right)}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 32.5%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\frac{-1}{2}, A\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   8. Simplified 32.5%

      \[\leadsto -\sqrt{F \cdot \color{blue}{\frac{-0.5}{A}}} \cdot \sqrt{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.25 \cdot 10^{-62}:\\ \;\;\;\;0 - \sqrt{2} \cdot \sqrt{F \cdot \frac{-0.5}{C}}\\ \mathbf{elif}\;C \leq 7.8 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + 1}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 0 - \sqrt{2}\\ \mathbf{if}\;C \leq -9.2 \cdot 10^{-60}:\\ \;\;\;\;0 - \sqrt{2} \cdot \sqrt{F \cdot \frac{-0.5}{C}}\\ \mathbf{elif}\;C \leq 1.4 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- 0.0 (sqrt 2.0))))
   (if (<= C -9.2e-60)
     (- 0.0 (* (sqrt 2.0) (sqrt (* F (/ -0.5 C)))))
     (if (<= C 1.4e-12)
       (* (sqrt (/ F B_m)) t_0)
       (* (sqrt (* F (/ -0.5 A))) t_0)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - sqrt(2.0);
	double tmp;
	if (C <= -9.2e-60) {
		tmp = 0.0 - (sqrt(2.0) * sqrt((F * (-0.5 / C))));
	} else if (C <= 1.4e-12) {
		tmp = sqrt((F / B_m)) * t_0;
	} else {
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
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 = 0.0d0 - sqrt(2.0d0)
    if (c <= (-9.2d-60)) then
        tmp = 0.0d0 - (sqrt(2.0d0) * sqrt((f * ((-0.5d0) / c))))
    else if (c <= 1.4d-12) then
        tmp = sqrt((f / b_m)) * t_0
    else
        tmp = sqrt((f * ((-0.5d0) / a))) * t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - Math.sqrt(2.0);
	double tmp;
	if (C <= -9.2e-60) {
		tmp = 0.0 - (Math.sqrt(2.0) * Math.sqrt((F * (-0.5 / C))));
	} else if (C <= 1.4e-12) {
		tmp = Math.sqrt((F / B_m)) * t_0;
	} else {
		tmp = Math.sqrt((F * (-0.5 / A))) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = 0.0 - math.sqrt(2.0)
	tmp = 0
	if C <= -9.2e-60:
		tmp = 0.0 - (math.sqrt(2.0) * math.sqrt((F * (-0.5 / C))))
	elif C <= 1.4e-12:
		tmp = math.sqrt((F / B_m)) * t_0
	else:
		tmp = math.sqrt((F * (-0.5 / A))) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(0.0 - sqrt(2.0))
	tmp = 0.0
	if (C <= -9.2e-60)
		tmp = Float64(0.0 - Float64(sqrt(2.0) * sqrt(Float64(F * Float64(-0.5 / C)))));
	elseif (C <= 1.4e-12)
		tmp = Float64(sqrt(Float64(F / B_m)) * t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = 0.0 - sqrt(2.0);
	tmp = 0.0;
	if (C <= -9.2e-60)
		tmp = 0.0 - (sqrt(2.0) * sqrt((F * (-0.5 / C))));
	elseif (C <= 1.4e-12)
		tmp = sqrt((F / B_m)) * t_0;
	else
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -9.2e-60], N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 1.4e-12], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if C < -9.2000000000000005e-60

   1. Initial program 11.5%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \left(\sqrt{2}\right)\right)\right) \]

   5. Simplified 19.2%

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

   6. Taylor expanded in C around -inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\frac{-1}{2}}{C}\right)}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 47.5%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\frac{-1}{2}, C\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   8. Simplified 47.5%

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

   if -9.2000000000000005e-60 < C < 1.4000000000000001e-12

   1. Initial program 25.6%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left(\sqrt{2}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 26.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 26.7%

