ABCF->ab-angle b

Percentage Accurate: 19.2% → 52.9%

Time: 25.9s

Alternatives: 15

Speedup: 5.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 52.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m * B_m) + (A * (C * -4.0));
	t_1 = A - (hypot(B_m, (A - C)) - 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 <= -Inf)
		tmp = sqrt((F * (t_1 / ((B_m * B_m) + (-4.0 * (A * C)))))) * (0.0 - sqrt(2.0));
	elseif (t_3 <= 0.0)
		tmp = sqrt(t_0) * (sqrt((t_1 * (2.0 * F))) / ((A * (4.0 * C)) - (B_m * B_m)));
	elseif (t_3 <= Inf)
		tmp = (((2.0 * F) ^ 0.5) * sqrt((t_1 * t_0))) / ((4.0 * (A * C)) - (B_m * B_m));
	else
		tmp = (sqrt((0.0 - (2.0 * F))) * sqrt(B_m)) / (0.0 - 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[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $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, (-Infinity)], N[(N[Sqrt[N[(F * N[(t$95$1 / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Power[N[(2.0 * F), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(t$95$1 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(0.0 - N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $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))) < -inf.0

   1. Initial program 3.1%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   5. Simplified 58.1%

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

   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))) < 0.0

   1. Initial program 52.2%

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

   3. Simplified 55.1%

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

   6. Applied egg-rr 59.4%

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

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

   1. Initial program 41.3%

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

   3. Simplified 54.0%

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

   6. Applied egg-rr 66.0%

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

   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 A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 18.8%

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

   5. Simplified 18.8%

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

   6. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 18.0%

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

   8. Simplified 18.0%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

      6. pow1/2 N/A

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

      7. pow1/2 N/A

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

      8. pow-prod-down N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

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

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

      14. neg-lowering-neg.f64 18.0%

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

   10. Applied egg-rr 18.0%

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

       1. unpow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

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

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

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

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

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

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

       8. neg-sub0 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(0 - F\right)\right)\right), \left(\sqrt{B}\right)\right), B\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(0, F\right)\right)\right), \left(\sqrt{B}\right)\right), B\right)\right) \]

       10. sqrt-lowering-sqrt.f64 26.8%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(0, F\right)\right)\right), \mathsf{sqrt.f64}\left(B\right)\right), B\right)\right) \]

   12. Applied egg-rr 26.8%

       \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot \left(0 - F\right)} \cdot \sqrt{B}}}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.1%

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

Alternative 2: 46.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m\\ \mathbf{if}\;B\_m \leq 4.3 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 3.7 \cdot 10^{+86}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B\_m, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0 - 2 \cdot F} \cdot \sqrt{B\_m}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* 4.0 (* A C)) (* B_m B_m))))
   (if (<= B_m 4.3e-181)
     (/ (sqrt (* -16.0 (* C (* F (* A C))))) t_0)
     (if (<= B_m 3.7e+86)
       (/
        (sqrt
         (*
          (+ (* B_m B_m) (* -4.0 (* A C)))
          (* (- A (- (hypot B_m (- A C)) C)) (* 2.0 F))))
        t_0)
       (/ (* (sqrt (- 0.0 (* 2.0 F))) (sqrt B_m)) (- 0.0 B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * (A * C)) - (B_m * B_m);
	double tmp;
	if (B_m <= 4.3e-181) {
		tmp = sqrt((-16.0 * (C * (F * (A * C))))) / t_0;
	} else if (B_m <= 3.7e+86) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A - (hypot(B_m, (A - C)) - C)) * (2.0 * F)))) / t_0;
	} else {
		tmp = (sqrt((0.0 - (2.0 * F))) * sqrt(B_m)) / (0.0 - 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 = (4.0 * (A * C)) - (B_m * B_m);
	double tmp;
	if (B_m <= 4.3e-181) {
		tmp = Math.sqrt((-16.0 * (C * (F * (A * C))))) / t_0;
	} else if (B_m <= 3.7e+86) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A - (Math.hypot(B_m, (A - C)) - C)) * (2.0 * F)))) / t_0;
	} else {
		tmp = (Math.sqrt((0.0 - (2.0 * F))) * Math.sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (4.0 * (A * C)) - (B_m * B_m)
	tmp = 0
	if B_m <= 4.3e-181:
		tmp = math.sqrt((-16.0 * (C * (F * (A * C))))) / t_0
	elif B_m <= 3.7e+86:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A - (math.hypot(B_m, (A - C)) - C)) * (2.0 * F)))) / t_0
	else:
		tmp = (math.sqrt((0.0 - (2.0 * F))) * math.sqrt(B_m)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 4.3e-181)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(C * Float64(F * Float64(A * C))))) / t_0);
	elseif (B_m <= 3.7e+86)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(A - Float64(hypot(B_m, Float64(A - C)) - C)) * Float64(2.0 * F)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(0.0 - Float64(2.0 * F))) * sqrt(B_m)) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * (A * C)) - (B_m * B_m);
	tmp = 0.0;
	if (B_m <= 4.3e-181)
		tmp = sqrt((-16.0 * (C * (F * (A * C))))) / t_0;
	elseif (B_m <= 3.7e+86)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A - (hypot(B_m, (A - C)) - C)) * (2.0 * F)))) / t_0;
	else
		tmp = (sqrt((0.0 - (2.0 * F))) * sqrt(B_m)) / (0.0 - 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[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.3e-181], N[(N[Sqrt[N[(-16.0 * N[(C * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 3.7e+86], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(0.0 - N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.3e-181

   1. Initial program 16.9%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in B around 0

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 11.9%

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

   7. Simplified 11.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 17.1%

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

   9. Applied egg-rr 17.1%

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

   if 4.3e-181 < B < 3.69999999999999992e86

   1. Initial program 35.0%

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

   3. Simplified 45.6%

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

   if 3.69999999999999992e86 < B

   1. Initial program 7.9%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 53.1%

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

   5. Simplified 53.1%

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

   6. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 51.7%

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

   8. Simplified 51.7%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

      6. pow1/2 N/A

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

      7. pow1/2 N/A

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

      8. pow-prod-down N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

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

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

      14. neg-lowering-neg.f64 51.6%

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

   10. Applied egg-rr 51.6%

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

       1. unpow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

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

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

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

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

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

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

       8. neg-sub0 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(0 - F\right)\right)\right), \left(\sqrt{B}\right)\right), B\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(0, F\right)\right)\right), \left(\sqrt{B}\right)\right), B\right)\right) \]

       10. sqrt-lowering-sqrt.f64 71.8%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(0, F\right)\right)\right), \mathsf{sqrt.f64}\left(B\right)\right), B\right)\right) \]

