ABCF->ab-angle a

Percentage Accurate: 19.6% → 58.8%

Time: 28.5s

Alternatives: 18

Speedup: 5.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.6% 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: 58.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\\ t_1 := B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{\frac{t\_0}{B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)}} \cdot \left(0 - \sqrt{2 \cdot F}\right)\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(\left(2 \cdot F\right) \cdot \left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A} + 2 \cdot C\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_0}}{t\_2 - B\_m \cdot B\_m} \cdot \sqrt{\left(2 \cdot F\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 -2e-208)
     (*
      (sqrt (/ t_0 (+ (* B_m B_m) (* A (* C -4.0)))))
      (- 0.0 (sqrt (* 2.0 F))))
     (if (<= t_3 4e-86)
       (/
        (sqrt (* t_1 (* (* 2.0 F) (+ (* -0.5 (/ (* B_m B_m) A)) (* 2.0 C)))))
        (- (* 4.0 (* A C)) (* B_m B_m)))
       (if (<= t_3 INFINITY)
         (* (/ (sqrt t_0) (- t_2 (* B_m B_m))) (sqrt (* (* 2.0 F) t_1)))
         (- 0.0 (/ (pow (* 2.0 F) 0.5) (sqrt B_m))))))))

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 <= -2e-208) {
		tmp = sqrt((t_0 / ((B_m * B_m) + (A * (C * -4.0))))) * (0.0 - sqrt((2.0 * F)));
	} else if (t_3 <= 4e-86) {
		tmp = sqrt((t_1 * ((2.0 * F) * ((-0.5 * ((B_m * B_m) / A)) + (2.0 * C))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt(t_0) / (t_2 - (B_m * B_m))) * sqrt(((2.0 * F) * t_1));
	} else {
		tmp = 0.0 - (pow((2.0 * F), 0.5) / sqrt(B_m));
	}
	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 <= -2e-208) {
		tmp = Math.sqrt((t_0 / ((B_m * B_m) + (A * (C * -4.0))))) * (0.0 - Math.sqrt((2.0 * F)));
	} else if (t_3 <= 4e-86) {
		tmp = Math.sqrt((t_1 * ((2.0 * F) * ((-0.5 * ((B_m * B_m) / A)) + (2.0 * C))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt(t_0) / (t_2 - (B_m * B_m))) * Math.sqrt(((2.0 * F) * t_1));
	} else {
		tmp = 0.0 - (Math.pow((2.0 * F), 0.5) / Math.sqrt(B_m));
	}
	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 <= -2e-208:
		tmp = math.sqrt((t_0 / ((B_m * B_m) + (A * (C * -4.0))))) * (0.0 - math.sqrt((2.0 * F)))
	elif t_3 <= 4e-86:
		tmp = math.sqrt((t_1 * ((2.0 * F) * ((-0.5 * ((B_m * B_m) / A)) + (2.0 * C))))) / ((4.0 * (A * C)) - (B_m * B_m))
	elif t_3 <= math.inf:
		tmp = (math.sqrt(t_0) / (t_2 - (B_m * B_m))) * math.sqrt(((2.0 * F) * t_1))
	else:
		tmp = 0.0 - (math.pow((2.0 * F), 0.5) / math.sqrt(B_m))
	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 <= -2e-208)
		tmp = Float64(sqrt(Float64(t_0 / Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))) * Float64(0.0 - sqrt(Float64(2.0 * F))));
	elseif (t_3 <= 4e-86)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(2.0 * F) * Float64(Float64(-0.5 * Float64(Float64(B_m * B_m) / A)) + Float64(2.0 * C))))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(t_0) / Float64(t_2 - Float64(B_m * B_m))) * sqrt(Float64(Float64(2.0 * F) * t_1)));
	else
		tmp = Float64(0.0 - Float64((Float64(2.0 * F) ^ 0.5) / sqrt(B_m)));
	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 <= -2e-208)
		tmp = sqrt((t_0 / ((B_m * B_m) + (A * (C * -4.0))))) * (0.0 - sqrt((2.0 * F)));
	elseif (t_3 <= 4e-86)
		tmp = sqrt((t_1 * ((2.0 * F) * ((-0.5 * ((B_m * B_m) / A)) + (2.0 * C))))) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (t_3 <= Inf)
		tmp = (sqrt(t_0) / (t_2 - (B_m * B_m))) * sqrt(((2.0 * F) * t_1));
	else
		tmp = 0.0 - (((2.0 * F) ^ 0.5) / sqrt(B_m));
	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, -2e-208], N[(N[Sqrt[N[(t$95$0 / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e-86], N[(N[Sqrt[N[(t$95$1 * N[(N[(2.0 * F), $MachinePrecision] * N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] / N[(t$95$2 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.0000000000000002e-208

   1. Initial program 35.6%

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

   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 63.0%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 84.6%

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

   if -2.0000000000000002e-208 < (/.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))) < 4.00000000000000034e-86

   1. Initial program 10.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 10.8%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

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

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. sub-neg N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

   6. Applied egg-rr 10.8%

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

   7. Taylor expanded in A around -inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-lowering-*.f64 43.9%

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

   9. Simplified 43.9%

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

   if 4.00000000000000034e-86 < (/.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 36.1%

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

   4. Applied egg-rr 74.5%

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

   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 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 3.0%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 3.9%

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

   8. Taylor expanded in B around inf

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

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

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

      2. /-lowering-/.f64 26.6%

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

   10. Simplified 26.6%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{1}}{\sqrt{B}}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}\right)\right) \]

       3. un-div-inv N/A

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

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

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

       5. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}}\right), \left(\sqrt{B}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot F\right), \frac{1}{2}\right), \left(\sqrt{B}\right)\right)\right) \]

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

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

       8. sqrt-lowering-sqrt.f64 26.6%

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

   12. Applied egg-rr 26.6%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}} \cdot \left(0 - \sqrt{2 \cdot F}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 4 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(-0.5 \cdot \frac{B \cdot B}{A} + 2 \cdot C\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \cdot \sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\\ \mathbf{if}\;B\_m \leq 1.4 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{t\_0} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 2.8 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\frac{t\_0}{B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)}} \cdot \left(0 - \sqrt{2 \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (+ A C) (hypot B_m (- A C)))))
   (if (<= B_m 1.4e-105)
     (*
      (sqrt t_0)
      (/
       (sqrt (* (* 2.0 F) (+ (* B_m B_m) (* -4.0 (* A C)))))
       (- (* (* 4.0 A) C) (* B_m B_m))))
     (if (<= B_m 2.8e+146)
       (*
        (sqrt (/ t_0 (+ (* B_m B_m) (* A (* C -4.0)))))
        (- 0.0 (sqrt (* 2.0 F))))
       (- 0.0 (/ (pow (* 2.0 F) 0.5) (sqrt B_m)))))))

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 tmp;
	if (B_m <= 1.4e-105) {
		tmp = sqrt(t_0) * (sqrt(((2.0 * F) * ((B_m * B_m) + (-4.0 * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m)));
	} else if (B_m <= 2.8e+146) {
		tmp = sqrt((t_0 / ((B_m * B_m) + (A * (C * -4.0))))) * (0.0 - sqrt((2.0 * F)));
	} else {
		tmp = 0.0 - (pow((2.0 * F), 0.5) / sqrt(B_m));
	}
	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 tmp;
	if (B_m <= 1.4e-105) {
		tmp = Math.sqrt(t_0) * (Math.sqrt(((2.0 * F) * ((B_m * B_m) + (-4.0 * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m)));
	} else if (B_m <= 2.8e+146) {
		tmp = Math.sqrt((t_0 / ((B_m * B_m) + (A * (C * -4.0))))) * (0.0 - Math.sqrt((2.0 * F)));
	} else {
		tmp = 0.0 - (Math.pow((2.0 * F), 0.5) / Math.sqrt(B_m));
	}
	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))
	tmp = 0
	if B_m <= 1.4e-105:
		tmp = math.sqrt(t_0) * (math.sqrt(((2.0 * F) * ((B_m * B_m) + (-4.0 * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m)))
	elif B_m <= 2.8e+146:
		tmp = math.sqrt((t_0 / ((B_m * B_m) + (A * (C * -4.0))))) * (0.0 - math.sqrt((2.0 * F)))
	else:
		tmp = 0.0 - (math.pow((2.0 * F), 0.5) / math.sqrt(B_m))
	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)))
	tmp = 0.0
	if (B_m <= 1.4e-105)
		tmp = Float64(sqrt(t_0) * Float64(sqrt(Float64(Float64(2.0 * F) * Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))));
	elseif (B_m <= 2.8e+146)
		tmp = Float64(sqrt(Float64(t_0 / Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))) * Float64(0.0 - sqrt(Float64(2.0 * F))));
	else
		tmp = Float64(0.0 - Float64((Float64(2.0 * F) ^ 0.5) / sqrt(B_m)));
	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));
	tmp = 0.0;
	if (B_m <= 1.4e-105)
		tmp = sqrt(t_0) * (sqrt(((2.0 * F) * ((B_m * B_m) + (-4.0 * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m)));
	elseif (B_m <= 2.8e+146)
		tmp = sqrt((t_0 / ((B_m * B_m) + (A * (C * -4.0))))) * (0.0 - sqrt((2.0 * F)));
	else
		tmp = 0.0 - (((2.0 * F) ^ 0.5) / sqrt(B_m));
	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]}, If[LessEqual[B$95$m, 1.4e-105], N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * 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]), $MachinePrecision], If[LessEqual[B$95$m, 2.8e+146], N[(N[Sqrt[N[(t$95$0 / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.4e-105

   1. Initial program 22.0%

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

   4. Applied egg-rr 41.1%

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

   if 1.4e-105 < B < 2.8000000000000001e146

   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 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 39.8%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 50.5%

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

   if 2.8000000000000001e146 < B

   1. Initial program 0.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 5.2%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 8.9%

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

   8. Taylor expanded in B around inf

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

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

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

      2. /-lowering-/.f64 74.6%

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

   10. Simplified 74.6%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{1}}{\sqrt{B}}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}\right)\right) \]

       3. un-div-inv N/A

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

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

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

       5. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}}\right), \left(\sqrt{B}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot F\right), \frac{1}{2}\right), \left(\sqrt{B}\right)\right)\right) \]