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

   if 1.4000000000000001e-12 < C

   1. Initial program 19.8%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \left(\sqrt{2}\right)\right)\right) \]

   5. Simplified 40.0%

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

   6. Taylor expanded in C around inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\frac{-1}{2}}{A}\right)}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 32.5%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\frac{-1}{2}, A\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   8. Simplified 32.5%

      \[\leadsto -\sqrt{F \cdot \color{blue}{\frac{-0.5}{A}}} \cdot \sqrt{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -9.2 \cdot 10^{-60}:\\ \;\;\;\;0 - \sqrt{2} \cdot \sqrt{F \cdot \frac{-0.5}{C}}\\ \mathbf{elif}\;C \leq 1.4 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 0 - \sqrt{2}\\ \mathbf{if}\;B\_m \leq 1.3 \cdot 10^{-145}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B\_m \cdot B\_m\right)}{C}}\right)\\ \mathbf{elif}\;B\_m \leq 1.18 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- 0.0 (sqrt 2.0))))
   (if (<= B_m 1.3e-145)
     (*
      0.25
      (*
       (/ 1.0 C)
       (sqrt (+ (* -16.0 (* C F)) (* 4.0 (/ (* F (* B_m B_m)) C))))))
     (if (<= B_m 1.18e-48)
       (* (sqrt (* -0.5 (/ F A))) t_0)
       (* (sqrt (/ F B_m)) t_0)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - sqrt(2.0);
	double tmp;
	if (B_m <= 1.3e-145) {
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	} else if (B_m <= 1.18e-48) {
		tmp = sqrt((-0.5 * (F / A))) * t_0;
	} else {
		tmp = sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
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 = 0.0d0 - sqrt(2.0d0)
    if (b_m <= 1.3d-145) then
        tmp = 0.25d0 * ((1.0d0 / c) * sqrt((((-16.0d0) * (c * f)) + (4.0d0 * ((f * (b_m * b_m)) / c)))))
    else if (b_m <= 1.18d-48) then
        tmp = sqrt(((-0.5d0) * (f / a))) * t_0
    else
        tmp = sqrt((f / b_m)) * t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 0.0 - Math.sqrt(2.0);
	double tmp;
	if (B_m <= 1.3e-145) {
		tmp = 0.25 * ((1.0 / C) * Math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	} else if (B_m <= 1.18e-48) {
		tmp = Math.sqrt((-0.5 * (F / A))) * t_0;
	} else {
		tmp = Math.sqrt((F / B_m)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = 0.0 - math.sqrt(2.0)
	tmp = 0
	if B_m <= 1.3e-145:
		tmp = 0.25 * ((1.0 / C) * math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))))
	elif B_m <= 1.18e-48:
		tmp = math.sqrt((-0.5 * (F / A))) * t_0
	else:
		tmp = math.sqrt((F / B_m)) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(0.0 - sqrt(2.0))
	tmp = 0.0
	if (B_m <= 1.3e-145)
		tmp = Float64(0.25 * Float64(Float64(1.0 / C) * sqrt(Float64(Float64(-16.0 * Float64(C * F)) + Float64(4.0 * Float64(Float64(F * Float64(B_m * B_m)) / C))))));
	elseif (B_m <= 1.18e-48)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / A))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = 0.0 - sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 1.3e-145)
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	elseif (B_m <= 1.18e-48)
		tmp = sqrt((-0.5 * (F / A))) * t_0;
	else
		tmp = sqrt((F / B_m)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.3e-145], N[(0.25 * N[(N[(1.0 / C), $MachinePrecision] * N[Sqrt[N[(N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.18e-48], N[(N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.3e-145

   1. Initial program 21.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 28.7%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 15.2%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(\frac{-2 \cdot \left(A \cdot C\right)}{A - C} + 2 \cdot A\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(0.5 \cdot \left(\frac{A \cdot C}{\left(A - C\right) \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)} + \frac{1}{A - C}\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \color{blue}{\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C}\right), \left(8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 10.1%

       \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \color{blue}{\left(2 \cdot \frac{\left(B \cdot B\right) \cdot \left(-0.5 \cdot \left(\left(B \cdot B\right) \cdot F\right) + 2 \cdot \left(\left(A \cdot A\right) \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around inf

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{1}{C}\right), \color{blue}{\left(\sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \left(\sqrt{\color{blue}{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}}\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\left(-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \left(\frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), C\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 19.3%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