   12. Applied egg-rr 71.8%

       \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot \left(0 - F\right)} \cdot \sqrt{B}}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.3 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{+86}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0 - 2 \cdot F} \cdot \sqrt{B}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m\\ \mathbf{if}\;B\_m \leq 5.8 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 2.9 \cdot 10^{+86}:\\ \;\;\;\;\frac{\sqrt{\left(A - \left(\mathsf{hypot}\left(B\_m, A - C\right) - C\right)\right) \cdot \left(\left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0 - 2 \cdot F} \cdot \sqrt{B\_m}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* 4.0 (* A C)) (* B_m B_m))))
   (if (<= B_m 5.8e-181)
     (/ (sqrt (* -16.0 (* C (* F (* A C))))) t_0)
     (if (<= B_m 2.9e+86)
       (/
        (sqrt
         (*
          (- A (- (hypot B_m (- A C)) C))
          (* (+ (* B_m B_m) (* A (* C -4.0))) (* 2.0 F))))
        t_0)
       (/ (* (sqrt (- 0.0 (* 2.0 F))) (sqrt B_m)) (- 0.0 B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * (A * C)) - (B_m * B_m);
	double tmp;
	if (B_m <= 5.8e-181) {
		tmp = sqrt((-16.0 * (C * (F * (A * C))))) / t_0;
	} else if (B_m <= 2.9e+86) {
		tmp = sqrt(((A - (hypot(B_m, (A - C)) - C)) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / t_0;
	} else {
		tmp = (sqrt((0.0 - (2.0 * F))) * sqrt(B_m)) / (0.0 - 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 = (4.0 * (A * C)) - (B_m * B_m);
	double tmp;
	if (B_m <= 5.8e-181) {
		tmp = Math.sqrt((-16.0 * (C * (F * (A * C))))) / t_0;
	} else if (B_m <= 2.9e+86) {
		tmp = Math.sqrt(((A - (Math.hypot(B_m, (A - C)) - C)) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / t_0;
	} else {
		tmp = (Math.sqrt((0.0 - (2.0 * F))) * Math.sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (4.0 * (A * C)) - (B_m * B_m)
	tmp = 0
	if B_m <= 5.8e-181:
		tmp = math.sqrt((-16.0 * (C * (F * (A * C))))) / t_0
	elif B_m <= 2.9e+86:
		tmp = math.sqrt(((A - (math.hypot(B_m, (A - C)) - C)) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / t_0
	else:
		tmp = (math.sqrt((0.0 - (2.0 * F))) * math.sqrt(B_m)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 5.8e-181)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(C * Float64(F * Float64(A * C))))) / t_0);
	elseif (B_m <= 2.9e+86)
		tmp = Float64(sqrt(Float64(Float64(A - Float64(hypot(B_m, Float64(A - C)) - C)) * Float64(Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))) * Float64(2.0 * F)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(0.0 - Float64(2.0 * F))) * sqrt(B_m)) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * (A * C)) - (B_m * B_m);
	tmp = 0.0;
	if (B_m <= 5.8e-181)
		tmp = sqrt((-16.0 * (C * (F * (A * C))))) / t_0;
	elseif (B_m <= 2.9e+86)
		tmp = sqrt(((A - (hypot(B_m, (A - C)) - C)) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / t_0;
	else
		tmp = (sqrt((0.0 - (2.0 * F))) * sqrt(B_m)) / (0.0 - 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[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 5.8e-181], N[(N[Sqrt[N[(-16.0 * N[(C * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 2.9e+86], N[(N[Sqrt[N[(N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(0.0 - N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 5.7999999999999996e-181

   1. Initial program 16.9%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in B around 0

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 11.9%

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

   7. Simplified 11.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 17.1%

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

   9. Applied egg-rr 17.1%

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

   if 5.7999999999999996e-181 < B < 2.8999999999999999e86

   1. Initial program 35.0%

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

   3. Simplified 45.6%

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

   6. Applied egg-rr 45.6%

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

   if 2.8999999999999999e86 < B

   1. Initial program 7.9%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 53.1%

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

   5. Simplified 53.1%

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

   6. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 51.7%

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

   8. Simplified 51.7%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

      6. pow1/2 N/A

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

      7. pow1/2 N/A

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

      8. pow-prod-down N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

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

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

      14. neg-lowering-neg.f64 51.6%

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

   10. Applied egg-rr 51.6%

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

       1. unpow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

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

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

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

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

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

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

       8. neg-sub0 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(0 - F\right)\right)\right), \left(\sqrt{B}\right)\right), B\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(0, F\right)\right)\right), \left(\sqrt{B}\right)\right), B\right)\right) \]

       10. sqrt-lowering-sqrt.f64 71.8%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(0, F\right)\right)\right), \mathsf{sqrt.f64}\left(B\right)\right), B\right)\right) \]

   12. Applied egg-rr 71.8%

       \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot \left(0 - F\right)} \cdot \sqrt{B}}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.8 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{+86}:\\ \;\;\;\;\frac{\sqrt{\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0 - 2 \cdot F} \cdot \sqrt{B}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.55 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot -16\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 1.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B\_m, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0 - 2 \cdot F} \cdot \sqrt{B\_m}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.55e-73)
   (/ (sqrt (* (* F (* A C)) (* C -16.0))) (- (* 4.0 (* A C)) (* B_m B_m)))
   (if (<= B_m 1.6e+32)
     (/
      (sqrt
       (*
        (+ (* B_m B_m) (* -4.0 (* A C)))
        (* (- A (- (hypot B_m (- A C)) C)) (* 2.0 F))))
      (- 0.0 (* B_m B_m)))
     (/ (* (sqrt (- 0.0 (* 2.0 F))) (sqrt B_m)) (- 0.0 B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.55e-73) {
		tmp = sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 1.6e+32) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A - (hypot(B_m, (A - C)) - C)) * (2.0 * F)))) / (0.0 - (B_m * B_m));
	} else {
		tmp = (sqrt((0.0 - (2.0 * F))) * sqrt(B_m)) / (0.0 - 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 <= 2.55e-73) {
		tmp = Math.sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 1.6e+32) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A - (Math.hypot(B_m, (A - C)) - C)) * (2.0 * F)))) / (0.0 - (B_m * B_m));
	} else {
		tmp = (Math.sqrt((0.0 - (2.0 * F))) * Math.sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 2.55e-73:
		tmp = math.sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m))
	elif B_m <= 1.6e+32:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A - (math.hypot(B_m, (A - C)) - C)) * (2.0 * F)))) / (0.0 - (B_m * B_m))
	else:
		tmp = (math.sqrt((0.0 - (2.0 * F))) * math.sqrt(B_m)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.55e-73)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(A * C)) * Float64(C * -16.0))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (B_m <= 1.6e+32)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(A - Float64(hypot(B_m, Float64(A - C)) - C)) * Float64(2.0 * F)))) / Float64(0.0 - Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(sqrt(Float64(0.0 - Float64(2.0 * F))) * sqrt(B_m)) / Float64(0.0 - 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 <= 2.55e-73)
		tmp = sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (B_m <= 1.6e+32)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A - (hypot(B_m, (A - C)) - C)) * (2.0 * F)))) / (0.0 - (B_m * B_m));
	else
		tmp = (sqrt((0.0 - (2.0 * F))) * sqrt(B_m)) / (0.0 - 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, 2.55e-73], N[(N[Sqrt[N[(N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(C * -16.0), $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, 1.6e+32], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(0.0 - N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.55e-73

   1. Initial program 17.4%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in B around 0

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 13.6%

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

   7. Simplified 13.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 11.6%

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

   9. Applied egg-rr 11.6%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

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

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

       5. associate-*l* N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. *-lowering-*.f64 18.3%

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

   11. Applied egg-rr 18.3%

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

   if 2.55e-73 < B < 1.5999999999999999e32

   1. Initial program 48.9%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), -4\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right), \left(0 - \color{blue}{{B}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), -4\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left({B}^{2}\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), -4\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \left(B \cdot \color{blue}{B}\right)\right)\right) \]

      5. *-lowering-*.f64 44.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), -4\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(B, \color{blue}{B}\right)\right)\right) \]

   7. Simplified 44.6%

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

   if 1.5999999999999999e32 < B

   1. Initial program 11.7%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 49.8%

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

   5. Simplified 49.8%

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

   6. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 48.7%

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

   8. Simplified 48.7%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

      6. pow1/2 N/A

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

      7. pow1/2 N/A

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

      8. pow-prod-down N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

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

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

      14. neg-lowering-neg.f64 48.6%

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

   10. Applied egg-rr 48.6%

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

       1. unpow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

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

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

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

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

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

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

       8. neg-sub0 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(0 - F\right)\right)\right), \left(\sqrt{B}\right)\right), B\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(0, F\right)\right)\right), \left(\sqrt{B}\right)\right), B\right)\right) \]

       10. sqrt-lowering-sqrt.f64 67.2%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(0, F\right)\right)\right), \mathsf{sqrt.f64}\left(B\right)\right), B\right)\right) \]