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

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

       8. sqrt-lowering-sqrt.f64 74.6%

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

   12. Applied egg-rr 74.6%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.4 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}} \cdot \left(0 - \sqrt{2 \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.4 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C \cdot -16\right)} \cdot \left|A\right|}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 2.8 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)}} \cdot \left(0 - \sqrt{2 \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.4e-139)
   (/ (* (sqrt (* F (* C -16.0))) (fabs A)) (- (* 4.0 (* A C)) (* B_m B_m)))
   (if (<= B_m 2.8e+146)
     (*
      (sqrt
       (/ (+ (+ A C) (hypot B_m (- A C))) (+ (* B_m B_m) (* A (* C -4.0)))))
      (- 0.0 (sqrt (* 2.0 F))))
     (- 0.0 (/ (pow (* 2.0 F) 0.5) (sqrt B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.4e-139) {
		tmp = (sqrt((F * (C * -16.0))) * fabs(A)) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 2.8e+146) {
		tmp = sqrt((((A + C) + hypot(B_m, (A - C))) / ((B_m * B_m) + (A * (C * -4.0))))) * (0.0 - sqrt((2.0 * F)));
	} else {
		tmp = 0.0 - (pow((2.0 * F), 0.5) / sqrt(B_m));
	}
	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 <= 1.4e-139) {
		tmp = (Math.sqrt((F * (C * -16.0))) * Math.abs(A)) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 2.8e+146) {
		tmp = Math.sqrt((((A + C) + Math.hypot(B_m, (A - C))) / ((B_m * B_m) + (A * (C * -4.0))))) * (0.0 - Math.sqrt((2.0 * F)));
	} else {
		tmp = 0.0 - (Math.pow((2.0 * F), 0.5) / Math.sqrt(B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.4e-139:
		tmp = (math.sqrt((F * (C * -16.0))) * math.fabs(A)) / ((4.0 * (A * C)) - (B_m * B_m))
	elif B_m <= 2.8e+146:
		tmp = math.sqrt((((A + C) + math.hypot(B_m, (A - C))) / ((B_m * B_m) + (A * (C * -4.0))))) * (0.0 - math.sqrt((2.0 * F)))
	else:
		tmp = 0.0 - (math.pow((2.0 * F), 0.5) / math.sqrt(B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.4e-139)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(C * -16.0))) * abs(A)) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (B_m <= 2.8e+146)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))) * Float64(0.0 - sqrt(Float64(2.0 * F))));
	else
		tmp = Float64(0.0 - Float64((Float64(2.0 * F) ^ 0.5) / sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.4e-139)
		tmp = (sqrt((F * (C * -16.0))) * abs(A)) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (B_m <= 2.8e+146)
		tmp = sqrt((((A + C) + hypot(B_m, (A - C))) / ((B_m * B_m) + (A * (C * -4.0))))) * (0.0 - sqrt((2.0 * F)));
	else
		tmp = 0.0 - (((2.0 * F) ^ 0.5) / sqrt(B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.4e-139], N[(N[(N[Sqrt[N[(F * N[(C * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[A], $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.8e+146], 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[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.3999999999999999e-139

   1. Initial program 21.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 28.0%

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

   5. Taylor expanded in B around 0

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

      1. associate-*r* N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-lowering-*.f64 10.1%

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

   7. Simplified 10.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. pow1/2 N/A

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

      14. rem-sqrt-square N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, -16\right)\right)\right), \left(\left|A\right|\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(A, C\right)}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      15. fabs-lowering-fabs.f64 17.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, -16\right)\right)\right), \mathsf{fabs.f64}\left(A\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(A, C\right)}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.9%

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

   if 1.3999999999999999e-139 < B < 2.8000000000000001e146

   1. Initial program 18.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 42.5%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 51.4%

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

   if 2.8000000000000001e146 < B

   1. Initial program 0.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 5.2%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 8.9%

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

   8. Taylor expanded in B around inf

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

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

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

      2. /-lowering-/.f64 74.6%

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

   10. Simplified 74.6%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{1}}{\sqrt{B}}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}\right)\right) \]

       3. un-div-inv N/A

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

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

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

       5. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}}\right), \left(\sqrt{B}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot F\right), \frac{1}{2}\right), \left(\sqrt{B}\right)\right)\right) \]

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

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

       8. sqrt-lowering-sqrt.f64 74.6%

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

   12. Applied egg-rr 74.6%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.4 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C \cdot -16\right)} \cdot \left|A\right|}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}} \cdot \left(0 - \sqrt{2 \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.8 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C \cdot -16\right)} \cdot \left|A\right|}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 9 \cdot 10^{+58}:\\ \;\;\;\;0 - \sqrt{\frac{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.8e-136)
   (/ (* (sqrt (* F (* C -16.0))) (fabs A)) (- (* 4.0 (* A C)) (* B_m B_m)))
   (if (<= B_m 9e+58)
     (-
      0.0
      (sqrt
       (/
        (* (* 2.0 F) (+ (+ A C) (hypot B_m (- A C))))
        (+ (* B_m B_m) (* A (* C -4.0))))))
     (- 0.0 (/ (pow (* 2.0 F) 0.5) (sqrt B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.8e-136) {
		tmp = (sqrt((F * (C * -16.0))) * fabs(A)) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 9e+58) {
		tmp = 0.0 - sqrt((((2.0 * F) * ((A + C) + hypot(B_m, (A - C)))) / ((B_m * B_m) + (A * (C * -4.0)))));
	} else {
		tmp = 0.0 - (pow((2.0 * F), 0.5) / sqrt(B_m));
	}
	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 <= 2.8e-136) {
		tmp = (Math.sqrt((F * (C * -16.0))) * Math.abs(A)) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 9e+58) {
		tmp = 0.0 - Math.sqrt((((2.0 * F) * ((A + C) + Math.hypot(B_m, (A - C)))) / ((B_m * B_m) + (A * (C * -4.0)))));
	} else {
		tmp = 0.0 - (Math.pow((2.0 * F), 0.5) / Math.sqrt(B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 2.8e-136:
		tmp = (math.sqrt((F * (C * -16.0))) * math.fabs(A)) / ((4.0 * (A * C)) - (B_m * B_m))
	elif B_m <= 9e+58:
		tmp = 0.0 - math.sqrt((((2.0 * F) * ((A + C) + math.hypot(B_m, (A - C)))) / ((B_m * B_m) + (A * (C * -4.0)))))
	else:
		tmp = 0.0 - (math.pow((2.0 * F), 0.5) / math.sqrt(B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.8e-136)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(C * -16.0))) * abs(A)) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (B_m <= 9e+58)
		tmp = Float64(0.0 - sqrt(Float64(Float64(Float64(2.0 * F) * Float64(Float64(A + C) + hypot(B_m, Float64(A - C)))) / Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))));
	else
		tmp = Float64(0.0 - Float64((Float64(2.0 * F) ^ 0.5) / sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 2.8e-136)
		tmp = (sqrt((F * (C * -16.0))) * abs(A)) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (B_m <= 9e+58)
		tmp = 0.0 - sqrt((((2.0 * F) * ((A + C) + hypot(B_m, (A - C)))) / ((B_m * B_m) + (A * (C * -4.0)))));
	else
		tmp = 0.0 - (((2.0 * F) ^ 0.5) / sqrt(B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2.8e-136], N[(N[(N[Sqrt[N[(F * N[(C * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[A], $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 9e+58], N[(0.0 - N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.8000000000000001e-136

   1. Initial program 21.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 28.0%

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

   5. Taylor expanded in B around 0

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

      1. associate-*r* N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-lowering-*.f64 10.1%

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

   7. Simplified 10.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. pow1/2 N/A

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

      14. rem-sqrt-square N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, -16\right)\right)\right), \left(\left|A\right|\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(A, C\right)}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      15. fabs-lowering-fabs.f64 17.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, -16\right)\right)\right), \mathsf{fabs.f64}\left(A\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(A, C\right)}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.9%

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

   if 2.8000000000000001e-136 < B < 8.9999999999999996e58

   1. Initial program 17.3%

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

   3. Taylor expanded in 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 40.3%

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

      1. sqrt-unprod N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. associate-+r+ N/A

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

      6. associate-*l* N/A

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

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

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

   7. Applied egg-rr 37.2%

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

   if 8.9999999999999996e58 < B

   1. Initial program 6.1%

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

   3. Taylor expanded in 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 16.5%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 25.2%

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

   8. Taylor expanded in B around inf

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

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

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

      2. /-lowering-/.f64 72.3%

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

   10. Simplified 72.3%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{1}}{\sqrt{B}}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}\right)\right) \]

       3. un-div-inv N/A

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

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

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

       5. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}}\right), \left(\sqrt{B}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot F\right), \frac{1}{2}\right), \left(\sqrt{B}\right)\right)\right) \]

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

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

       8. sqrt-lowering-sqrt.f64 72.2%

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

   12. Applied egg-rr 72.2%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.8 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C \cdot -16\right)} \cdot \left|A\right|}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+58}:\\ \;\;\;\;0 - \sqrt{\frac{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 0 - \sqrt{2 \cdot F}\\ \mathbf{if}\;C \leq -4.2 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\frac{-0.5}{C}} \cdot t\_0\\ \mathbf{elif}\;C \leq 7.5 \cdot 10^{+99}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot C}{B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- 0.0 (sqrt (* 2.0 F)))))
   (if (<= C -4.2e-60)
     (* (sqrt (/ -0.5 C)) t_0)
     (if (<= C 7.5e+99)
       (- 0.0 (/ (pow (* 2.0 F) 0.5) (sqrt B_m)))
       (* (sqrt (/ (* 2.0 C) (+ (* B_m B_m) (* A (* C -4.0))))) 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 * F));
	double tmp;
	if (C <= -4.2e-60) {
		tmp = sqrt((-0.5 / C)) * t_0;
	} else if (C <= 7.5e+99) {
		tmp = 0.0 - (pow((2.0 * F), 0.5) / sqrt(B_m));
	} else {
		tmp = sqrt(((2.0 * C) / ((B_m * B_m) + (A * (C * -4.0))))) * 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 * f))
    if (c <= (-4.2d-60)) then
        tmp = sqrt(((-0.5d0) / c)) * t_0
    else if (c <= 7.5d+99) then
        tmp = 0.0d0 - (((2.0d0 * f) ** 0.5d0) / sqrt(b_m))
    else
        tmp = sqrt(((2.0d0 * c) / ((b_m * b_m) + (a * (c * (-4.0d0)))))) * 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 * F));
	double tmp;
	if (C <= -4.2e-60) {
		tmp = Math.sqrt((-0.5 / C)) * t_0;
	} else if (C <= 7.5e+99) {
		tmp = 0.0 - (Math.pow((2.0 * F), 0.5) / Math.sqrt(B_m));
	} else {
		tmp = Math.sqrt(((2.0 * C) / ((B_m * B_m) + (A * (C * -4.0))))) * 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 * F))
	tmp = 0
	if C <= -4.2e-60:
		tmp = math.sqrt((-0.5 / C)) * t_0
	elif C <= 7.5e+99:
		tmp = 0.0 - (math.pow((2.0 * F), 0.5) / math.sqrt(B_m))
	else:
		tmp = math.sqrt(((2.0 * C) / ((B_m * B_m) + (A * (C * -4.0))))) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(0.0 - sqrt(Float64(2.0 * F)))
	tmp = 0.0
	if (C <= -4.2e-60)
		tmp = Float64(sqrt(Float64(-0.5 / C)) * t_0);
	elseif (C <= 7.5e+99)
		tmp = Float64(0.0 - Float64((Float64(2.0 * F) ^ 0.5) / sqrt(B_m)));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) / Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))) * 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 * F));
	tmp = 0.0;
	if (C <= -4.2e-60)
		tmp = sqrt((-0.5 / C)) * t_0;
	elseif (C <= 7.5e+99)
		tmp = 0.0 - (((2.0 * F) ^ 0.5) / sqrt(B_m));
	else
		tmp = sqrt(((2.0 * C) / ((B_m * B_m) + (A * (C * -4.0))))) * 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[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -4.2e-60], N[(N[Sqrt[N[(-0.5 / C), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[C, 7.5e+99], N[(0.0 - N[(N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if C < -4.19999999999999982e-60

   1. Initial program 6.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 19.1%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 20.7%

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

   8. Taylor expanded in A around inf

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

      1. /-lowering-/.f64 46.7%

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

   10. Simplified 46.7%

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

   if -4.19999999999999982e-60 < C < 7.49999999999999963e99

   1. Initial program 28.1%

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

   3. Taylor expanded in 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 31.0%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 41.4%