   13. Simplified 19.3%

       \[\leadsto \color{blue}{0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{\left(B \cdot B\right) \cdot F}{C}}\right)} \]

   if 1.3e-145 < B < 1.18000000000000007e-48

   1. Initial program 19.5%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \left(\sqrt{2}\right)\right)\right) \]

   5. Simplified 41.5%

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

   6. Taylor expanded in C around inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{F}{A}\right)}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{F}{A}\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. /-lowering-/.f64 26.9%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(F, A\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   8. Simplified 26.9%

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

   if 1.18000000000000007e-48 < B

   1. Initial program 17.6%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left(\sqrt{2}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 55.4%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 55.4%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.3 \cdot 10^{-145}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B \cdot B\right)}{C}}\right)\\ \mathbf{elif}\;B \leq 1.18 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.35 \cdot 10^{-145}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B\_m \cdot B\_m\right)}{C}}\right)\\ \mathbf{elif}\;B\_m \leq 4.9 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(C \cdot \left(C \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.35e-145)
   (*
    0.25
    (* (/ 1.0 C) (sqrt (+ (* -16.0 (* C F)) (* 4.0 (/ (* F (* B_m B_m)) C))))))
   (if (<= B_m 4.9e-51)
     (/ (sqrt (* -16.0 (* A (* C (* C F))))) (- (* (* 4.0 A) C) (* B_m B_m)))
     (* (sqrt (/ F B_m)) (- 0.0 (sqrt 2.0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.35e-145) {
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	} else if (B_m <= 4.9e-51) {
		tmp = sqrt((-16.0 * (A * (C * (C * F))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	}
	return tmp;
}

Fortran

B_m = abs(b)
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 <= 1.35d-145) then
        tmp = 0.25d0 * ((1.0d0 / c) * sqrt((((-16.0d0) * (c * f)) + (4.0d0 * ((f * (b_m * b_m)) / c)))))
    else if (b_m <= 4.9d-51) then
        tmp = sqrt(((-16.0d0) * (a * (c * (c * f))))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = sqrt((f / b_m)) * (0.0d0 - sqrt(2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.35e-145) {
		tmp = 0.25 * ((1.0 / C) * Math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	} else if (B_m <= 4.9e-51) {
		tmp = Math.sqrt((-16.0 * (A * (C * (C * F))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.sqrt((F / B_m)) * (0.0 - Math.sqrt(2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.35e-145:
		tmp = 0.25 * ((1.0 / C) * math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))))
	elif B_m <= 4.9e-51:
		tmp = math.sqrt((-16.0 * (A * (C * (C * F))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.sqrt((F / B_m)) * (0.0 - math.sqrt(2.0))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.35e-145)
		tmp = Float64(0.25 * Float64(Float64(1.0 / C) * sqrt(Float64(Float64(-16.0 * Float64(C * F)) + Float64(4.0 * Float64(Float64(F * Float64(B_m * B_m)) / C))))));
	elseif (B_m <= 4.9e-51)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(A * Float64(C * Float64(C * F))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(0.0 - sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.35e-145)
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	elseif (B_m <= 4.9e-51)
		tmp = sqrt((-16.0 * (A * (C * (C * F))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = sqrt((F / B_m)) * (0.0 - sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.35e-145], N[(0.25 * N[(N[(1.0 / C), $MachinePrecision] * N[Sqrt[N[(N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.9e-51], N[(N[Sqrt[N[(-16.0 * N[(A * N[(C * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.35e-145

   1. Initial program 21.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 28.7%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 15.2%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(\frac{-2 \cdot \left(A \cdot C\right)}{A - C} + 2 \cdot A\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(0.5 \cdot \left(\frac{A \cdot C}{\left(A - C\right) \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)} + \frac{1}{A - C}\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \color{blue}{\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C}\right), \left(8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 10.1%

       \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \color{blue}{\left(2 \cdot \frac{\left(B \cdot B\right) \cdot \left(-0.5 \cdot \left(\left(B \cdot B\right) \cdot F\right) + 2 \cdot \left(\left(A \cdot A\right) \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around inf

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{1}{C}\right), \color{blue}{\left(\sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \left(\sqrt{\color{blue}{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}}\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\left(-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \left(\frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), C\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 19.3%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

   13. Simplified 19.3%

       \[\leadsto \color{blue}{0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{\left(B \cdot B\right) \cdot F}{C}}\right)} \]

   if 1.35e-145 < B < 4.89999999999999974e-51

   1. Initial program 19.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 26.6%

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

   6. Applied egg-rr 23.2%

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

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-8 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 19.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 19.8%