   12. Applied egg-rr 67.2%

       \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot \left(0 - F\right)} \cdot \sqrt{B}}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.55 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot -16\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{0 - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0 - 2 \cdot F} \cdot \sqrt{B}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m\\ \mathbf{if}\;B\_m \leq 5.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot -16\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 1.6 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B\_m, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0 - 2 \cdot F} \cdot \sqrt{B\_m}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* 4.0 (* A C)) (* B_m B_m))))
   (if (<= B_m 5.6e-74)
     (/ (sqrt (* (* F (* A C)) (* C -16.0))) t_0)
     (if (<= B_m 1.6e+33)
       (/
        (sqrt (* (* B_m B_m) (* (- A (- (hypot B_m (- A C)) C)) (* 2.0 F))))
        t_0)
       (/ (* (sqrt (- 0.0 (* 2.0 F))) (sqrt B_m)) (- 0.0 B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * (A * C)) - (B_m * B_m);
	double tmp;
	if (B_m <= 5.6e-74) {
		tmp = sqrt(((F * (A * C)) * (C * -16.0))) / t_0;
	} else if (B_m <= 1.6e+33) {
		tmp = sqrt(((B_m * B_m) * ((A - (hypot(B_m, (A - C)) - C)) * (2.0 * F)))) / t_0;
	} else {
		tmp = (sqrt((0.0 - (2.0 * F))) * sqrt(B_m)) / (0.0 - 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 = (4.0 * (A * C)) - (B_m * B_m);
	double tmp;
	if (B_m <= 5.6e-74) {
		tmp = Math.sqrt(((F * (A * C)) * (C * -16.0))) / t_0;
	} else if (B_m <= 1.6e+33) {
		tmp = Math.sqrt(((B_m * B_m) * ((A - (Math.hypot(B_m, (A - C)) - C)) * (2.0 * F)))) / t_0;
	} else {
		tmp = (Math.sqrt((0.0 - (2.0 * F))) * Math.sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (4.0 * (A * C)) - (B_m * B_m)
	tmp = 0
	if B_m <= 5.6e-74:
		tmp = math.sqrt(((F * (A * C)) * (C * -16.0))) / t_0
	elif B_m <= 1.6e+33:
		tmp = math.sqrt(((B_m * B_m) * ((A - (math.hypot(B_m, (A - C)) - C)) * (2.0 * F)))) / t_0
	else:
		tmp = (math.sqrt((0.0 - (2.0 * F))) * math.sqrt(B_m)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 5.6e-74)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(A * C)) * Float64(C * -16.0))) / t_0);
	elseif (B_m <= 1.6e+33)
		tmp = Float64(sqrt(Float64(Float64(B_m * B_m) * Float64(Float64(A - Float64(hypot(B_m, Float64(A - C)) - C)) * Float64(2.0 * F)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(0.0 - Float64(2.0 * F))) * sqrt(B_m)) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * (A * C)) - (B_m * B_m);
	tmp = 0.0;
	if (B_m <= 5.6e-74)
		tmp = sqrt(((F * (A * C)) * (C * -16.0))) / t_0;
	elseif (B_m <= 1.6e+33)
		tmp = sqrt(((B_m * B_m) * ((A - (hypot(B_m, (A - C)) - C)) * (2.0 * F)))) / t_0;
	else
		tmp = (sqrt((0.0 - (2.0 * F))) * sqrt(B_m)) / (0.0 - 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[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 5.6e-74], N[(N[Sqrt[N[(N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(C * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 1.6e+33], N[(N[Sqrt[N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(0.0 - N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 5.59999999999999976e-74

   1. Initial program 17.4%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in B around 0

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 13.6%

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

   7. Simplified 13.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 11.6%

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

   9. Applied egg-rr 11.6%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

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

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

       5. associate-*l* N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. *-lowering-*.f64 18.3%

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

   11. Applied egg-rr 18.3%

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

   if 5.59999999999999976e-74 < B < 1.60000000000000009e33

   1. Initial program 48.9%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in B around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 44.5%

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

   7. Simplified 44.5%

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

   if 1.60000000000000009e33 < B

   1. Initial program 11.7%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 49.8%

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

   5. Simplified 49.8%

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

   6. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 48.7%

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

   8. Simplified 48.7%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

      6. pow1/2 N/A

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

      7. pow1/2 N/A

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

      8. pow-prod-down N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

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

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

      14. neg-lowering-neg.f64 48.6%

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

   10. Applied egg-rr 48.6%

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

       1. unpow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

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

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

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

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

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

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

       8. neg-sub0 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(0 - F\right)\right)\right), \left(\sqrt{B}\right)\right), B\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(0, F\right)\right)\right), \left(\sqrt{B}\right)\right), B\right)\right) \]

       10. sqrt-lowering-sqrt.f64 67.2%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(0, F\right)\right)\right), \mathsf{sqrt.f64}\left(B\right)\right), B\right)\right) \]

   12. Applied egg-rr 67.2%

       \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot \left(0 - F\right)} \cdot \sqrt{B}}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot -16\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0 - 2 \cdot F} \cdot \sqrt{B}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.05 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot -16\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 2 \cdot 10^{+118}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B\_m, C\right)\right)\right)\right)}^{0.5}}{0 - B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0 - 2 \cdot F} \cdot \sqrt{B\_m}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.05e-73)
   (/ (sqrt (* (* F (* A C)) (* C -16.0))) (- (* 4.0 (* A C)) (* B_m B_m)))
   (if (<= B_m 2e+118)
     (/ (pow (* 2.0 (* F (- C (hypot B_m C)))) 0.5) (- 0.0 B_m))
     (/ (* (sqrt (- 0.0 (* 2.0 F))) (sqrt B_m)) (- 0.0 B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.05e-73) {
		tmp = sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 2e+118) {
		tmp = pow((2.0 * (F * (C - hypot(B_m, C)))), 0.5) / (0.0 - B_m);
	} else {
		tmp = (sqrt((0.0 - (2.0 * F))) * sqrt(B_m)) / (0.0 - 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 <= 1.05e-73) {
		tmp = Math.sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 2e+118) {
		tmp = Math.pow((2.0 * (F * (C - Math.hypot(B_m, C)))), 0.5) / (0.0 - B_m);
	} else {
		tmp = (Math.sqrt((0.0 - (2.0 * F))) * Math.sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.05e-73:
		tmp = math.sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m))
	elif B_m <= 2e+118:
		tmp = math.pow((2.0 * (F * (C - math.hypot(B_m, C)))), 0.5) / (0.0 - B_m)
	else:
		tmp = (math.sqrt((0.0 - (2.0 * F))) * math.sqrt(B_m)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.05e-73)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(A * C)) * Float64(C * -16.0))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (B_m <= 2e+118)
		tmp = Float64((Float64(2.0 * Float64(F * Float64(C - hypot(B_m, C)))) ^ 0.5) / Float64(0.0 - B_m));
	else
		tmp = Float64(Float64(sqrt(Float64(0.0 - Float64(2.0 * F))) * sqrt(B_m)) / Float64(0.0 - 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 <= 1.05e-73)
		tmp = sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (B_m <= 2e+118)
		tmp = ((2.0 * (F * (C - hypot(B_m, C)))) ^ 0.5) / (0.0 - B_m);
	else
		tmp = (sqrt((0.0 - (2.0 * F))) * sqrt(B_m)) / (0.0 - 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, 1.05e-73], N[(N[Sqrt[N[(N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(C * -16.0), $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, 2e+118], N[(N[Power[N[(2.0 * N[(F * N[(C - N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(0.0 - N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.0499999999999999e-73

   1. Initial program 17.4%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in B around 0

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 13.6%

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

   7. Simplified 13.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 11.6%

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

   9. Applied egg-rr 11.6%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

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

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

       5. associate-*l* N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. *-lowering-*.f64 18.3%

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

   11. Applied egg-rr 18.3%

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

   if 1.0499999999999999e-73 < B < 1.99999999999999993e118

   1. Initial program 45.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 41.4%

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

   5. Simplified 41.4%

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. sqrt-lowering-sqrt.f64 41.4%