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

   8. Taylor expanded in B around inf

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

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

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

      2. /-lowering-/.f64 28.3%

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

   10. Simplified 28.3%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{1}}{\sqrt{B}}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}\right)\right) \]

       3. un-div-inv N/A

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

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

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

       5. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}}\right), \left(\sqrt{B}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot F\right), \frac{1}{2}\right), \left(\sqrt{B}\right)\right)\right) \]

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

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

       8. sqrt-lowering-sqrt.f64 28.3%

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

   12. Applied egg-rr 28.3%

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

   if 7.49999999999999963e99 < C

   1. Initial program 7.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 26.8%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 40.2%

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

   8. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 42.1%

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

   10. Simplified 42.1%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -4.2 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\frac{-0.5}{C}} \cdot \left(0 - \sqrt{2 \cdot F}\right)\\ \mathbf{elif}\;C \leq 7.5 \cdot 10^{+99}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot C}{B \cdot B + A \cdot \left(C \cdot -4\right)}} \cdot \left(0 - \sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 0 - \sqrt{2 \cdot F}\\ \mathbf{if}\;C \leq -4.2 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\frac{-0.5}{C}} \cdot t\_0\\ \mathbf{elif}\;C \leq 1.3 \cdot 10^{+104}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B\_m}}\\ \mathbf{elif}\;C \leq 1.45 \cdot 10^{+228}:\\ \;\;\;\;\frac{-1}{\frac{B\_m \cdot B\_m - A \cdot \left(4 \cdot C\right)}{C \cdot \sqrt{F \cdot \left(A \cdot -16\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-0.5}{A}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- 0.0 (sqrt (* 2.0 F)))))
   (if (<= C -4.2e-60)
     (* (sqrt (/ -0.5 C)) t_0)
     (if (<= C 1.3e+104)
       (- 0.0 (/ (pow (* 2.0 F) 0.5) (sqrt B_m)))
       (if (<= C 1.45e+228)
         (/
          -1.0
          (/ (- (* B_m B_m) (* A (* 4.0 C))) (* C (sqrt (* F (* A -16.0))))))
         (* (sqrt (/ -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 * F));
	double tmp;
	if (C <= -4.2e-60) {
		tmp = sqrt((-0.5 / C)) * t_0;
	} else if (C <= 1.3e+104) {
		tmp = 0.0 - (pow((2.0 * F), 0.5) / sqrt(B_m));
	} else if (C <= 1.45e+228) {
		tmp = -1.0 / (((B_m * B_m) - (A * (4.0 * C))) / (C * sqrt((F * (A * -16.0)))));
	} else {
		tmp = sqrt((-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 * f))
    if (c <= (-4.2d-60)) then
        tmp = sqrt(((-0.5d0) / c)) * t_0
    else if (c <= 1.3d+104) then
        tmp = 0.0d0 - (((2.0d0 * f) ** 0.5d0) / sqrt(b_m))
    else if (c <= 1.45d+228) then
        tmp = (-1.0d0) / (((b_m * b_m) - (a * (4.0d0 * c))) / (c * sqrt((f * (a * (-16.0d0))))))
    else
        tmp = sqrt(((-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 * F));
	double tmp;
	if (C <= -4.2e-60) {
		tmp = Math.sqrt((-0.5 / C)) * t_0;
	} else if (C <= 1.3e+104) {
		tmp = 0.0 - (Math.pow((2.0 * F), 0.5) / Math.sqrt(B_m));
	} else if (C <= 1.45e+228) {
		tmp = -1.0 / (((B_m * B_m) - (A * (4.0 * C))) / (C * Math.sqrt((F * (A * -16.0)))));
	} else {
		tmp = Math.sqrt((-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 * F))
	tmp = 0
	if C <= -4.2e-60:
		tmp = math.sqrt((-0.5 / C)) * t_0
	elif C <= 1.3e+104:
		tmp = 0.0 - (math.pow((2.0 * F), 0.5) / math.sqrt(B_m))
	elif C <= 1.45e+228:
		tmp = -1.0 / (((B_m * B_m) - (A * (4.0 * C))) / (C * math.sqrt((F * (A * -16.0)))))
	else:
		tmp = math.sqrt((-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(Float64(2.0 * F)))
	tmp = 0.0
	if (C <= -4.2e-60)
		tmp = Float64(sqrt(Float64(-0.5 / C)) * t_0);
	elseif (C <= 1.3e+104)
		tmp = Float64(0.0 - Float64((Float64(2.0 * F) ^ 0.5) / sqrt(B_m)));
	elseif (C <= 1.45e+228)
		tmp = Float64(-1.0 / Float64(Float64(Float64(B_m * B_m) - Float64(A * Float64(4.0 * C))) / Float64(C * sqrt(Float64(F * Float64(A * -16.0))))));
	else
		tmp = Float64(sqrt(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 * F));
	tmp = 0.0;
	if (C <= -4.2e-60)
		tmp = sqrt((-0.5 / C)) * t_0;
	elseif (C <= 1.3e+104)
		tmp = 0.0 - (((2.0 * F) ^ 0.5) / sqrt(B_m));
	elseif (C <= 1.45e+228)
		tmp = -1.0 / (((B_m * B_m) - (A * (4.0 * C))) / (C * sqrt((F * (A * -16.0)))));
	else
		tmp = sqrt((-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[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -4.2e-60], N[(N[Sqrt[N[(-0.5 / C), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[C, 1.3e+104], N[(0.0 - N[(N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 1.45e+228], N[(-1.0 / N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(C * N[Sqrt[N[(F * N[(A * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-0.5 / A), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if C < -4.19999999999999982e-60

   1. Initial program 6.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 19.1%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 20.7%

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

   8. Taylor expanded in A around inf

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

      1. /-lowering-/.f64 46.7%

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

   10. Simplified 46.7%

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

   if -4.19999999999999982e-60 < C < 1.3e104

   1. Initial program 28.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 31.5%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 41.9%

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

   8. Taylor expanded in B around inf

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

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

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

      2. /-lowering-/.f64 28.2%

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

   10. Simplified 28.2%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{1}}{\sqrt{B}}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}\right)\right) \]

       3. un-div-inv N/A

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

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

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

       5. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}}\right), \left(\sqrt{B}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot F\right), \frac{1}{2}\right), \left(\sqrt{B}\right)\right)\right) \]

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

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

       8. sqrt-lowering-sqrt.f64 28.2%

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

   12. Applied egg-rr 28.2%

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

   if 1.3e104 < C < 1.45000000000000001e228

   1. Initial program 9.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 32.9%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 6.3%

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

   7. Simplified 6.3%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. clear-num N/A

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

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

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

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

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

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

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

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

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

      15. associate-*r* N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

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

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

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

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

   9. Applied egg-rr 2.3%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. sqrt-prod N/A

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

       4. pow2 N/A

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

       5. sqrt-pow1 N/A

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

       6. metadata-eval N/A

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

       7. metadata-eval N/A

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

       8. pow-flip N/A

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

       9. inv-pow N/A

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

       10. remove-double-div N/A

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

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

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

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

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

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

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

       14. *-lowering-*.f64 49.1%

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

   11. Applied egg-rr 49.1%

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

   if 1.45000000000000001e228 < C

   1. Initial program 2.1%

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

   3. Taylor expanded in 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 25.7%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 45.0%

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

   8. Taylor expanded in A around -inf

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

      1. /-lowering-/.f64 48.0%

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

   10. Simplified 48.0%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \sqrt{\color{blue}{\frac{-0.5}{A}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -4.2 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\frac{-0.5}{C}} \cdot \left(0 - \sqrt{2 \cdot F}\right)\\ \mathbf{elif}\;C \leq 1.3 \cdot 10^{+104}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}\\ \mathbf{elif}\;C \leq 1.45 \cdot 10^{+228}:\\ \;\;\;\;\frac{-1}{\frac{B \cdot B - A \cdot \left(4 \cdot C\right)}{C \cdot \sqrt{F \cdot \left(A \cdot -16\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-0.5}{A}} \cdot \left(0 - \sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq -6.5 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{elif}\;C \leq 4.5 \cdot 10^{+104}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B\_m}}\\ \mathbf{elif}\;C \leq 3.5 \cdot 10^{+228}:\\ \;\;\;\;\frac{-1}{\frac{B\_m \cdot B\_m - A \cdot \left(4 \cdot C\right)}{C \cdot \sqrt{F \cdot \left(A \cdot -16\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C -6.5e+63)
   (* (sqrt (* -0.5 (/ F C))) (- 0.0 (sqrt 2.0)))
   (if (<= C 4.5e+104)
     (- 0.0 (/ (pow (* 2.0 F) 0.5) (sqrt B_m)))
     (if (<= C 3.5e+228)
       (/
        -1.0
        (/ (- (* B_m B_m) (* A (* 4.0 C))) (* C (sqrt (* F (* A -16.0))))))
       (- 0.0 (* (sqrt 2.0) (sqrt (* -0.5 (/ F A)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -6.5e+63) {
		tmp = sqrt((-0.5 * (F / C))) * (0.0 - sqrt(2.0));
	} else if (C <= 4.5e+104) {
		tmp = 0.0 - (pow((2.0 * F), 0.5) / sqrt(B_m));
	} else if (C <= 3.5e+228) {
		tmp = -1.0 / (((B_m * B_m) - (A * (4.0 * C))) / (C * sqrt((F * (A * -16.0)))));
	} else {
		tmp = 0.0 - (sqrt(2.0) * sqrt((-0.5 * (F / A))));
	}
	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 (c <= (-6.5d+63)) then
        tmp = sqrt(((-0.5d0) * (f / c))) * (0.0d0 - sqrt(2.0d0))
    else if (c <= 4.5d+104) then
        tmp = 0.0d0 - (((2.0d0 * f) ** 0.5d0) / sqrt(b_m))
    else if (c <= 3.5d+228) then
        tmp = (-1.0d0) / (((b_m * b_m) - (a * (4.0d0 * c))) / (c * sqrt((f * (a * (-16.0d0))))))
    else
        tmp = 0.0d0 - (sqrt(2.0d0) * sqrt(((-0.5d0) * (f / a))))
    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 (C <= -6.5e+63) {
		tmp = Math.sqrt((-0.5 * (F / C))) * (0.0 - Math.sqrt(2.0));
	} else if (C <= 4.5e+104) {
		tmp = 0.0 - (Math.pow((2.0 * F), 0.5) / Math.sqrt(B_m));
	} else if (C <= 3.5e+228) {
		tmp = -1.0 / (((B_m * B_m) - (A * (4.0 * C))) / (C * Math.sqrt((F * (A * -16.0)))));
	} else {
		tmp = 0.0 - (Math.sqrt(2.0) * Math.sqrt((-0.5 * (F / A))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= -6.5e+63:
		tmp = math.sqrt((-0.5 * (F / C))) * (0.0 - math.sqrt(2.0))
	elif C <= 4.5e+104:
		tmp = 0.0 - (math.pow((2.0 * F), 0.5) / math.sqrt(B_m))
	elif C <= 3.5e+228:
		tmp = -1.0 / (((B_m * B_m) - (A * (4.0 * C))) / (C * math.sqrt((F * (A * -16.0)))))
	else:
		tmp = 0.0 - (math.sqrt(2.0) * math.sqrt((-0.5 * (F / A))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= -6.5e+63)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * Float64(0.0 - sqrt(2.0)));
	elseif (C <= 4.5e+104)
		tmp = Float64(0.0 - Float64((Float64(2.0 * F) ^ 0.5) / sqrt(B_m)));
	elseif (C <= 3.5e+228)
		tmp = Float64(-1.0 / Float64(Float64(Float64(B_m * B_m) - Float64(A * Float64(4.0 * C))) / Float64(C * sqrt(Float64(F * Float64(A * -16.0))))));
	else
		tmp = Float64(0.0 - Float64(sqrt(2.0) * sqrt(Float64(-0.5 * Float64(F / A)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= -6.5e+63)
		tmp = sqrt((-0.5 * (F / C))) * (0.0 - sqrt(2.0));
	elseif (C <= 4.5e+104)
		tmp = 0.0 - (((2.0 * F) ^ 0.5) / sqrt(B_m));
	elseif (C <= 3.5e+228)
		tmp = -1.0 / (((B_m * B_m) - (A * (4.0 * C))) / (C * sqrt((F * (A * -16.0)))));
	else
		tmp = 0.0 - (sqrt(2.0) * sqrt((-0.5 * (F / A))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, -6.5e+63], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 4.5e+104], N[(0.0 - N[(N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 3.5e+228], N[(-1.0 / N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(C * N[Sqrt[N[(F * N[(A * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if C < -6.49999999999999992e63