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

       1. sqrt-unprod N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(-8 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right) \cdot 2}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-8 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right) \cdot 2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -8\right) \cdot 2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot \left(-8 \cdot 2\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(\left(C \cdot C\right) \cdot F\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot \left(C \cdot F\right)\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, \left(C \cdot F\right)\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. *-lowering-*.f64 23.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, F\right)\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   11. Applied egg-rr 23.3%

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

   if 4.89999999999999974e-51 < B

   1. Initial program 17.6%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left(\sqrt{2}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 55.4%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 55.4%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.35 \cdot 10^{-145}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B \cdot B\right)}{C}}\right)\\ \mathbf{elif}\;B \leq 4.9 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(C \cdot \left(C \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(0 - \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 22.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;A \leq -3.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(C \cdot \left(C \cdot F\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;A \leq 4 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B\_m \cdot B\_m\right)}{C}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* (* 4.0 A) C) (* B_m B_m))))
   (if (<= A -3.9e-106)
     (/ (sqrt (* -16.0 (* A (* C (* C F))))) t_0)
     (if (<= A 4e-295)
       (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
       (*
        0.25
        (*
         (/ 1.0 C)
         (sqrt (+ (* -16.0 (* C F)) (* 4.0 (/ (* F (* B_m B_m)) C))))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (A <= -3.9e-106) {
		tmp = sqrt((-16.0 * (A * (C * (C * F))))) / t_0;
	} else if (A <= 4e-295) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	}
	return tmp;
}

Fortran

B_m = abs(b)
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) - (b_m * b_m)
    if (a <= (-3.9d-106)) then
        tmp = sqrt(((-16.0d0) * (a * (c * (c * f))))) / t_0
    else if (a <= 4d-295) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else
        tmp = 0.25d0 * ((1.0d0 / c) * sqrt((((-16.0d0) * (c * f)) + (4.0d0 * ((f * (b_m * b_m)) / c)))))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (A <= -3.9e-106) {
		tmp = Math.sqrt((-16.0 * (A * (C * (C * F))))) / t_0;
	} else if (A <= 4e-295) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = 0.25 * ((1.0 / C) * Math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = ((4.0 * A) * C) - (B_m * B_m)
	tmp = 0
	if A <= -3.9e-106:
		tmp = math.sqrt((-16.0 * (A * (C * (C * F))))) / t_0
	elif A <= 4e-295:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	else:
		tmp = 0.25 * ((1.0 / C) * math.sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))
	tmp = 0.0
	if (A <= -3.9e-106)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(A * Float64(C * Float64(C * F))))) / t_0);
	elseif (A <= 4e-295)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = Float64(0.25 * Float64(Float64(1.0 / C) * sqrt(Float64(Float64(-16.0 * Float64(C * F)) + Float64(4.0 * Float64(Float64(F * Float64(B_m * B_m)) / C))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = ((4.0 * A) * C) - (B_m * B_m);
	tmp = 0.0;
	if (A <= -3.9e-106)
		tmp = sqrt((-16.0 * (A * (C * (C * F))))) / t_0;
	elseif (A <= 4e-295)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	else
		tmp = 0.25 * ((1.0 / C) * sqrt(((-16.0 * (C * F)) + (4.0 * ((F * (B_m * B_m)) / C)))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -3.9e-106], N[(N[Sqrt[N[(-16.0 * N[(A * N[(C * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[A, 4e-295], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(0.25 * N[(N[(1.0 / C), $MachinePrecision] * N[Sqrt[N[(N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if A < -3.9000000000000001e-106

   1. Initial program 13.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 14.4%

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

   6. Applied egg-rr 11.4%

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

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-8 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 21.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 21.9%

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

       1. sqrt-unprod N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(-8 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right) \cdot 2}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-8 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right) \cdot 2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -8\right) \cdot 2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot \left(-8 \cdot 2\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(\left(C \cdot C\right) \cdot F\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot \left(C \cdot F\right)\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, \left(C \cdot F\right)\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. *-lowering-*.f64 26.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, F\right)\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   11. Applied egg-rr 26.3%