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

   7. Applied egg-rr 41.4%

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

      1. distribute-lft-neg-out N/A

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

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

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 41.6%

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

   if 1.99999999999999993e118 < B

   1. Initial program 0.4%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 51.3%

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

   5. Simplified 51.3%

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

   6. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 51.4%

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

   8. Simplified 51.4%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

      6. pow1/2 N/A

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

      7. pow1/2 N/A

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

      8. pow-prod-down N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

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

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

      14. neg-lowering-neg.f64 51.3%

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

   10. Applied egg-rr 51.3%

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

       1. unpow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

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

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

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

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

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

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

       8. neg-sub0 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(0 - F\right)\right)\right), \left(\sqrt{B}\right)\right), B\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(0, F\right)\right)\right), \left(\sqrt{B}\right)\right), B\right)\right) \]

       10. sqrt-lowering-sqrt.f64 75.0%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(0, F\right)\right)\right), \mathsf{sqrt.f64}\left(B\right)\right), B\right)\right) \]

   12. Applied egg-rr 75.0%

       \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot \left(0 - F\right)} \cdot \sqrt{B}}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.05 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot -16\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 2 \cdot 10^{+118}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{0 - B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0 - 2 \cdot F} \cdot \sqrt{B}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot -16\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0 - 2 \cdot F} \cdot \sqrt{B\_m}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5e-73)
   (/ (sqrt (* (* F (* A C)) (* C -16.0))) (- (* 4.0 (* A C)) (* B_m B_m)))
   (/ (* (sqrt (- 0.0 (* 2.0 F))) (sqrt B_m)) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5e-73) {
		tmp = sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = (sqrt((0.0 - (2.0 * F))) * sqrt(B_m)) / (0.0 - 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 <= 5d-73) then
        tmp = sqrt(((f * (a * c)) * (c * (-16.0d0)))) / ((4.0d0 * (a * c)) - (b_m * b_m))
    else
        tmp = (sqrt((0.0d0 - (2.0d0 * f))) * sqrt(b_m)) / (0.0d0 - 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 <= 5e-73) {
		tmp = Math.sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = (Math.sqrt((0.0 - (2.0 * F))) * Math.sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 5e-73:
		tmp = math.sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m))
	else:
		tmp = (math.sqrt((0.0 - (2.0 * F))) * math.sqrt(B_m)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5e-73)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(A * C)) * Float64(C * -16.0))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(sqrt(Float64(0.0 - Float64(2.0 * F))) * sqrt(B_m)) / Float64(0.0 - 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 <= 5e-73)
		tmp = sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m));
	else
		tmp = (sqrt((0.0 - (2.0 * F))) * sqrt(B_m)) / (0.0 - 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, 5e-73], N[(N[Sqrt[N[(N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(C * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(0.0 - N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.9999999999999998e-73

   1. Initial program 17.4%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in B around 0

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 13.6%

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

   7. Simplified 13.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 11.6%

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

   9. Applied egg-rr 11.6%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

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

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

       5. associate-*l* N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. *-lowering-*.f64 18.3%

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

   11. Applied egg-rr 18.3%

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

   if 4.9999999999999998e-73 < B

   1. Initial program 20.9%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 46.8%

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

   5. Simplified 46.8%

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

   6. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 43.0%

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

   8. Simplified 43.0%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

      6. pow1/2 N/A

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

      7. pow1/2 N/A

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

      8. pow-prod-down N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

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

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

      14. neg-lowering-neg.f64 43.0%

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

   10. Applied egg-rr 43.0%

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

       1. unpow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

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

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

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

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

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

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

       8. neg-sub0 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(0 - F\right)\right)\right), \left(\sqrt{B}\right)\right), B\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(0, F\right)\right)\right), \left(\sqrt{B}\right)\right), B\right)\right) \]

       10. sqrt-lowering-sqrt.f64 56.9%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(0, F\right)\right)\right), \mathsf{sqrt.f64}\left(B\right)\right), B\right)\right) \]

   12. Applied egg-rr 56.9%

       \[\leadsto -\frac{\color{blue}{\sqrt{2 \cdot \left(0 - F\right)} \cdot \sqrt{B}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot -16\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0 - 2 \cdot F} \cdot \sqrt{B}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 31.8% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot -16\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 2 \cdot 10^{+162}:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B\_m \cdot F\right)\right)}^{0.5}}{0 - B\_m}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{F \cdot -2 + \frac{2 \cdot \left(A \cdot F\right)}{B\_m}}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.8e-74)
   (/ (sqrt (* (* F (* A C)) (* C -16.0))) (- (* 4.0 (* A C)) (* B_m B_m)))
   (if (<= B_m 2e+162)
     (/ (pow (* -2.0 (* B_m F)) 0.5) (- 0.0 B_m))
     (- 0.0 (sqrt (/ (+ (* F -2.0) (/ (* 2.0 (* A F)) B_m)) 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-74) {
		tmp = sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 2e+162) {
		tmp = pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	} else {
		tmp = 0.0 - sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / 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-74) then
        tmp = sqrt(((f * (a * c)) * (c * (-16.0d0)))) / ((4.0d0 * (a * c)) - (b_m * b_m))
    else if (b_m <= 2d+162) then
        tmp = (((-2.0d0) * (b_m * f)) ** 0.5d0) / (0.0d0 - b_m)
    else
        tmp = 0.0d0 - sqrt((((f * (-2.0d0)) + ((2.0d0 * (a * f)) / b_m)) / 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-74) {
		tmp = Math.sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 2e+162) {
		tmp = Math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	} else {
		tmp = 0.0 - Math.sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 3.8e-74:
		tmp = math.sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m))
	elif B_m <= 2e+162:
		tmp = math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m)
	else:
		tmp = 0.0 - math.sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.8e-74)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(A * C)) * Float64(C * -16.0))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (B_m <= 2e+162)
		tmp = Float64((Float64(-2.0 * Float64(B_m * F)) ^ 0.5) / Float64(0.0 - B_m));
	else
		tmp = Float64(0.0 - sqrt(Float64(Float64(Float64(F * -2.0) + Float64(Float64(2.0 * Float64(A * F)) / B_m)) / 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-74)
		tmp = sqrt(((F * (A * C)) * (C * -16.0))) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (B_m <= 2e+162)
		tmp = ((-2.0 * (B_m * F)) ^ 0.5) / (0.0 - B_m);
	else
		tmp = 0.0 - sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / 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-74], N[(N[Sqrt[N[(N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(C * -16.0), $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, 2e+162], N[(N[Power[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Sqrt[N[(N[(N[(F * -2.0), $MachinePrecision] + N[(N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.7999999999999996e-74

   1. Initial program 17.4%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in B around 0

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 13.6%

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

   7. Simplified 13.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 11.6%

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

   9. Applied egg-rr 11.6%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

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

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

       5. associate-*l* N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. *-lowering-*.f64 18.3%

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

   11. Applied egg-rr 18.3%

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

   if 3.7999999999999996e-74 < B < 1.9999999999999999e162

   1. Initial program 35.5%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 46.5%

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

   5. Simplified 46.5%

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

   6. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 40.2%

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

   8. Simplified 40.2%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

      6. pow1/2 N/A

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

      7. pow1/2 N/A

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

      8. pow-prod-down N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

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

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

      14. neg-lowering-neg.f64 40.2%

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

   10. Applied egg-rr 40.2%

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

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}, \frac{1}{2}\right), B\right)\right) \]
   12. Step-by-step derivation

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 40.2%

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

   13. Simplified 40.2%

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

   if 1.9999999999999999e162 < B

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in B around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

      9. associate-*r/ N/A

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

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

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

      11. associate-*r* N/A

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

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

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

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

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      7. associate-*r/ N/A

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

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

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(A \cdot F\right)\right), B\right)\right), B\right)\right)\right) \]

      10. *-lowering-*.f64 59.1%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right), B\right)\right), B\right)\right)\right) \]