   1. Initial program 0.9%

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

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   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 8.9%

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

   6. Taylor expanded in A around inf

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

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

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

      2. /-lowering-/.f64 34.8%

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

   8. Simplified 34.8%

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

   if -6.49999999999999992e63 < C < 4.4999999999999998e104

   1. Initial program 27.2%

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

   3. Taylor expanded in 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 33.0%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 43.0%

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

   8. Taylor expanded in B around inf

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

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

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

      2. /-lowering-/.f64 27.0%

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

   10. Simplified 27.0%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{1}}{\sqrt{B}}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}\right)\right) \]

       3. un-div-inv N/A

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

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

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

       5. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}}\right), \left(\sqrt{B}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot F\right), \frac{1}{2}\right), \left(\sqrt{B}\right)\right)\right) \]

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

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

       8. sqrt-lowering-sqrt.f64 27.0%

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

   12. Applied egg-rr 27.0%

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

   if 4.4999999999999998e104 < C < 3.5000000000000002e228

   1. Initial program 9.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 32.9%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 6.3%

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

   7. Simplified 6.3%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. clear-num N/A

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

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

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

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

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

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

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

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

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

      15. associate-*r* N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

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

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

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

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

   9. Applied egg-rr 2.3%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. sqrt-prod N/A

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

       4. pow2 N/A

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

       5. sqrt-pow1 N/A

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

       6. metadata-eval N/A

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

       7. metadata-eval N/A

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

       8. pow-flip N/A

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

       9. inv-pow N/A

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

       10. remove-double-div N/A

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

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

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

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

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

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

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

       14. *-lowering-*.f64 49.1%

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

   11. Applied egg-rr 49.1%

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

   if 3.5000000000000002e228 < C

   1. Initial program 2.1%

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

   3. Taylor expanded in 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 25.7%

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

   6. Taylor expanded in A 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 44.9%

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

   8. Simplified 44.9%

      \[\leadsto -\sqrt{\color{blue}{-0.5 \cdot \frac{F}{A}}} \cdot \sqrt{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -6.5 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{elif}\;C \leq 4.5 \cdot 10^{+104}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}\\ \mathbf{elif}\;C \leq 3.5 \cdot 10^{+228}:\\ \;\;\;\;\frac{-1}{\frac{B \cdot B - A \cdot \left(4 \cdot C\right)}{C \cdot \sqrt{F \cdot \left(A \cdot -16\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C \cdot -16\right)} \cdot \left|A\right|}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 95000000:\\ \;\;\;\;\sqrt{\frac{-0.5}{A}} \cdot \left(0 - \sqrt{2 \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.4e-137)
   (/ (* (sqrt (* F (* C -16.0))) (fabs A)) (- (* 4.0 (* A C)) (* B_m B_m)))
   (if (<= B_m 95000000.0)
     (* (sqrt (/ -0.5 A)) (- 0.0 (sqrt (* 2.0 F))))
     (- 0.0 (/ (pow (* 2.0 F) 0.5) (sqrt B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.4e-137) {
		tmp = (sqrt((F * (C * -16.0))) * fabs(A)) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 95000000.0) {
		tmp = sqrt((-0.5 / A)) * (0.0 - sqrt((2.0 * F)));
	} else {
		tmp = 0.0 - (pow((2.0 * F), 0.5) / sqrt(B_m));
	}
	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 <= 3.4d-137) then
        tmp = (sqrt((f * (c * (-16.0d0)))) * abs(a)) / ((4.0d0 * (a * c)) - (b_m * b_m))
    else if (b_m <= 95000000.0d0) then
        tmp = sqrt(((-0.5d0) / a)) * (0.0d0 - sqrt((2.0d0 * f)))
    else
        tmp = 0.0d0 - (((2.0d0 * f) ** 0.5d0) / sqrt(b_m))
    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 <= 3.4e-137) {
		tmp = (Math.sqrt((F * (C * -16.0))) * Math.abs(A)) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 95000000.0) {
		tmp = Math.sqrt((-0.5 / A)) * (0.0 - Math.sqrt((2.0 * F)));
	} else {
		tmp = 0.0 - (Math.pow((2.0 * F), 0.5) / Math.sqrt(B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 3.4e-137:
		tmp = (math.sqrt((F * (C * -16.0))) * math.fabs(A)) / ((4.0 * (A * C)) - (B_m * B_m))
	elif B_m <= 95000000.0:
		tmp = math.sqrt((-0.5 / A)) * (0.0 - math.sqrt((2.0 * F)))
	else:
		tmp = 0.0 - (math.pow((2.0 * F), 0.5) / math.sqrt(B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.4e-137)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(C * -16.0))) * abs(A)) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (B_m <= 95000000.0)
		tmp = Float64(sqrt(Float64(-0.5 / A)) * Float64(0.0 - sqrt(Float64(2.0 * F))));
	else
		tmp = Float64(0.0 - Float64((Float64(2.0 * F) ^ 0.5) / sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 3.4e-137)
		tmp = (sqrt((F * (C * -16.0))) * abs(A)) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (B_m <= 95000000.0)
		tmp = sqrt((-0.5 / A)) * (0.0 - sqrt((2.0 * F)));
	else
		tmp = 0.0 - (((2.0 * F) ^ 0.5) / sqrt(B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 3.4e-137], N[(N[(N[Sqrt[N[(F * N[(C * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[A], $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 95000000.0], N[(N[Sqrt[N[(-0.5 / A), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.40000000000000014e-137

   1. Initial program 21.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 28.0%

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

   5. Taylor expanded in B around 0

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

      1. associate-*r* N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-lowering-*.f64 10.1%

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

   7. Simplified 10.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. pow1/2 N/A

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

      14. rem-sqrt-square N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, -16\right)\right)\right), \left(\left|A\right|\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(A, C\right)}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      15. fabs-lowering-fabs.f64 17.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, -16\right)\right)\right), \mathsf{fabs.f64}\left(A\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(A, C\right)}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.9%

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

   if 3.40000000000000014e-137 < B < 9.5e7

   1. Initial program 17.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 36.4%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 42.4%

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

   8. Taylor expanded in A around -inf

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

      1. /-lowering-/.f64 38.6%

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

   10. Simplified 38.6%

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

   if 9.5e7 < B

   1. Initial program 7.3%

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

   3. Taylor expanded in 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 21.5%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 28.6%

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

   8. Taylor expanded in B around inf

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

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

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

      2. /-lowering-/.f64 63.8%

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

   10. Simplified 63.8%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{1}}{\sqrt{B}}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}\right)\right) \]

       3. un-div-inv N/A

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

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

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

       5. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}}\right), \left(\sqrt{B}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot F\right), \frac{1}{2}\right), \left(\sqrt{B}\right)\right)\right) \]

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

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

       8. sqrt-lowering-sqrt.f64 63.8%

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

   12. Applied egg-rr 63.8%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C \cdot -16\right)} \cdot \left|A\right|}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 95000000:\\ \;\;\;\;\sqrt{\frac{-0.5}{A}} \cdot \left(0 - \sqrt{2 \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.32 \cdot 10^{-276}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 64000000:\\ \;\;\;\;\sqrt{\frac{-0.5}{A}} \cdot \left(0 - \sqrt{2 \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.32e-276)
   (/
    (sqrt (* (+ (* B_m B_m) (* -4.0 (* A C))) (* (* 2.0 F) (* 2.0 A))))
    (- (* 4.0 (* A C)) (* B_m B_m)))
   (if (<= B_m 64000000.0)
     (* (sqrt (/ -0.5 A)) (- 0.0 (sqrt (* 2.0 F))))
     (- 0.0 (/ (pow (* 2.0 F) 0.5) (sqrt B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.32e-276) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (2.0 * A)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 64000000.0) {
		tmp = sqrt((-0.5 / A)) * (0.0 - sqrt((2.0 * F)));
	} else {
		tmp = 0.0 - (pow((2.0 * F), 0.5) / sqrt(B_m));
	}
	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 <= 1.32d-276) then
        tmp = sqrt((((b_m * b_m) + ((-4.0d0) * (a * c))) * ((2.0d0 * f) * (2.0d0 * a)))) / ((4.0d0 * (a * c)) - (b_m * b_m))
    else if (b_m <= 64000000.0d0) then
        tmp = sqrt(((-0.5d0) / a)) * (0.0d0 - sqrt((2.0d0 * f)))
    else
        tmp = 0.0d0 - (((2.0d0 * f) ** 0.5d0) / sqrt(b_m))
    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 <= 1.32e-276) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (2.0 * A)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 64000000.0) {
		tmp = Math.sqrt((-0.5 / A)) * (0.0 - Math.sqrt((2.0 * F)));
	} else {
		tmp = 0.0 - (Math.pow((2.0 * F), 0.5) / Math.sqrt(B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.32e-276:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (2.0 * A)))) / ((4.0 * (A * C)) - (B_m * B_m))
	elif B_m <= 64000000.0:
		tmp = math.sqrt((-0.5 / A)) * (0.0 - math.sqrt((2.0 * F)))
	else:
		tmp = 0.0 - (math.pow((2.0 * F), 0.5) / math.sqrt(B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.32e-276)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(2.0 * F) * Float64(2.0 * A)))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (B_m <= 64000000.0)
		tmp = Float64(sqrt(Float64(-0.5 / A)) * Float64(0.0 - sqrt(Float64(2.0 * F))));
	else
		tmp = Float64(0.0 - Float64((Float64(2.0 * F) ^ 0.5) / sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.32e-276)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (2.0 * A)))) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (B_m <= 64000000.0)
		tmp = sqrt((-0.5 / A)) * (0.0 - sqrt((2.0 * F)));
	else
		tmp = 0.0 - (((2.0 * F) ^ 0.5) / sqrt(B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.32e-276], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 64000000.0], N[(N[Sqrt[N[(-0.5 / A), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.31999999999999992e-276

   1. Initial program 21.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 27.8%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

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

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. sub-neg N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