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

   if -3.9000000000000001e-106 < A < 4.00000000000000024e-295

   1. Initial program 33.9%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 39.9%

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

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 19.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 19.2%

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

   if 4.00000000000000024e-295 < A

   1. Initial program 19.3%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 28.8%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.5%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(\frac{-2 \cdot \left(A \cdot C\right)}{A - C} + 2 \cdot A\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(0.5 \cdot \left(\frac{A \cdot C}{\left(A - C\right) \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)} + \frac{1}{A - C}\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \color{blue}{\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot \left({B}^{2} \cdot F\right) + 2 \cdot \left({A}^{2} \cdot F\right)\right)}{C}\right), \left(8 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 11.6%

       \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) + \color{blue}{\left(2 \cdot \frac{\left(B \cdot B\right) \cdot \left(-0.5 \cdot \left(\left(B \cdot B\right) \cdot F\right) + 2 \cdot \left(\left(A \cdot A\right) \cdot F\right)\right)}{C} + 8 \cdot \left(A \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around inf

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\frac{1}{C}\right), \color{blue}{\left(\sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}\right)}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \left(\sqrt{\color{blue}{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}}}\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\left(-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(4 \cdot \frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \left(\frac{{B}^{2} \cdot F}{C}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), C\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 29.0%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, C\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), C\right)\right)\right)\right)\right)\right) \]

   13. Simplified 29.0%

       \[\leadsto \color{blue}{0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{\left(B \cdot B\right) \cdot F}{C}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(C \cdot \left(C \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;A \leq 4 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{1}{C} \cdot \sqrt{-16 \cdot \left(C \cdot F\right) + 4 \cdot \frac{F \cdot \left(B \cdot B\right)}{C}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 17.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;A \leq -2.3 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(C \cdot \left(C \cdot F\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;A \leq 4.5 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(A \cdot F\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* (* 4.0 A) C) (* B_m B_m))))
   (if (<= A -2.3e-107)
     (/ (sqrt (* -16.0 (* A (* C (* C F))))) t_0)
     (if (<= A 4.5e-294)
       (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
       (/ (sqrt (* -16.0 (* (* A C) (* A F)))) t_0)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (A <= -2.3e-107) {
		tmp = sqrt((-16.0 * (A * (C * (C * F))))) / t_0;
	} else if (A <= 4.5e-294) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = sqrt((-16.0 * ((A * C) * (A * F)))) / t_0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
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) - (b_m * b_m)
    if (a <= (-2.3d-107)) then
        tmp = sqrt(((-16.0d0) * (a * (c * (c * f))))) / t_0
    else if (a <= 4.5d-294) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else
        tmp = sqrt(((-16.0d0) * ((a * c) * (a * f)))) / t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (A <= -2.3e-107) {
		tmp = Math.sqrt((-16.0 * (A * (C * (C * F))))) / t_0;
	} else if (A <= 4.5e-294) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = Math.sqrt((-16.0 * ((A * C) * (A * F)))) / t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = ((4.0 * A) * C) - (B_m * B_m)
	tmp = 0
	if A <= -2.3e-107:
		tmp = math.sqrt((-16.0 * (A * (C * (C * F))))) / t_0
	elif A <= 4.5e-294:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	else:
		tmp = math.sqrt((-16.0 * ((A * C) * (A * F)))) / t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))
	tmp = 0.0
	if (A <= -2.3e-107)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(A * Float64(C * Float64(C * F))))) / t_0);
	elseif (A <= 4.5e-294)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(A * C) * Float64(A * F)))) / t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = ((4.0 * A) * C) - (B_m * B_m);
	tmp = 0.0;
	if (A <= -2.3e-107)
		tmp = sqrt((-16.0 * (A * (C * (C * F))))) / t_0;
	elseif (A <= 4.5e-294)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	else
		tmp = sqrt((-16.0 * ((A * C) * (A * F)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -2.3e-107], N[(N[Sqrt[N[(-16.0 * N[(A * N[(C * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[A, 4.5e-294], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(-16.0 * N[(N[(A * C), $MachinePrecision] * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if A < -2.30000000000000003e-107

   1. Initial program 13.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 14.4%

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

   6. Applied egg-rr 11.4%

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

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-8 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 21.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 21.9%