   10. Simplified 59.1%

       \[\leadsto \color{blue}{-\sqrt{\frac{-2 \cdot F + \frac{2 \cdot \left(A \cdot F\right)}{B}}{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot -16\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 2 \cdot 10^{+162}:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B \cdot F\right)\right)}^{0.5}}{0 - B}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{F \cdot -2 + \frac{2 \cdot \left(A \cdot F\right)}{B}}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 2.3 \cdot 10^{+162}:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B\_m \cdot F\right)\right)}^{0.5}}{0 - B\_m}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{F \cdot -2 + \frac{2 \cdot \left(A \cdot F\right)}{B\_m}}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 6.8e-74)
   (/ (sqrt (* -16.0 (* C (* F (* A C))))) (- (* 4.0 (* A C)) (* B_m B_m)))
   (if (<= B_m 2.3e+162)
     (/ (pow (* -2.0 (* B_m F)) 0.5) (- 0.0 B_m))
     (- 0.0 (sqrt (/ (+ (* F -2.0) (/ (* 2.0 (* A F)) B_m)) B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6.8e-74) {
		tmp = sqrt((-16.0 * (C * (F * (A * C))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 2.3e+162) {
		tmp = pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	} else {
		tmp = 0.0 - sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / 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 <= 6.8d-74) then
        tmp = sqrt(((-16.0d0) * (c * (f * (a * c))))) / ((4.0d0 * (a * c)) - (b_m * b_m))
    else if (b_m <= 2.3d+162) then
        tmp = (((-2.0d0) * (b_m * f)) ** 0.5d0) / (0.0d0 - b_m)
    else
        tmp = 0.0d0 - sqrt((((f * (-2.0d0)) + ((2.0d0 * (a * f)) / b_m)) / 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 <= 6.8e-74) {
		tmp = Math.sqrt((-16.0 * (C * (F * (A * C))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 2.3e+162) {
		tmp = Math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	} else {
		tmp = 0.0 - Math.sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 6.8e-74:
		tmp = math.sqrt((-16.0 * (C * (F * (A * C))))) / ((4.0 * (A * C)) - (B_m * B_m))
	elif B_m <= 2.3e+162:
		tmp = math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m)
	else:
		tmp = 0.0 - math.sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 6.8e-74)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(C * Float64(F * Float64(A * C))))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (B_m <= 2.3e+162)
		tmp = Float64((Float64(-2.0 * Float64(B_m * F)) ^ 0.5) / Float64(0.0 - B_m));
	else
		tmp = Float64(0.0 - sqrt(Float64(Float64(Float64(F * -2.0) + Float64(Float64(2.0 * Float64(A * F)) / B_m)) / 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 <= 6.8e-74)
		tmp = sqrt((-16.0 * (C * (F * (A * C))))) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (B_m <= 2.3e+162)
		tmp = ((-2.0 * (B_m * F)) ^ 0.5) / (0.0 - B_m);
	else
		tmp = 0.0 - sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / 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, 6.8e-74], N[(N[Sqrt[N[(-16.0 * N[(C * N[(F * N[(A * 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[B$95$m, 2.3e+162], N[(N[Power[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Sqrt[N[(N[(N[(F * -2.0), $MachinePrecision] + N[(N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.8000000000000001e-74

   1. Initial program 17.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 23.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   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. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot {C}^{2}\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot {C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left({C}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 13.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 13.6%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(\left(A \cdot C\right) \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot C\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(F \cdot \left(A \cdot C\right)\right), C\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(A \cdot C\right)\right), C\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(C \cdot A\right)\right), C\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 18.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, A\right)\right), C\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 18.3%

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left(F \cdot \left(C \cdot A\right)\right) \cdot C\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]

   if 6.8000000000000001e-74 < B < 2.29999999999999994e162

   1. Initial program 35.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 46.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 46.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]

   6. Taylor expanded in C around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(B \cdot F\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-1, \left(B \cdot F\right)\right)\right)\right) \]

      2. *-lowering-*.f64 40.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, F\right)\right)\right)\right) \]

   8. Simplified 40.2%

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{-1 \cdot \left(B \cdot F\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right), B\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({2}^{\frac{1}{2}} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right), B\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({2}^{\frac{1}{2}} \cdot {\left(-1 \cdot \left(B \cdot F\right)\right)}^{\frac{1}{2}}\right), B\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot \left(-1 \cdot \left(B \cdot F\right)\right)\right)}^{\frac{1}{2}}\right), B\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot \left(-1 \cdot \left(B \cdot F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(-1 \cdot \left(B \cdot F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{neg}\left(B \cdot F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(B \cdot \left(\mathsf{neg}\left(F\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, \left(\mathsf{neg}\left(F\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      14. neg-lowering-neg.f64 40.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, \mathsf{neg.f64}\left(F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   10. Applied egg-rr 40.2%

       \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(B \cdot \left(-F\right)\right)\right)}^{0.5}}{B}} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}, \frac{1}{2}\right), B\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right), \frac{1}{2}\right), B\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \left(F \cdot B\right)\right), \frac{1}{2}\right), B\right)\right) \]

       3. *-lowering-*.f64 40.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(F, B\right)\right), \frac{1}{2}\right), B\right)\right) \]

   13. Simplified 40.2%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(F \cdot B\right)\right)}}^{0.5}}{B} \]

   if 2.29999999999999994e162 < B

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({B}^{3} \cdot \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{3}\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\left(-2 \cdot F\right), \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(\frac{2 \cdot \left(F \cdot \left(A + C\right)\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(2 \cdot \left(F \cdot \left(A + C\right)\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(\left(2 \cdot F\right) \cdot \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot F\right), \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(C + A\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      15. +-lowering-+.f64 0.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \mathsf{+.f64}\left(C, A\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 0.0%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(-2 \cdot F + \frac{\left(2 \cdot F\right) \cdot \left(C + A\right)}{B}\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}\right), B\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot F\right), \left(2 \cdot \frac{A \cdot F}{B}\right)\right), B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(2 \cdot \frac{A \cdot F}{B}\right)\right), B\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(\frac{2 \cdot \left(A \cdot F\right)}{B}\right)\right), B\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(2 \cdot \left(A \cdot F\right)\right), B\right)\right), B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(A \cdot F\right)\right), B\right)\right), B\right)\right)\right) \]

      10. *-lowering-*.f64 59.1%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right), B\right)\right), B\right)\right)\right) \]