   6. Applied egg-rr 27.2%

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

   7. Taylor expanded in A around inf

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

      1. *-lowering-*.f64 12.2%

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

   9. Simplified 12.2%

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

   if 1.31999999999999992e-276 < B < 6.4e7

   1. Initial program 21.1%

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

   3. Taylor expanded in 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 27.5%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 32.7%

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

   8. Taylor expanded in A around -inf

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

      1. /-lowering-/.f64 32.5%

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

   10. Simplified 32.5%

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

   if 6.4e7 < B

   1. Initial program 7.3%

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

   3. Taylor expanded in 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 21.5%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 28.6%

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

   8. Taylor expanded in B around inf

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

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

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

      2. /-lowering-/.f64 63.8%

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

   10. Simplified 63.8%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{1}}{\sqrt{B}}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}\right)\right) \]

       3. un-div-inv N/A

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

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

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

       5. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}}\right), \left(\sqrt{B}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot F\right), \frac{1}{2}\right), \left(\sqrt{B}\right)\right)\right) \]

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

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

       8. sqrt-lowering-sqrt.f64 63.8%

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

   12. Applied egg-rr 63.8%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.32 \cdot 10^{-276}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 64000000:\\ \;\;\;\;\sqrt{\frac{-0.5}{A}} \cdot \left(0 - \sqrt{2 \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7.6 \cdot 10^{-137}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot A + -0.5 \cdot \frac{B\_m \cdot B\_m}{C}\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 6.5 \cdot 10^{-91}:\\ \;\;\;\;0 - \sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7.6e-137)
   (/
    (sqrt
     (*
      (+ (* B_m B_m) (* -4.0 (* A C)))
      (* (* 2.0 F) (+ (* 2.0 A) (* -0.5 (/ (* B_m B_m) C))))))
    (- (* 4.0 (* A C)) (* B_m B_m)))
   (if (<= B_m 6.5e-91)
     (- 0.0 (* (sqrt 2.0) (sqrt (* -0.5 (/ F A)))))
     (- 0.0 (/ (pow (* 2.0 F) 0.5) (sqrt B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.6e-137) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * A) + (-0.5 * ((B_m * B_m) / C)))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 6.5e-91) {
		tmp = 0.0 - (sqrt(2.0) * sqrt((-0.5 * (F / A))));
	} else {
		tmp = 0.0 - (pow((2.0 * F), 0.5) / sqrt(B_m));
	}
	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 <= 7.6d-137) then
        tmp = sqrt((((b_m * b_m) + ((-4.0d0) * (a * c))) * ((2.0d0 * f) * ((2.0d0 * a) + ((-0.5d0) * ((b_m * b_m) / c)))))) / ((4.0d0 * (a * c)) - (b_m * b_m))
    else if (b_m <= 6.5d-91) then
        tmp = 0.0d0 - (sqrt(2.0d0) * sqrt(((-0.5d0) * (f / a))))
    else
        tmp = 0.0d0 - (((2.0d0 * f) ** 0.5d0) / sqrt(b_m))
    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 <= 7.6e-137) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * A) + (-0.5 * ((B_m * B_m) / C)))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 6.5e-91) {
		tmp = 0.0 - (Math.sqrt(2.0) * Math.sqrt((-0.5 * (F / A))));
	} else {
		tmp = 0.0 - (Math.pow((2.0 * F), 0.5) / Math.sqrt(B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 7.6e-137:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * A) + (-0.5 * ((B_m * B_m) / C)))))) / ((4.0 * (A * C)) - (B_m * B_m))
	elif B_m <= 6.5e-91:
		tmp = 0.0 - (math.sqrt(2.0) * math.sqrt((-0.5 * (F / A))))
	else:
		tmp = 0.0 - (math.pow((2.0 * F), 0.5) / math.sqrt(B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7.6e-137)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(2.0 * F) * Float64(Float64(2.0 * A) + Float64(-0.5 * Float64(Float64(B_m * B_m) / C)))))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (B_m <= 6.5e-91)
		tmp = Float64(0.0 - Float64(sqrt(2.0) * sqrt(Float64(-0.5 * Float64(F / A)))));
	else
		tmp = Float64(0.0 - Float64((Float64(2.0 * F) ^ 0.5) / sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 7.6e-137)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * A) + (-0.5 * ((B_m * B_m) / C)))))) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (B_m <= 6.5e-91)
		tmp = 0.0 - (sqrt(2.0) * sqrt((-0.5 * (F / A))));
	else
		tmp = 0.0 - (((2.0 * F) ^ 0.5) / sqrt(B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 7.6e-137], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(N[(2.0 * A), $MachinePrecision] + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6.5e-91], N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 7.59999999999999997e-137

   1. Initial program 21.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 28.0%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

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

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. sub-neg N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

   6. Applied egg-rr 27.5%

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

   7. Taylor expanded in C around -inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-lowering-*.f64 13.9%

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

   9. Simplified 13.9%

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

   if 7.59999999999999997e-137 < B < 6.5000000000000001e-91

   1. Initial program 19.2%

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

   3. Taylor expanded in 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 42.9%

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

   6. Taylor expanded in A 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 35.1%

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

   8. Simplified 35.1%

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

   if 6.5000000000000001e-91 < B

   1. Initial program 9.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 24.2%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 30.7%

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

   8. Taylor expanded in B around inf

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

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

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

      2. /-lowering-/.f64 53.0%

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

   10. Simplified 53.0%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{1}}{\sqrt{B}}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}\right)\right) \]

       3. un-div-inv N/A

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

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

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

       5. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}}\right), \left(\sqrt{B}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot F\right), \frac{1}{2}\right), \left(\sqrt{B}\right)\right)\right) \]

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

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

       8. sqrt-lowering-sqrt.f64 53.0%

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

   12. Applied egg-rr 53.0%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.6 \cdot 10^{-137}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot A + -0.5 \cdot \frac{B \cdot B}{C}\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-91}:\\ \;\;\;\;0 - \sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot A + -0.5 \cdot \frac{B\_m \cdot B\_m}{C}\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.8e-133)
   (/
    (sqrt
     (*
      (+ (* B_m B_m) (* -4.0 (* A C)))
      (* (* 2.0 F) (+ (* 2.0 A) (* -0.5 (/ (* B_m B_m) C))))))
    (- (* 4.0 (* A C)) (* B_m B_m)))
   (- 0.0 (/ (pow (* 2.0 F) 0.5) (sqrt B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.8e-133) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * A) + (-0.5 * ((B_m * B_m) / C)))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = 0.0 - (pow((2.0 * F), 0.5) / sqrt(B_m));
	}
	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 <= 3.8d-133) then
        tmp = sqrt((((b_m * b_m) + ((-4.0d0) * (a * c))) * ((2.0d0 * f) * ((2.0d0 * a) + ((-0.5d0) * ((b_m * b_m) / c)))))) / ((4.0d0 * (a * c)) - (b_m * b_m))
    else
        tmp = 0.0d0 - (((2.0d0 * f) ** 0.5d0) / sqrt(b_m))
    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 <= 3.8e-133) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * A) + (-0.5 * ((B_m * B_m) / C)))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = 0.0 - (Math.pow((2.0 * F), 0.5) / Math.sqrt(B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 3.8e-133:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * A) + (-0.5 * ((B_m * B_m) / C)))))) / ((4.0 * (A * C)) - (B_m * B_m))
	else:
		tmp = 0.0 - (math.pow((2.0 * F), 0.5) / math.sqrt(B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.8e-133)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(2.0 * F) * Float64(Float64(2.0 * A) + Float64(-0.5 * Float64(Float64(B_m * B_m) / C)))))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	else
		tmp = Float64(0.0 - Float64((Float64(2.0 * F) ^ 0.5) / sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 3.8e-133)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * A) + (-0.5 * ((B_m * B_m) / C)))))) / ((4.0 * (A * C)) - (B_m * B_m));
	else
		tmp = 0.0 - (((2.0 * F) ^ 0.5) / sqrt(B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 3.8e-133], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(N[(2.0 * A), $MachinePrecision] + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.8000000000000003e-133

   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 27.8%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

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

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. sub-neg N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

   6. Applied egg-rr 27.3%

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

   7. Taylor expanded in C around -inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-lowering-*.f64 14.5%

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

   9. Simplified 14.5%

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

   if 3.8000000000000003e-133 < B

   1. Initial program 11.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 26.9%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 33.7%

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

   8. Taylor expanded in B around inf

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

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

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

      2. /-lowering-/.f64 49.3%

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

   10. Simplified 49.3%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{1}}{\sqrt{B}}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}\right)\right) \]

       3. un-div-inv N/A

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

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

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

       5. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot F\right)}^{\frac{1}{2}}\right), \left(\sqrt{B}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot F\right), \frac{1}{2}\right), \left(\sqrt{B}\right)\right)\right) \]

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

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

       8. sqrt-lowering-sqrt.f64 49.3%

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

   12. Applied egg-rr 49.3%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot A + -0.5 \cdot \frac{B \cdot B}{C}\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{{\left(2 \cdot F\right)}^{0.5}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot A + -0.5 \cdot \frac{B\_m \cdot B\_m}{C}\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(0 - {B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.5e-133)
   (/
    (sqrt
     (*
      (+ (* B_m B_m) (* -4.0 (* A C)))
      (* (* 2.0 F) (+ (* 2.0 A) (* -0.5 (/ (* B_m B_m) C))))))
    (- (* 4.0 (* A C)) (* B_m B_m)))
   (* (sqrt (* 2.0 F)) (- 0.0 (pow B_m -0.5)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.5e-133) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * A) + (-0.5 * ((B_m * B_m) / C)))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = sqrt((2.0 * F)) * (0.0 - pow(B_m, -0.5));
	}
	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 <= 4.5d-133) then
        tmp = sqrt((((b_m * b_m) + ((-4.0d0) * (a * c))) * ((2.0d0 * f) * ((2.0d0 * a) + ((-0.5d0) * ((b_m * b_m) / c)))))) / ((4.0d0 * (a * c)) - (b_m * b_m))
    else
        tmp = sqrt((2.0d0 * f)) * (0.0d0 - (b_m ** (-0.5d0)))
    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 <= 4.5e-133) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * A) + (-0.5 * ((B_m * B_m) / C)))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = Math.sqrt((2.0 * F)) * (0.0 - Math.pow(B_m, -0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 4.5e-133:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * A) + (-0.5 * ((B_m * B_m) / C)))))) / ((4.0 * (A * C)) - (B_m * B_m))
	else:
		tmp = math.sqrt((2.0 * F)) * (0.0 - math.pow(B_m, -0.5))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4.5e-133)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(2.0 * F) * Float64(Float64(2.0 * A) + Float64(-0.5 * Float64(Float64(B_m * B_m) / C)))))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(0.0 - (B_m ^ -0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 4.5e-133)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * A) + (-0.5 * ((B_m * B_m) / C)))))) / ((4.0 * (A * C)) - (B_m * B_m));
	else
		tmp = sqrt((2.0 * F)) * (0.0 - (B_m ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 4.5e-133], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(N[(2.0 * A), $MachinePrecision] + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Power[B$95$m, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.50000000000000009e-133

   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 27.8%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

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

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. sub-neg N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

   6. Applied egg-rr 27.3%

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

   7. Taylor expanded in C around -inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-lowering-*.f64 14.5%

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

   9. Simplified 14.5%

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

   if 4.50000000000000009e-133 < B

   1. Initial program 11.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 26.9%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*r* N/A