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

       1. sqrt-unprod N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(-8 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right) \cdot 2}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-8 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right) \cdot 2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -8\right) \cdot 2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot \left(-8 \cdot 2\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(\left(C \cdot C\right) \cdot F\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot \left(C \cdot F\right)\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, \left(C \cdot F\right)\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. *-lowering-*.f64 26.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, F\right)\right)\right), -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   11. Applied egg-rr 26.3%

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

   if -2.30000000000000003e-107 < A < 4.49999999999999981e-294

   1. Initial program 33.3%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 39.1%

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

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 19.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 19.0%

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

   if 4.49999999999999981e-294 < A

   1. Initial program 19.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 29.0%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 14.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 14.4%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(\left(A \cdot A\right) \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(F \cdot A\right) \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(F \cdot A\right), \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, A\right), \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 20.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, A\right), \mathsf{*.f64}\left(A, C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 20.0%

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left(F \cdot A\right) \cdot \left(A \cdot C\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.3 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(C \cdot \left(C \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;A \leq 4.5 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(A \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 16.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;A \leq -3.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{t\_0}\\ \mathbf{elif}\;A \leq 2.6 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(A \cdot F\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* (* 4.0 A) C) (* B_m B_m))))
   (if (<= A -3.5e-106)
     (/ (sqrt (* (* A -16.0) (* F (* C C)))) t_0)
     (if (<= A 2.6e-294)
       (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
       (/ (sqrt (* -16.0 (* (* A C) (* A F)))) t_0)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (A <= -3.5e-106) {
		tmp = sqrt(((A * -16.0) * (F * (C * C)))) / t_0;
	} else if (A <= 2.6e-294) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = sqrt((-16.0 * ((A * C) * (A * F)))) / t_0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
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) - (b_m * b_m)
    if (a <= (-3.5d-106)) then
        tmp = sqrt(((a * (-16.0d0)) * (f * (c * c)))) / t_0
    else if (a <= 2.6d-294) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else
        tmp = sqrt(((-16.0d0) * ((a * c) * (a * f)))) / t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (A <= -3.5e-106) {
		tmp = Math.sqrt(((A * -16.0) * (F * (C * C)))) / t_0;
	} else if (A <= 2.6e-294) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = Math.sqrt((-16.0 * ((A * C) * (A * F)))) / t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = ((4.0 * A) * C) - (B_m * B_m)
	tmp = 0
	if A <= -3.5e-106:
		tmp = math.sqrt(((A * -16.0) * (F * (C * C)))) / t_0
	elif A <= 2.6e-294:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	else:
		tmp = math.sqrt((-16.0 * ((A * C) * (A * F)))) / t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))
	tmp = 0.0
	if (A <= -3.5e-106)
		tmp = Float64(sqrt(Float64(Float64(A * -16.0) * Float64(F * Float64(C * C)))) / t_0);
	elseif (A <= 2.6e-294)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(A * C) * Float64(A * F)))) / t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = ((4.0 * A) * C) - (B_m * B_m);
	tmp = 0.0;
	if (A <= -3.5e-106)
		tmp = sqrt(((A * -16.0) * (F * (C * C)))) / t_0;
	elseif (A <= 2.6e-294)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	else
		tmp = sqrt((-16.0 * ((A * C) * (A * F)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -3.5e-106], N[(N[Sqrt[N[(N[(A * -16.0), $MachinePrecision] * N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[A, 2.6e-294], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(-16.0 * N[(N[(A * C), $MachinePrecision] * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if A < -3.5e-106

   1. Initial program 13.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 14.4%

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

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-16 \cdot A\right) \cdot \left({C}^{2} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-16 \cdot A\right), \left({C}^{2} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \left({C}^{2} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 22.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 22.0%

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

   if -3.5e-106 < A < 2.5999999999999999e-294

   1. Initial program 33.3%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 39.1%

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

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 19.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 19.0%

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

   if 2.5999999999999999e-294 < A

   1. Initial program 19.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 29.0%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 14.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 14.4%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(\left(A \cdot A\right) \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(F \cdot A\right) \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(F \cdot A\right), \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, A\right), \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 20.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, A\right), \mathsf{*.f64}\left(A, C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 20.0%