   10. Simplified 59.1%

       \[\leadsto \color{blue}{-\sqrt{\frac{-2 \cdot F + \frac{2 \cdot \left(A \cdot F\right)}{B}}{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{+162}:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B \cdot F\right)\right)}^{0.5}}{0 - B}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{F \cdot -2 + \frac{2 \cdot \left(A \cdot F\right)}{B}}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.4% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{0.25}{A}}{C} \cdot \sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(C + C\right)\right)\right)}\\ \mathbf{elif}\;B\_m \leq 2.55 \cdot 10^{+162}:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B\_m \cdot F\right)\right)}^{0.5}}{0 - B\_m}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{F \cdot -2 + \frac{2 \cdot \left(A \cdot F\right)}{B\_m}}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7e-73)
   (* (/ (/ 0.25 A) C) (sqrt (* (* A -8.0) (* C (* F (+ C C))))))
   (if (<= B_m 2.55e+162)
     (/ (pow (* -2.0 (* B_m F)) 0.5) (- 0.0 B_m))
     (- 0.0 (sqrt (/ (+ (* F -2.0) (/ (* 2.0 (* A F)) B_m)) B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7e-73) {
		tmp = ((0.25 / A) / C) * sqrt(((A * -8.0) * (C * (F * (C + C)))));
	} else if (B_m <= 2.55e+162) {
		tmp = pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	} else {
		tmp = 0.0 - sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / 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 <= 7d-73) then
        tmp = ((0.25d0 / a) / c) * sqrt(((a * (-8.0d0)) * (c * (f * (c + c)))))
    else if (b_m <= 2.55d+162) then
        tmp = (((-2.0d0) * (b_m * f)) ** 0.5d0) / (0.0d0 - b_m)
    else
        tmp = 0.0d0 - sqrt((((f * (-2.0d0)) + ((2.0d0 * (a * f)) / b_m)) / 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 <= 7e-73) {
		tmp = ((0.25 / A) / C) * Math.sqrt(((A * -8.0) * (C * (F * (C + C)))));
	} else if (B_m <= 2.55e+162) {
		tmp = Math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	} else {
		tmp = 0.0 - Math.sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 7e-73:
		tmp = ((0.25 / A) / C) * math.sqrt(((A * -8.0) * (C * (F * (C + C)))))
	elif B_m <= 2.55e+162:
		tmp = math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m)
	else:
		tmp = 0.0 - math.sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7e-73)
		tmp = Float64(Float64(Float64(0.25 / A) / C) * sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(C + C))))));
	elseif (B_m <= 2.55e+162)
		tmp = Float64((Float64(-2.0 * Float64(B_m * F)) ^ 0.5) / Float64(0.0 - B_m));
	else
		tmp = Float64(0.0 - sqrt(Float64(Float64(Float64(F * -2.0) + Float64(Float64(2.0 * Float64(A * F)) / B_m)) / 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 <= 7e-73)
		tmp = ((0.25 / A) / C) * sqrt(((A * -8.0) * (C * (F * (C + C)))));
	elseif (B_m <= 2.55e+162)
		tmp = ((-2.0 * (B_m * F)) ^ 0.5) / (0.0 - B_m);
	else
		tmp = 0.0 - sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / 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, 7e-73], N[(N[(N[(0.25 / A), $MachinePrecision] / C), $MachinePrecision] * N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.55e+162], N[(N[Power[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Sqrt[N[(N[(N[(F * -2.0), $MachinePrecision] + N[(N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.9999999999999995e-73

   1. Initial program 17.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 23.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 23.2%

      \[\leadsto \color{blue}{\frac{1}{A \cdot \left(C \cdot 4\right) - B \cdot B} \cdot \sqrt{2 \cdot \left(\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}} \]

   7. Taylor expanded in C around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{\frac{1}{16} \cdot \frac{{B}^{2}}{{A}^{2} \cdot C} + \frac{1}{4} \cdot \frac{1}{A}}{C}\right)}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{16} \cdot \frac{{B}^{2}}{{A}^{2} \cdot C} + \frac{1}{4} \cdot \frac{1}{A}\right), C\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{16} \cdot \frac{{B}^{2}}{{A}^{2} \cdot C}\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{16} \cdot {B}^{2}}{{A}^{2} \cdot C}\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{16} \cdot {B}^{2}\right), \left({A}^{2} \cdot C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \left({B}^{2}\right)\right), \left({A}^{2} \cdot C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \left(B \cdot B\right)\right), \left({A}^{2} \cdot C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \left({A}^{2} \cdot C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\left({A}^{2}\right), C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\left(A \cdot A\right), C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right)\right), \left(\frac{\frac{1}{4} \cdot 1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right)\right), \left(\frac{\frac{1}{4}}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 12.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right)\right), \mathsf{/.f64}\left(\frac{1}{4}, A\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 12.7%

      \[\leadsto \color{blue}{\frac{\frac{0.0625 \cdot \left(B \cdot B\right)}{\left(A \cdot A\right) \cdot C} + \frac{0.25}{A}}{C}} \cdot \sqrt{2 \cdot \left(\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \]

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{1}{4}}{A}\right)}, C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 15.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 15.1%

       \[\leadsto \frac{\color{blue}{\frac{0.25}{A}}}{C} \cdot \sqrt{2 \cdot \left(\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \]

   13. Taylor expanded in A around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)\right)}\right)\right) \]
   14. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\left(\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)\right)\right) \]

       2. rem-square-sqrt N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\left(\left(\left(\sqrt{-8} \cdot \sqrt{-8}\right) \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\left(\left({\left(\sqrt{-8}\right)}^{2} \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\left(\left(A \cdot {\left(\sqrt{-8}\right)}^{2}\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot {\left(\sqrt{-8}\right)}^{2}\right), \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(\sqrt{-8} \cdot \sqrt{-8}\right)\right), \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)\right)\right) \]

       7. rem-square-sqrt N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot -8\right), \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, -8\right), \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, -8\right), \mathsf{*.f64}\left(C, \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, -8\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \left(C - -1 \cdot C\right)\right)\right)\right)\right)\right) \]

       11. cancel-sign-sub-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, -8\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \left(C + \left(\mathsf{neg}\left(-1\right)\right) \cdot C\right)\right)\right)\right)\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, -8\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \left(C + 1 \cdot C\right)\right)\right)\right)\right)\right) \]

       13. *-lft-identity N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, -8\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \left(C + C\right)\right)\right)\right)\right)\right) \]

       14. +-lowering-+.f64 17.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, -8\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(C, C\right)\right)\right)\right)\right)\right) \]

   15. Simplified 17.4%

       \[\leadsto \frac{\frac{0.25}{A}}{C} \cdot \sqrt{\color{blue}{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(C + C\right)\right)\right)}} \]

   if 6.9999999999999995e-73 < B < 2.5499999999999999e162

   1. Initial program 35.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 46.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 46.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]

   6. Taylor expanded in C around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(B \cdot F\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-1, \left(B \cdot F\right)\right)\right)\right) \]

      2. *-lowering-*.f64 40.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, F\right)\right)\right)\right) \]

   8. Simplified 40.2%

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{-1 \cdot \left(B \cdot F\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right), B\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({2}^{\frac{1}{2}} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right), B\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({2}^{\frac{1}{2}} \cdot {\left(-1 \cdot \left(B \cdot F\right)\right)}^{\frac{1}{2}}\right), B\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot \left(-1 \cdot \left(B \cdot F\right)\right)\right)}^{\frac{1}{2}}\right), B\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot \left(-1 \cdot \left(B \cdot F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(-1 \cdot \left(B \cdot F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{neg}\left(B \cdot F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(B \cdot \left(\mathsf{neg}\left(F\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, \left(\mathsf{neg}\left(F\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      14. neg-lowering-neg.f64 40.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, \mathsf{neg.f64}\left(F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   10. Applied egg-rr 40.2%

       \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(B \cdot \left(-F\right)\right)\right)}^{0.5}}{B}} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}, \frac{1}{2}\right), B\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right), \frac{1}{2}\right), B\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \left(F \cdot B\right)\right), \frac{1}{2}\right), B\right)\right) \]

       3. *-lowering-*.f64 40.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(F, B\right)\right), \frac{1}{2}\right), B\right)\right) \]

   13. Simplified 40.2%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(F \cdot B\right)\right)}}^{0.5}}{B} \]

   if 2.5499999999999999e162 < B

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({B}^{3} \cdot \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{3}\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\left(-2 \cdot F\right), \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(\frac{2 \cdot \left(F \cdot \left(A + C\right)\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(2 \cdot \left(F \cdot \left(A + C\right)\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(\left(2 \cdot F\right) \cdot \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot F\right), \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(C + A\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      15. +-lowering-+.f64 0.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \mathsf{+.f64}\left(C, A\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 0.0%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(-2 \cdot F + \frac{\left(2 \cdot F\right) \cdot \left(C + A\right)}{B}\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}\right), B\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot F\right), \left(2 \cdot \frac{A \cdot F}{B}\right)\right), B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(2 \cdot \frac{A \cdot F}{B}\right)\right), B\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(\frac{2 \cdot \left(A \cdot F\right)}{B}\right)\right), B\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(2 \cdot \left(A \cdot F\right)\right), B\right)\right), B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(A \cdot F\right)\right), B\right)\right), B\right)\right)\right) \]

      10. *-lowering-*.f64 59.1%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right), B\right)\right), B\right)\right)\right) \]