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

   7. Applied egg-rr 33.7%

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

   8. Taylor expanded in B around inf

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

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

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

      2. /-lowering-/.f64 49.3%

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

   10. Simplified 49.3%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. inv-pow N/A

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

       2. sqrt-pow1 N/A

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

       3. metadata-eval N/A

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

       4. pow-lowering-pow.f64 49.3%

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

   12. Applied egg-rr 49.3%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot A + -0.5 \cdot \frac{B \cdot B}{C}\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(0 - {B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 31.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6.2 \cdot 10^{-282}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 6 \cdot 10^{+19}:\\ \;\;\;\;\frac{C \cdot {\left(F \cdot \left(A \cdot -16\right)\right)}^{0.5}}{A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;0 - {\left(\frac{2 \cdot F}{B\_m}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 6.2e-282)
   (/
    (sqrt (* (+ (* B_m B_m) (* -4.0 (* A C))) (* (* 2.0 F) (* 2.0 A))))
    (- (* 4.0 (* A C)) (* B_m B_m)))
   (if (<= B_m 6e+19)
     (/ (* C (pow (* F (* A -16.0)) 0.5)) (- (* A (* 4.0 C)) (* B_m B_m)))
     (- 0.0 (pow (/ (* 2.0 F) B_m) 0.5)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6.2e-282) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (2.0 * A)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 6e+19) {
		tmp = (C * pow((F * (A * -16.0)), 0.5)) / ((A * (4.0 * C)) - (B_m * B_m));
	} else {
		tmp = 0.0 - pow(((2.0 * F) / B_m), 0.5);
	}
	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 <= 6.2d-282) then
        tmp = sqrt((((b_m * b_m) + ((-4.0d0) * (a * c))) * ((2.0d0 * f) * (2.0d0 * a)))) / ((4.0d0 * (a * c)) - (b_m * b_m))
    else if (b_m <= 6d+19) then
        tmp = (c * ((f * (a * (-16.0d0))) ** 0.5d0)) / ((a * (4.0d0 * c)) - (b_m * b_m))
    else
        tmp = 0.0d0 - (((2.0d0 * f) / b_m) ** 0.5d0)
    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 <= 6.2e-282) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (2.0 * A)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 6e+19) {
		tmp = (C * Math.pow((F * (A * -16.0)), 0.5)) / ((A * (4.0 * C)) - (B_m * B_m));
	} else {
		tmp = 0.0 - Math.pow(((2.0 * F) / B_m), 0.5);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 6.2e-282:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (2.0 * A)))) / ((4.0 * (A * C)) - (B_m * B_m))
	elif B_m <= 6e+19:
		tmp = (C * math.pow((F * (A * -16.0)), 0.5)) / ((A * (4.0 * C)) - (B_m * B_m))
	else:
		tmp = 0.0 - math.pow(((2.0 * F) / B_m), 0.5)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 6.2e-282)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(2.0 * F) * Float64(2.0 * A)))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (B_m <= 6e+19)
		tmp = Float64(Float64(C * (Float64(F * Float64(A * -16.0)) ^ 0.5)) / Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m)));
	else
		tmp = Float64(0.0 - (Float64(Float64(2.0 * F) / B_m) ^ 0.5));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 6.2e-282)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (2.0 * A)))) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (B_m <= 6e+19)
		tmp = (C * ((F * (A * -16.0)) ^ 0.5)) / ((A * (4.0 * C)) - (B_m * B_m));
	else
		tmp = 0.0 - (((2.0 * F) / B_m) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 6.2e-282], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6e+19], N[(N[(C * N[Power[N[(F * N[(A * -16.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Power[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.20000000000000027e-282

   1. Initial program 21.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 28.0%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

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

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. sub-neg N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

   6. Applied egg-rr 27.3%

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

   7. Taylor expanded in A around inf

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

      1. *-lowering-*.f64 12.3%

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

   9. Simplified 12.3%

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

   if 6.20000000000000027e-282 < B < 6e19

   1. Initial program 19.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.6%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 17.8%

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

   7. Simplified 17.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. clear-num N/A

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(A \cdot \left(C \cdot 4\right)\right), \left(B \cdot B\right)\right), \left(\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot 4\right)\right), \left(B \cdot B\right)\right), \left(\sqrt{\color{blue}{-16} \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \left(B \cdot B\right)\right), \left(\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \left(\sqrt{-16 \cdot \color{blue}{\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}\right)\right)\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)\right)\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(\left(-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)\right) \cdot F\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)\right)\right)\right) \]

      19. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(\left(-16 \cdot A\right) \cdot \left(C \cdot C\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(\left(A \cdot -16\right) \cdot \left(C \cdot C\right)\right)\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A \cdot \left(-16 \cdot \left(C \cdot C\right)\right)\right)\right)\right)\right)\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(A, \left(-16 \cdot \left(C \cdot C\right)\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 17.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{A \cdot \left(C \cdot 4\right) - B \cdot B}{\sqrt{F \cdot \left(A \cdot \left(-16 \cdot \left(C \cdot C\right)\right)\right)}}}} \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{\sqrt{F \cdot \left(A \cdot \left(-16 \cdot \left(C \cdot C\right)\right)\right)}}{\color{blue}{A \cdot \left(C \cdot 4\right) - B \cdot B}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{F \cdot \left(A \cdot \left(-16 \cdot \left(C \cdot C\right)\right)\right)}\right), \color{blue}{\left(A \cdot \left(C \cdot 4\right) - B \cdot B\right)}\right) \]

       3. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot \left(-16 \cdot \left(C \cdot C\right)\right)\right)\right)}^{\frac{1}{2}}\right), \left(\color{blue}{A \cdot \left(C \cdot 4\right)} - B \cdot B\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(\left(A \cdot -16\right) \cdot \left(C \cdot C\right)\right)\right)}^{\frac{1}{2}}\right), \left(A \cdot \left(C \cdot 4\right) - B \cdot B\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(F \cdot \left(A \cdot -16\right)\right) \cdot \left(C \cdot C\right)\right)}^{\frac{1}{2}}\right), \left(\color{blue}{A} \cdot \left(C \cdot 4\right) - B \cdot B\right)\right) \]

       6. unpow-prod-down N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot {\left(C \cdot C\right)}^{\frac{1}{2}}\right), \left(\color{blue}{A \cdot \left(C \cdot 4\right)} - B \cdot B\right)\right) \]

       7. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot \sqrt{C \cdot C}\right), \left(A \cdot \color{blue}{\left(C \cdot 4\right)} - B \cdot B\right)\right) \]

       8. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot \sqrt{{C}^{2}}\right), \left(A \cdot \left(\color{blue}{C} \cdot 4\right) - B \cdot B\right)\right) \]

       9. sqrt-pow1 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot {C}^{\left(\frac{2}{2}\right)}\right), \left(A \cdot \color{blue}{\left(C \cdot 4\right)} - B \cdot B\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot {C}^{1}\right), \left(A \cdot \left(C \cdot \color{blue}{4}\right) - B \cdot B\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot {C}^{\left(\mathsf{neg}\left(-1\right)\right)}\right), \left(A \cdot \left(C \cdot \color{blue}{4}\right) - B \cdot B\right)\right) \]

       12. pow-flip N/A

           \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot \frac{1}{{C}^{-1}}\right), \left(A \cdot \color{blue}{\left(C \cdot 4\right)} - B \cdot B\right)\right) \]

       13. inv-pow N/A

           \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot \frac{1}{\frac{1}{C}}\right), \left(A \cdot \left(C \cdot \color{blue}{4}\right) - B \cdot B\right)\right) \]

       14. remove-double-div N/A

           \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot C\right), \left(A \cdot \color{blue}{\left(C \cdot 4\right)} - B \cdot B\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}}\right), C\right), \left(\color{blue}{A \cdot \left(C \cdot 4\right)} - B \cdot B\right)\right) \]

       16. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(F \cdot \left(A \cdot -16\right)\right), \frac{1}{2}\right), C\right), \left(\color{blue}{A} \cdot \left(C \cdot 4\right) - B \cdot B\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \left(A \cdot -16\right)\right), \frac{1}{2}\right), C\right), \left(A \cdot \left(C \cdot 4\right) - B \cdot B\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(A, -16\right)\right), \frac{1}{2}\right), C\right), \left(A \cdot \left(C \cdot 4\right) - B \cdot B\right)\right) \]

       19. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(A, -16\right)\right), \frac{1}{2}\right), C\right), \mathsf{\_.f64}\left(\left(A \cdot \left(C \cdot 4\right)\right), \color{blue}{\left(B \cdot B\right)}\right)\right) \]

   11. Applied egg-rr 28.9%

       \[\leadsto \color{blue}{\frac{{\left(F \cdot \left(A \cdot -16\right)\right)}^{0.5} \cdot C}{A \cdot \left(C \cdot 4\right) - B \cdot B}} \]

   if 6e19 < B

   1. Initial program 7.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 20.5%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\right)\right) \]

      2. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(\frac{F \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(\frac{F \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(F \cdot \frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \left({F}^{\frac{1}{2}} \cdot {\left(\frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left(\sqrt{2} \cdot {F}^{\frac{1}{2}}\right) \cdot {\left(\frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

   7. Applied egg-rr 28.7%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \color{blue}{\left(\sqrt{\frac{1}{B}}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{B}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 67.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, B\right)\right)\right)\right) \]

   10. Simplified 67.7%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-unprod N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\left(2 \cdot F\right) \cdot \frac{1}{B}}\right)\right) \]

       2. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right)\right) \]

       3. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right), \frac{1}{2}\right)\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{2 \cdot F}{B}\right), \frac{1}{2}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot F\right), B\right), \frac{1}{2}\right)\right) \]

       6. *-lowering-*.f64 56.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, F\right), B\right), \frac{1}{2}\right)\right) \]

   12. Applied egg-rr 56.3%

       \[\leadsto -\color{blue}{{\left(\frac{2 \cdot F}{B}\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{-282}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{+19}:\\ \;\;\;\;\frac{C \cdot {\left(F \cdot \left(A \cdot -16\right)\right)}^{0.5}}{A \cdot \left(4 \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;0 - {\left(\frac{2 \cdot F}{B}\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 32.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.3 \cdot 10^{+19}:\\ \;\;\;\;\frac{C \cdot {\left(F \cdot \left(A \cdot -16\right)\right)}^{0.5}}{A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;0 - {\left(\frac{2 \cdot F}{B\_m}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.3e+19)
   (/ (* C (pow (* F (* A -16.0)) 0.5)) (- (* A (* 4.0 C)) (* B_m B_m)))
   (- 0.0 (pow (/ (* 2.0 F) B_m) 0.5))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.3e+19) {
		tmp = (C * pow((F * (A * -16.0)), 0.5)) / ((A * (4.0 * C)) - (B_m * B_m));
	} else {
		tmp = 0.0 - pow(((2.0 * F) / B_m), 0.5);
	}
	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 <= 2.3d+19) then
        tmp = (c * ((f * (a * (-16.0d0))) ** 0.5d0)) / ((a * (4.0d0 * c)) - (b_m * b_m))
    else
        tmp = 0.0d0 - (((2.0d0 * f) / b_m) ** 0.5d0)
    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 <= 2.3e+19) {
		tmp = (C * Math.pow((F * (A * -16.0)), 0.5)) / ((A * (4.0 * C)) - (B_m * B_m));
	} else {
		tmp = 0.0 - Math.pow(((2.0 * F) / B_m), 0.5);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 2.3e+19:
		tmp = (C * math.pow((F * (A * -16.0)), 0.5)) / ((A * (4.0 * C)) - (B_m * B_m))
	else:
		tmp = 0.0 - math.pow(((2.0 * F) / B_m), 0.5)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.3e+19)
		tmp = Float64(Float64(C * (Float64(F * Float64(A * -16.0)) ^ 0.5)) / Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m)));
	else
		tmp = Float64(0.0 - (Float64(Float64(2.0 * F) / B_m) ^ 0.5));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 2.3e+19)
		tmp = (C * ((F * (A * -16.0)) ^ 0.5)) / ((A * (4.0 * C)) - (B_m * B_m));
	else
		tmp = 0.0 - (((2.0 * F) / B_m) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2.3e+19], N[(N[(C * N[Power[N[(F * N[(A * -16.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Power[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.3e19