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left(F \cdot A\right) \cdot \left(A \cdot C\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 20.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;A \leq 2.6 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(A \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 15.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;B\_m \leq 1.9 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(A \cdot F\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* (* 4.0 A) C) (* B_m B_m))))
   (if (<= B_m 1.9e-60)
     (/ (sqrt (* -16.0 (* (* A C) (* A F)))) t_0)
     (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 1.9e-60) {
		tmp = sqrt((-16.0 * ((A * C) * (A * F)))) / t_0;
	} else {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
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) - (b_m * b_m)
    if (b_m <= 1.9d-60) then
        tmp = sqrt(((-16.0d0) * ((a * c) * (a * f)))) / t_0
    else
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 1.9e-60) {
		tmp = Math.sqrt((-16.0 * ((A * C) * (A * F)))) / t_0;
	} else {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = ((4.0 * A) * C) - (B_m * B_m)
	tmp = 0
	if B_m <= 1.9e-60:
		tmp = math.sqrt((-16.0 * ((A * C) * (A * F)))) / t_0
	else:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 1.9e-60)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(A * C) * Float64(A * F)))) / t_0);
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = ((4.0 * A) * C) - (B_m * B_m);
	tmp = 0.0;
	if (B_m <= 1.9e-60)
		tmp = sqrt((-16.0 * ((A * C) * (A * F)))) / t_0;
	else
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.9e-60], N[(N[Sqrt[N[(-16.0 * N[(N[(A * C), $MachinePrecision] * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.89999999999999997e-60

   1. Initial program 21.4%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 28.2%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 12.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 12.3%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(\left(A \cdot A\right) \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(F \cdot A\right) \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(F \cdot A\right), \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, A\right), \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 16.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, A\right), \mathsf{*.f64}\left(A, C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 16.1%

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

   if 1.89999999999999997e-60 < B

   1. Initial program 17.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 21.8%

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

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 16.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 16.0%

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

4. Final simplification 16.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.9 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(A \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 13.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(A \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (/ (sqrt (* -16.0 (* (* A C) (* A F)))) (- (* (* 4.0 A) C) (* B_m B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt((-16.0 * ((A * C) * (A * F)))) / (((4.0 * A) * C) - (B_m * B_m));
}

Fortran

B_m = abs(b)
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(((-16.0d0) * ((a * c) * (a * f)))) / (((4.0d0 * a) * c) - (b_m * b_m))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((-16.0 * ((A * C) * (A * F)))) / (((4.0 * A) * C) - (B_m * B_m));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt((-16.0 * ((A * C) * (A * F)))) / (((4.0 * A) * C) - (B_m * B_m))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(-16.0 * Float64(Float64(A * C) * Float64(A * F)))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt((-16.0 * ((A * C) * (A * F)))) / (((4.0 * A) * C) - (B_m * B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(-16.0 * N[(N[(A * C), $MachinePrecision] * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.3%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

3. Simplified 26.3%

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

5. Taylor expanded in B around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. *-lowering-*.f64 9.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

7. Simplified 9.7%

   \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(\left(A \cdot A\right) \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(F \cdot A\right) \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(F \cdot A\right), \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, A\right), \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. *-lowering-*.f64 12.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, A\right), \mathsf{*.f64}\left(A, C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

9. Applied egg-rr 12.8%

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

10. Final simplification 12.8%

    \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(A \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
11. Add Preprocessing

Alternative 17: 12.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \frac{\sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (/ (sqrt (* -16.0 (* A (* F (* A C))))) (- (* (* 4.0 A) C) (* B_m B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt((-16.0 * (A * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
}

Fortran

B_m = abs(b)
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(((-16.0d0) * (a * (f * (a * c))))) / (((4.0d0 * a) * c) - (b_m * b_m))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((-16.0 * (A * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt((-16.0 * (A * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(-16.0 * Float64(A * Float64(F * Float64(A * C))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt((-16.0 * (A * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(-16.0 * N[(A * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.3%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

3. Simplified 26.3%

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

5. Taylor expanded in B around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. *-lowering-*.f64 9.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

7. Simplified 9.7%

   \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
8. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left(\left(A \cdot C\right) \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(A \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   5. *-lowering-*.f64 12.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

9. Applied egg-rr 12.3%

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

10. Final simplification 12.3%

    \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(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))))