   10. Simplified 59.1%

       \[\leadsto \color{blue}{-\sqrt{\frac{-2 \cdot F + \frac{2 \cdot \left(A \cdot F\right)}{B}}{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{0.25}{A}}{C} \cdot \sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(C + C\right)\right)\right)}\\ \mathbf{elif}\;B \leq 2.55 \cdot 10^{+162}:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B \cdot F\right)\right)}^{0.5}}{0 - B}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{F \cdot -2 + \frac{2 \cdot \left(A \cdot F\right)}{B}}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{0.25}{A}}{C} \cdot \sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}\\ \mathbf{elif}\;B\_m \leq 2.5 \cdot 10^{+162}:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B\_m \cdot F\right)\right)}^{0.5}}{0 - B\_m}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{F \cdot -2 + \frac{2 \cdot \left(A \cdot F\right)}{B\_m}}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.1e-73)
   (* (/ (/ 0.25 A) C) (sqrt (* (* A -16.0) (* F (* C C)))))
   (if (<= B_m 2.5e+162)
     (/ (pow (* -2.0 (* B_m F)) 0.5) (- 0.0 B_m))
     (- 0.0 (sqrt (/ (+ (* F -2.0) (/ (* 2.0 (* A F)) B_m)) B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.1e-73) {
		tmp = ((0.25 / A) / C) * sqrt(((A * -16.0) * (F * (C * C))));
	} else if (B_m <= 2.5e+162) {
		tmp = pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	} else {
		tmp = 0.0 - sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / 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 <= 1.1d-73) then
        tmp = ((0.25d0 / a) / c) * sqrt(((a * (-16.0d0)) * (f * (c * c))))
    else if (b_m <= 2.5d+162) then
        tmp = (((-2.0d0) * (b_m * f)) ** 0.5d0) / (0.0d0 - b_m)
    else
        tmp = 0.0d0 - sqrt((((f * (-2.0d0)) + ((2.0d0 * (a * f)) / b_m)) / 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 <= 1.1e-73) {
		tmp = ((0.25 / A) / C) * Math.sqrt(((A * -16.0) * (F * (C * C))));
	} else if (B_m <= 2.5e+162) {
		tmp = Math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	} else {
		tmp = 0.0 - Math.sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.1e-73:
		tmp = ((0.25 / A) / C) * math.sqrt(((A * -16.0) * (F * (C * C))))
	elif B_m <= 2.5e+162:
		tmp = math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m)
	else:
		tmp = 0.0 - math.sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.1e-73)
		tmp = Float64(Float64(Float64(0.25 / A) / C) * sqrt(Float64(Float64(A * -16.0) * Float64(F * Float64(C * C)))));
	elseif (B_m <= 2.5e+162)
		tmp = Float64((Float64(-2.0 * Float64(B_m * F)) ^ 0.5) / Float64(0.0 - B_m));
	else
		tmp = Float64(0.0 - sqrt(Float64(Float64(Float64(F * -2.0) + Float64(Float64(2.0 * Float64(A * F)) / B_m)) / 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 <= 1.1e-73)
		tmp = ((0.25 / A) / C) * sqrt(((A * -16.0) * (F * (C * C))));
	elseif (B_m <= 2.5e+162)
		tmp = ((-2.0 * (B_m * F)) ^ 0.5) / (0.0 - B_m);
	else
		tmp = 0.0 - sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / 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, 1.1e-73], N[(N[(N[(0.25 / A), $MachinePrecision] / C), $MachinePrecision] * N[Sqrt[N[(N[(A * -16.0), $MachinePrecision] * N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.5e+162], N[(N[Power[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Sqrt[N[(N[(N[(F * -2.0), $MachinePrecision] + N[(N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.1e-73

   1. Initial program 17.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 23.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 23.2%

      \[\leadsto \color{blue}{\frac{1}{A \cdot \left(C \cdot 4\right) - B \cdot B} \cdot \sqrt{2 \cdot \left(\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}} \]

   7. Taylor expanded in C around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{\frac{1}{16} \cdot \frac{{B}^{2}}{{A}^{2} \cdot C} + \frac{1}{4} \cdot \frac{1}{A}}{C}\right)}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{16} \cdot \frac{{B}^{2}}{{A}^{2} \cdot C} + \frac{1}{4} \cdot \frac{1}{A}\right), C\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{16} \cdot \frac{{B}^{2}}{{A}^{2} \cdot C}\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{16} \cdot {B}^{2}}{{A}^{2} \cdot C}\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{16} \cdot {B}^{2}\right), \left({A}^{2} \cdot C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \left({B}^{2}\right)\right), \left({A}^{2} \cdot C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \left(B \cdot B\right)\right), \left({A}^{2} \cdot C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \left({A}^{2} \cdot C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\left({A}^{2}\right), C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\left(A \cdot A\right), C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right)\right), \left(\frac{\frac{1}{4} \cdot 1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right)\right), \left(\frac{\frac{1}{4}}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 12.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right)\right), \mathsf{/.f64}\left(\frac{1}{4}, A\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 12.7%

      \[\leadsto \color{blue}{\frac{\frac{0.0625 \cdot \left(B \cdot B\right)}{\left(A \cdot A\right) \cdot C} + \frac{0.25}{A}}{C}} \cdot \sqrt{2 \cdot \left(\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \]

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{1}{4}}{A}\right)}, C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 15.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 15.1%

       \[\leadsto \frac{\color{blue}{\frac{0.25}{A}}}{C} \cdot \sqrt{2 \cdot \left(\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \]

   13. Taylor expanded in B around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right)\right) \]
   14. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\left(\left(-16 \cdot A\right) \cdot \left({C}^{2} \cdot F\right)\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-16 \cdot A\right), \left({C}^{2} \cdot F\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \left({C}^{2} \cdot F\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \left(F \cdot {C}^{2}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \mathsf{*.f64}\left(F, \left({C}^{2}\right)\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \mathsf{*.f64}\left(F, \left(C \cdot C\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 14.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, A\right), \mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, C\right)\right)\right)\right)\right) \]

   15. Simplified 14.6%

       \[\leadsto \frac{\frac{0.25}{A}}{C} \cdot \sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}} \]

   if 1.1e-73 < B < 2.4999999999999998e162

   1. Initial program 35.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 46.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 46.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]

   6. Taylor expanded in C around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(B \cdot F\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-1, \left(B \cdot F\right)\right)\right)\right) \]

      2. *-lowering-*.f64 40.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, F\right)\right)\right)\right) \]

   8. Simplified 40.2%

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{-1 \cdot \left(B \cdot F\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right), B\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({2}^{\frac{1}{2}} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right), B\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({2}^{\frac{1}{2}} \cdot {\left(-1 \cdot \left(B \cdot F\right)\right)}^{\frac{1}{2}}\right), B\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot \left(-1 \cdot \left(B \cdot F\right)\right)\right)}^{\frac{1}{2}}\right), B\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot \left(-1 \cdot \left(B \cdot F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(-1 \cdot \left(B \cdot F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{neg}\left(B \cdot F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(B \cdot \left(\mathsf{neg}\left(F\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, \left(\mathsf{neg}\left(F\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      14. neg-lowering-neg.f64 40.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, \mathsf{neg.f64}\left(F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   10. Applied egg-rr 40.2%

       \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(B \cdot \left(-F\right)\right)\right)}^{0.5}}{B}} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}, \frac{1}{2}\right), B\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right), \frac{1}{2}\right), B\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \left(F \cdot B\right)\right), \frac{1}{2}\right), B\right)\right) \]

       3. *-lowering-*.f64 40.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(F, B\right)\right), \frac{1}{2}\right), B\right)\right) \]