   1. Initial program 20.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 27.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 9.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 9.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left(A \cdot C\right) - B \cdot B\right), \color{blue}{\left(\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot A\right) \cdot C - B \cdot B\right), \left(\sqrt{\color{blue}{-16} \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(A \cdot 4\right) \cdot C - B \cdot B\right), \left(\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(C \cdot \left(A \cdot 4\right) - B \cdot B\right), \left(\sqrt{\color{blue}{-16} \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(C \cdot \left(A \cdot 4\right)\right), \left(B \cdot B\right)\right), \left(\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(A \cdot 4\right) \cdot C\right), \left(B \cdot B\right)\right), \left(\sqrt{\color{blue}{-16} \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(A \cdot \left(4 \cdot C\right)\right), \left(B \cdot B\right)\right), \left(\sqrt{\color{blue}{-16} \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(A \cdot \left(C \cdot 4\right)\right), \left(B \cdot B\right)\right), \left(\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot 4\right)\right), \left(B \cdot B\right)\right), \left(\sqrt{\color{blue}{-16} \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \left(B \cdot B\right)\right), \left(\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \left(\sqrt{-16 \cdot \color{blue}{\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}\right)\right)\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)\right)\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(\left(-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)\right) \cdot F\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)\right)\right)\right) \]

      19. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(\left(-16 \cdot A\right) \cdot \left(C \cdot C\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(\left(A \cdot -16\right) \cdot \left(C \cdot C\right)\right)\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A \cdot \left(-16 \cdot \left(C \cdot C\right)\right)\right)\right)\right)\right)\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(A, \left(-16 \cdot \left(C \cdot C\right)\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 9.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{A \cdot \left(C \cdot 4\right) - B \cdot B}{\sqrt{F \cdot \left(A \cdot \left(-16 \cdot \left(C \cdot C\right)\right)\right)}}}} \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{\sqrt{F \cdot \left(A \cdot \left(-16 \cdot \left(C \cdot C\right)\right)\right)}}{\color{blue}{A \cdot \left(C \cdot 4\right) - B \cdot B}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{F \cdot \left(A \cdot \left(-16 \cdot \left(C \cdot C\right)\right)\right)}\right), \color{blue}{\left(A \cdot \left(C \cdot 4\right) - B \cdot B\right)}\right) \]

       3. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot \left(-16 \cdot \left(C \cdot C\right)\right)\right)\right)}^{\frac{1}{2}}\right), \left(\color{blue}{A \cdot \left(C \cdot 4\right)} - B \cdot B\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(\left(A \cdot -16\right) \cdot \left(C \cdot C\right)\right)\right)}^{\frac{1}{2}}\right), \left(A \cdot \left(C \cdot 4\right) - B \cdot B\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(F \cdot \left(A \cdot -16\right)\right) \cdot \left(C \cdot C\right)\right)}^{\frac{1}{2}}\right), \left(\color{blue}{A} \cdot \left(C \cdot 4\right) - B \cdot B\right)\right) \]

       6. unpow-prod-down N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot {\left(C \cdot C\right)}^{\frac{1}{2}}\right), \left(\color{blue}{A \cdot \left(C \cdot 4\right)} - B \cdot B\right)\right) \]

       7. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot \sqrt{C \cdot C}\right), \left(A \cdot \color{blue}{\left(C \cdot 4\right)} - B \cdot B\right)\right) \]

       8. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot \sqrt{{C}^{2}}\right), \left(A \cdot \left(\color{blue}{C} \cdot 4\right) - B \cdot B\right)\right) \]

       9. sqrt-pow1 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot {C}^{\left(\frac{2}{2}\right)}\right), \left(A \cdot \color{blue}{\left(C \cdot 4\right)} - B \cdot B\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot {C}^{1}\right), \left(A \cdot \left(C \cdot \color{blue}{4}\right) - B \cdot B\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot {C}^{\left(\mathsf{neg}\left(-1\right)\right)}\right), \left(A \cdot \left(C \cdot \color{blue}{4}\right) - B \cdot B\right)\right) \]

       12. pow-flip N/A

           \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot \frac{1}{{C}^{-1}}\right), \left(A \cdot \color{blue}{\left(C \cdot 4\right)} - B \cdot B\right)\right) \]

       13. inv-pow N/A

           \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot \frac{1}{\frac{1}{C}}\right), \left(A \cdot \left(C \cdot \color{blue}{4}\right) - B \cdot B\right)\right) \]

       14. remove-double-div N/A

           \[\leadsto \mathsf{/.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}} \cdot C\right), \left(A \cdot \color{blue}{\left(C \cdot 4\right)} - B \cdot B\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(F \cdot \left(A \cdot -16\right)\right)}^{\frac{1}{2}}\right), C\right), \left(\color{blue}{A \cdot \left(C \cdot 4\right)} - B \cdot B\right)\right) \]

       16. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(F \cdot \left(A \cdot -16\right)\right), \frac{1}{2}\right), C\right), \left(\color{blue}{A} \cdot \left(C \cdot 4\right) - B \cdot B\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \left(A \cdot -16\right)\right), \frac{1}{2}\right), C\right), \left(A \cdot \left(C \cdot 4\right) - B \cdot B\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(A, -16\right)\right), \frac{1}{2}\right), C\right), \left(A \cdot \left(C \cdot 4\right) - B \cdot B\right)\right) \]

       19. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(A, -16\right)\right), \frac{1}{2}\right), C\right), \mathsf{\_.f64}\left(\left(A \cdot \left(C \cdot 4\right)\right), \color{blue}{\left(B \cdot B\right)}\right)\right) \]

   11. Applied egg-rr 17.1%

       \[\leadsto \color{blue}{\frac{{\left(F \cdot \left(A \cdot -16\right)\right)}^{0.5} \cdot C}{A \cdot \left(C \cdot 4\right) - B \cdot B}} \]

   if 2.3e19 < B

   1. Initial program 7.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 20.5%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\right)\right) \]

      2. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(\frac{F \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(\frac{F \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(F \cdot \frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \left({F}^{\frac{1}{2}} \cdot {\left(\frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left(\sqrt{2} \cdot {F}^{\frac{1}{2}}\right) \cdot {\left(\frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

   7. Applied egg-rr 28.7%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \color{blue}{\left(\sqrt{\frac{1}{B}}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{B}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 67.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, B\right)\right)\right)\right) \]

   10. Simplified 67.7%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-unprod N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\left(2 \cdot F\right) \cdot \frac{1}{B}}\right)\right) \]

       2. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right)\right) \]

       3. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right), \frac{1}{2}\right)\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{2 \cdot F}{B}\right), \frac{1}{2}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot F\right), B\right), \frac{1}{2}\right)\right) \]

       6. *-lowering-*.f64 56.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, F\right), B\right), \frac{1}{2}\right)\right) \]

   12. Applied egg-rr 56.3%

       \[\leadsto -\color{blue}{{\left(\frac{2 \cdot F}{B}\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.3 \cdot 10^{+19}:\\ \;\;\;\;\frac{C \cdot {\left(F \cdot \left(A \cdot -16\right)\right)}^{0.5}}{A \cdot \left(4 \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;0 - {\left(\frac{2 \cdot F}{B}\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;A \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;0 - {\left(\frac{2 \cdot F}{B\_m}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{4 \cdot F}{\frac{B\_m \cdot B\_m + \frac{A}{\frac{-0.25}{C}}}{A}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A 1.5e+58)
   (- 0.0 (pow (/ (* 2.0 F) B_m) 0.5))
   (- 0.0 (sqrt (/ (* 4.0 F) (/ (+ (* B_m B_m) (/ A (/ -0.25 C))) A))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= 1.5e+58) {
		tmp = 0.0 - pow(((2.0 * F) / B_m), 0.5);
	} else {
		tmp = 0.0 - sqrt(((4.0 * F) / (((B_m * B_m) + (A / (-0.25 / C))) / A)));
	}
	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 (a <= 1.5d+58) then
        tmp = 0.0d0 - (((2.0d0 * f) / b_m) ** 0.5d0)
    else
        tmp = 0.0d0 - sqrt(((4.0d0 * f) / (((b_m * b_m) + (a / ((-0.25d0) / c))) / a)))
    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 (A <= 1.5e+58) {
		tmp = 0.0 - Math.pow(((2.0 * F) / B_m), 0.5);
	} else {
		tmp = 0.0 - Math.sqrt(((4.0 * F) / (((B_m * B_m) + (A / (-0.25 / C))) / A)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if A <= 1.5e+58:
		tmp = 0.0 - math.pow(((2.0 * F) / B_m), 0.5)
	else:
		tmp = 0.0 - math.sqrt(((4.0 * F) / (((B_m * B_m) + (A / (-0.25 / C))) / A)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= 1.5e+58)
		tmp = Float64(0.0 - (Float64(Float64(2.0 * F) / B_m) ^ 0.5));
	else
		tmp = Float64(0.0 - sqrt(Float64(Float64(4.0 * F) / Float64(Float64(Float64(B_m * B_m) + Float64(A / Float64(-0.25 / C))) / A))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (A <= 1.5e+58)
		tmp = 0.0 - (((2.0 * F) / B_m) ^ 0.5);
	else
		tmp = 0.0 - sqrt(((4.0 * F) / (((B_m * B_m) + (A / (-0.25 / C))) / A)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, 1.5e+58], N[(0.0 - N[Power[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Sqrt[N[(N[(4.0 * F), $MachinePrecision] / N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A / N[(-0.25 / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < 1.5000000000000001e58

   1. Initial program 17.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 25.7%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\right)\right) \]

      2. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(\frac{F \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(\frac{F \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(F \cdot \frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \left({F}^{\frac{1}{2}} \cdot {\left(\frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left(\sqrt{2} \cdot {F}^{\frac{1}{2}}\right) \cdot {\left(\frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

   7. Applied egg-rr 33.6%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \color{blue}{\left(\sqrt{\frac{1}{B}}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{B}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 21.1%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, B\right)\right)\right)\right) \]

   10. Simplified 21.1%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-unprod N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\left(2 \cdot F\right) \cdot \frac{1}{B}}\right)\right) \]

       2. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right)\right) \]

       3. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right), \frac{1}{2}\right)\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{2 \cdot F}{B}\right), \frac{1}{2}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot F\right), B\right), \frac{1}{2}\right)\right) \]

       6. *-lowering-*.f64 17.8%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, F\right), B\right), \frac{1}{2}\right)\right) \]

   12. Applied egg-rr 17.8%

       \[\leadsto -\color{blue}{{\left(\frac{2 \cdot F}{B}\right)}^{0.5}} \]

   if 1.5000000000000001e58 < A

   1. Initial program 20.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 30.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\right)\right) \]

      2. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(\frac{F \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(\frac{F \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(F \cdot \frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \left({F}^{\frac{1}{2}} \cdot {\left(\frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left(\sqrt{2} \cdot {F}^{\frac{1}{2}}\right) \cdot {\left(\frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

   7. Applied egg-rr 41.4%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}}} \]

   8. Taylor expanded in A around inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 \cdot A\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 41.4%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 41.4%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \sqrt{\frac{\color{blue}{2 \cdot A}}{B \cdot B + A \cdot \left(C \cdot -4\right)}} \]
   11. Step-by-step derivation

       1. sqrt-unprod N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\left(2 \cdot F\right) \cdot \frac{2 \cdot A}{B \cdot B + A \cdot \left(C \cdot -4\right)}}\right)\right) \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(2 \cdot F\right) \cdot \frac{2 \cdot A}{B \cdot B + A \cdot \left(C \cdot -4\right)}\right)\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(2 \cdot F\right) \cdot \left(2 \cdot \frac{A}{B \cdot B + A \cdot \left(C \cdot -4\right)}\right)\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot F\right) \cdot 2\right) \cdot \frac{A}{B \cdot B + A \cdot \left(C \cdot -4\right)}\right)\right)\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot F\right) \cdot 2\right) \cdot \frac{1}{\frac{B \cdot B + A \cdot \left(C \cdot -4\right)}{A}}\right)\right)\right) \]

       6. un-div-inv N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(2 \cdot F\right) \cdot 2}{\frac{B \cdot B + A \cdot \left(C \cdot -4\right)}{A}}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(2 \cdot F\right) \cdot 2\right), \left(\frac{B \cdot B + A \cdot \left(C \cdot -4\right)}{A}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(F \cdot 2\right) \cdot 2\right), \left(\frac{B \cdot B + A \cdot \left(C \cdot -4\right)}{A}\right)\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(F \cdot \left(2 \cdot 2\right)\right), \left(\frac{B \cdot B + A \cdot \left(C \cdot -4\right)}{A}\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(F \cdot 4\right), \left(\frac{B \cdot B + A \cdot \left(C \cdot -4\right)}{A}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 4\right), \left(\frac{B \cdot B + A \cdot \left(C \cdot -4\right)}{A}\right)\right)\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 4\right), \mathsf{/.f64}\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right), A\right)\right)\right)\right) \]

   12. Applied egg-rr 34.9%

       \[\leadsto -\color{blue}{\sqrt{\frac{F \cdot 4}{\frac{B \cdot B + \frac{A}{\frac{-0.25}{C}}}{A}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;0 - {\left(\frac{2 \cdot F}{B}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{4 \cdot F}{\frac{B \cdot B + \frac{A}{\frac{-0.25}{C}}}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.3 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;0 - {\left(\frac{2 \cdot F}{B\_m}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.3e-139)
   (/ (sqrt (* (* -16.0 (* A A)) (* C F))) (* 4.0 (* A C)))
   (- 0.0 (pow (/ (* 2.0 F) B_m) 0.5))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.3e-139) {
		tmp = sqrt(((-16.0 * (A * A)) * (C * F))) / (4.0 * (A * C));
	} else {
		tmp = 0.0 - pow(((2.0 * F) / B_m), 0.5);
	}
	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 <= 1.3d-139) then
        tmp = sqrt((((-16.0d0) * (a * a)) * (c * f))) / (4.0d0 * (a * c))
    else
        tmp = 0.0d0 - (((2.0d0 * f) / b_m) ** 0.5d0)
    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 <= 1.3e-139) {
		tmp = Math.sqrt(((-16.0 * (A * A)) * (C * F))) / (4.0 * (A * C));
	} else {
		tmp = 0.0 - Math.pow(((2.0 * F) / B_m), 0.5);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.3e-139:
		tmp = math.sqrt(((-16.0 * (A * A)) * (C * F))) / (4.0 * (A * C))
	else:
		tmp = 0.0 - math.pow(((2.0 * F) / B_m), 0.5)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.3e-139)
		tmp = Float64(sqrt(Float64(Float64(-16.0 * Float64(A * A)) * Float64(C * F))) / Float64(4.0 * Float64(A * C)));
	else
		tmp = Float64(0.0 - (Float64(Float64(2.0 * F) / B_m) ^ 0.5));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.3e-139)
		tmp = sqrt(((-16.0 * (A * A)) * (C * F))) / (4.0 * (A * C));
	else
		tmp = 0.0 - (((2.0 * F) / B_m) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.3e-139], N[(N[Sqrt[N[(N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Power[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.2999999999999999e-139

   1. Initial program 21.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 28.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-16 \cdot {A}^{2}\right), \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2}\right)\right), \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot A\right)\right), \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]

   8. Taylor expanded in A around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right)\right), \color{blue}{\left(4 \cdot \left(A \cdot C\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right)\right), \mathsf{*.f64}\left(4, \color{blue}{\left(A \cdot C\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right)\right), \mathsf{*.f64}\left(4, \left(C \cdot \color{blue}{A}\right)\right)\right) \]

      3. *-lowering-*.f64 10.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(C, \color{blue}{A}\right)\right)\right) \]

   10. Simplified 10.4%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}{\color{blue}{4 \cdot \left(C \cdot A\right)}} \]

   if 1.2999999999999999e-139 < B

   1. Initial program 10.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   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 26.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\right)\right) \]

      2. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(\frac{F \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(\frac{F \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(F \cdot \frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \left({F}^{\frac{1}{2}} \cdot {\left(\frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left(\sqrt{2} \cdot {F}^{\frac{1}{2}}\right) \cdot {\left(\frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

   7. Applied egg-rr 33.3%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \color{blue}{\left(\sqrt{\frac{1}{B}}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{B}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 48.9%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, B\right)\right)\right)\right) \]

   10. Simplified 48.9%

       \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
   11. Step-by-step derivation

       1. sqrt-unprod N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\left(2 \cdot F\right) \cdot \frac{1}{B}}\right)\right) \]

       2. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right)\right) \]

       3. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right), \frac{1}{2}\right)\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{2 \cdot F}{B}\right), \frac{1}{2}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot F\right), B\right), \frac{1}{2}\right)\right) \]

       6. *-lowering-*.f64 40.8%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, F\right), B\right), \frac{1}{2}\right)\right) \]

   12. Applied egg-rr 40.8%

       \[\leadsto -\color{blue}{{\left(\frac{2 \cdot F}{B}\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.3 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;0 - {\left(\frac{2 \cdot F}{B}\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 27.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ 0 - {\left(\frac{2 \cdot F}{B\_m}\right)}^{0.5} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (- 0.0 (pow (/ (* 2.0 F) B_m) 0.5)))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return 0.0 - pow(((2.0 * F) / B_m), 0.5);
}

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 = 0.0d0 - (((2.0d0 * f) / b_m) ** 0.5d0)
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return 0.0 - Math.pow(((2.0 * F) / B_m), 0.5);
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return 0.0 - math.pow(((2.0 * F) / B_m), 0.5)

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(0.0 - (Float64(Float64(2.0 * F) / B_m) ^ 0.5))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = 0.0 - (((2.0 * F) / B_m) ^ 0.5);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(0.0 - N[Power[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 26.7%

   \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\right)\right) \]

   2. pow1/2 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(\frac{F \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

   3. associate-+r+ N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(\frac{F \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(F \cdot \frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

   5. unpow-prod-down N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \left({F}^{\frac{1}{2}} \cdot {\left(\frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\left(\sqrt{2} \cdot {F}^{\frac{1}{2}}\right) \cdot {\left(\frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

7. Applied egg-rr 35.2%

   \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}}} \]

8. Taylor expanded in B around inf

   \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \color{blue}{\left(\sqrt{\frac{1}{B}}\right)}\right)\right) \]
9. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{B}\right)\right)\right)\right) \]

   2. /-lowering-/.f64 19.1%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, B\right)\right)\right)\right) \]

10. Simplified 19.1%

    \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
11. Step-by-step derivation

    1. sqrt-unprod N/A

       \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\left(2 \cdot F\right) \cdot \frac{1}{B}}\right)\right) \]

    2. pow1/2 N/A

       \[\leadsto \mathsf{neg.f64}\left(\left({\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right)\right) \]

    3. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right), \frac{1}{2}\right)\right) \]

    4. un-div-inv N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{2 \cdot F}{B}\right), \frac{1}{2}\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot F\right), B\right), \frac{1}{2}\right)\right) \]

    6. *-lowering-*.f64 15.9%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, F\right), B\right), \frac{1}{2}\right)\right) \]

12. Applied egg-rr 15.9%

    \[\leadsto -\color{blue}{{\left(\frac{2 \cdot F}{B}\right)}^{0.5}} \]

13. Final simplification 15.9%

    \[\leadsto 0 - {\left(\frac{2 \cdot F}{B}\right)}^{0.5} \]
14. Add Preprocessing

Alternative 18: 27.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ 0 - \sqrt{\frac{2 \cdot F}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (- 0.0 (sqrt (/ (* 2.0 F) B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return 0.0 - sqrt(((2.0 * F) / 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 = 0.0d0 - sqrt(((2.0d0 * f) / b_m))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return 0.0 - Math.sqrt(((2.0 * F) / B_m));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return 0.0 - math.sqrt(((2.0 * F) / B_m))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(0.0 - sqrt(Float64(Float64(2.0 * F) / B_m)))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = 0.0 - sqrt(((2.0 * F) / B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(0.0 - N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 26.7%

   \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\right)\right) \]

   2. pow1/2 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(\frac{F \cdot \left(A + \left(C + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

   3. associate-+r+ N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(\frac{F \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot {\left(F \cdot \frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

   5. unpow-prod-down N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \left({F}^{\frac{1}{2}} \cdot {\left(\frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\left(\sqrt{2} \cdot {F}^{\frac{1}{2}}\right) \cdot {\left(\frac{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)}^{\frac{1}{2}}\right)\right) \]

7. Applied egg-rr 35.2%

   \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}}} \]

8. Taylor expanded in B around inf

   \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \color{blue}{\left(\sqrt{\frac{1}{B}}\right)}\right)\right) \]
9. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{B}\right)\right)\right)\right) \]

   2. /-lowering-/.f64 19.1%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, B\right)\right)\right)\right) \]

10. Simplified 19.1%

    \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
11. Step-by-step derivation

    1. sqrt-unprod N/A

       \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\left(2 \cdot F\right) \cdot \frac{1}{B}}\right)\right) \]

    2. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)\right)\right) \]

    3. un-div-inv N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{2 \cdot F}{B}\right)\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot F\right), B\right)\right)\right) \]

    5. *-lowering-*.f64 15.7%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, F\right), B\right)\right)\right) \]

12. Applied egg-rr 15.7%

    \[\leadsto -\color{blue}{\sqrt{\frac{2 \cdot F}{B}}} \]

13. Final simplification 15.7%

    \[\leadsto 0 - \sqrt{\frac{2 \cdot F}{B}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(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))))