   13. Simplified 40.2%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(F \cdot B\right)\right)}}^{0.5}}{B} \]

   if 2.4999999999999998e162 < B

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({B}^{3} \cdot \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{3}\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\left(-2 \cdot F\right), \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(\frac{2 \cdot \left(F \cdot \left(A + C\right)\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(2 \cdot \left(F \cdot \left(A + C\right)\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(\left(2 \cdot F\right) \cdot \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot F\right), \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(C + A\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      15. +-lowering-+.f64 0.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \mathsf{+.f64}\left(C, A\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 0.0%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(-2 \cdot F + \frac{\left(2 \cdot F\right) \cdot \left(C + A\right)}{B}\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}\right), B\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot F\right), \left(2 \cdot \frac{A \cdot F}{B}\right)\right), B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(2 \cdot \frac{A \cdot F}{B}\right)\right), B\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(\frac{2 \cdot \left(A \cdot F\right)}{B}\right)\right), B\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(2 \cdot \left(A \cdot F\right)\right), B\right)\right), B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(A \cdot F\right)\right), B\right)\right), B\right)\right)\right) \]

      10. *-lowering-*.f64 59.1%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right), B\right)\right), B\right)\right)\right) \]

   10. Simplified 59.1%

       \[\leadsto \color{blue}{-\sqrt{\frac{-2 \cdot F + \frac{2 \cdot \left(A \cdot F\right)}{B}}{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{0.25}{A}}{C} \cdot \sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{+162}:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B \cdot F\right)\right)}^{0.5}}{0 - B}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{F \cdot -2 + \frac{2 \cdot \left(A \cdot F\right)}{B}}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.1% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{+15}:\\ \;\;\;\;0 - \sqrt{\frac{F \cdot -2 + \frac{2 \cdot \left(A \cdot F\right)}{B\_m}}{B\_m}}\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{2 \cdot \left(B\_m \cdot \left(0 - F\right)\right)} \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{A}}{C} \cdot \sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -2.3e+15)
   (- 0.0 (sqrt (/ (+ (* F -2.0) (/ (* 2.0 (* A F)) B_m)) B_m)))
   (if (<= F -3.8e-302)
     (* (sqrt (* 2.0 (* B_m (- 0.0 F)))) (/ -1.0 B_m))
     (* (/ (/ 0.25 A) C) (sqrt (* -16.0 (* F (* C (* A A)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2.3e+15) {
		tmp = 0.0 - sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m));
	} else if (F <= -3.8e-302) {
		tmp = sqrt((2.0 * (B_m * (0.0 - F)))) * (-1.0 / B_m);
	} else {
		tmp = ((0.25 / A) / C) * sqrt((-16.0 * (F * (C * (A * 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 (f <= (-2.3d+15)) then
        tmp = 0.0d0 - sqrt((((f * (-2.0d0)) + ((2.0d0 * (a * f)) / b_m)) / b_m))
    else if (f <= (-3.8d-302)) then
        tmp = sqrt((2.0d0 * (b_m * (0.0d0 - f)))) * ((-1.0d0) / b_m)
    else
        tmp = ((0.25d0 / a) / c) * sqrt(((-16.0d0) * (f * (c * (a * 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 (F <= -2.3e+15) {
		tmp = 0.0 - Math.sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m));
	} else if (F <= -3.8e-302) {
		tmp = Math.sqrt((2.0 * (B_m * (0.0 - F)))) * (-1.0 / B_m);
	} else {
		tmp = ((0.25 / A) / C) * Math.sqrt((-16.0 * (F * (C * (A * A)))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= -2.3e+15:
		tmp = 0.0 - math.sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m))
	elif F <= -3.8e-302:
		tmp = math.sqrt((2.0 * (B_m * (0.0 - F)))) * (-1.0 / B_m)
	else:
		tmp = ((0.25 / A) / C) * math.sqrt((-16.0 * (F * (C * (A * A)))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -2.3e+15)
		tmp = Float64(0.0 - sqrt(Float64(Float64(Float64(F * -2.0) + Float64(Float64(2.0 * Float64(A * F)) / B_m)) / B_m)));
	elseif (F <= -3.8e-302)
		tmp = Float64(sqrt(Float64(2.0 * Float64(B_m * Float64(0.0 - F)))) * Float64(-1.0 / B_m));
	else
		tmp = Float64(Float64(Float64(0.25 / A) / C) * sqrt(Float64(-16.0 * Float64(F * Float64(C * Float64(A * A))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -2.3e+15)
		tmp = 0.0 - sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m));
	elseif (F <= -3.8e-302)
		tmp = sqrt((2.0 * (B_m * (0.0 - F)))) * (-1.0 / B_m);
	else
		tmp = ((0.25 / A) / C) * sqrt((-16.0 * (F * (C * (A * A)))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -2.3e+15], N[(0.0 - N[Sqrt[N[(N[(N[(F * -2.0), $MachinePrecision] + N[(N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.8e-302], N[(N[Sqrt[N[(2.0 * N[(B$95$m * N[(0.0 - F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 / A), $MachinePrecision] / C), $MachinePrecision] * N[Sqrt[N[(-16.0 * N[(F * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.3e15

   1. Initial program 15.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 17.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({B}^{3} \cdot \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{3}\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\left(-2 \cdot F\right), \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(\frac{2 \cdot \left(F \cdot \left(A + C\right)\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(2 \cdot \left(F \cdot \left(A + C\right)\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(\left(2 \cdot F\right) \cdot \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot F\right), \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(C + A\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      15. +-lowering-+.f64 3.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \mathsf{+.f64}\left(C, A\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 3.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(-2 \cdot F + \frac{\left(2 \cdot F\right) \cdot \left(C + A\right)}{B}\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}\right), B\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot F\right), \left(2 \cdot \frac{A \cdot F}{B}\right)\right), B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(2 \cdot \frac{A \cdot F}{B}\right)\right), B\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(\frac{2 \cdot \left(A \cdot F\right)}{B}\right)\right), B\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(2 \cdot \left(A \cdot F\right)\right), B\right)\right), B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(A \cdot F\right)\right), B\right)\right), B\right)\right)\right) \]

      10. *-lowering-*.f64 20.7%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right), B\right)\right), B\right)\right)\right) \]

   10. Simplified 20.7%

       \[\leadsto \color{blue}{-\sqrt{\frac{-2 \cdot F + \frac{2 \cdot \left(A \cdot F\right)}{B}}{B}}} \]

   if -2.3e15 < F < -3.8e-302

   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 A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 26.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 26.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]

   6. Taylor expanded in C around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(B \cdot F\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-1, \left(B \cdot F\right)\right)\right)\right) \]

      2. *-lowering-*.f64 24.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, F\right)\right)\right)\right) \]

   8. Simplified 24.5%

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{-1 \cdot \left(B \cdot F\right)}} \]
   9. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(-1 \cdot \left(B \cdot F\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(\mathsf{neg}\left(B \cdot F\right)\right)}^{\frac{1}{2}}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(\mathsf{neg}\left(F \cdot B\right)\right)}^{\frac{1}{2}}\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(\left(\mathsf{neg}\left(F\right)\right) \cdot B\right)}^{\frac{1}{2}}\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(\mathsf{neg}\left(F\right)\right)}^{\frac{1}{2}} \cdot \color{blue}{{B}^{\frac{1}{2}}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\left({\left(\mathsf{neg}\left(F\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left({B}^{\frac{1}{2}}\right)}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(F\right)\right), \frac{1}{2}\right), \left({\color{blue}{B}}^{\frac{1}{2}}\right)\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(F\right), \frac{1}{2}\right), \left({B}^{\frac{1}{2}}\right)\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(F\right), \frac{1}{2}\right), \left(\sqrt{B}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 24.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(F\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(B\right)\right)\right) \]

   10. Applied egg-rr 24.2%

       \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\left({\left(-F\right)}^{0.5} \cdot \sqrt{B}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \left(\color{blue}{{\left(\mathsf{neg}\left(F\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{B}\right) \]

       2. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \left({\left(\mathsf{neg}\left(F\right)\right)}^{\frac{1}{2}} \cdot \sqrt{B}\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2} \cdot \left({\left(\mathsf{neg}\left(F\right)\right)}^{\frac{1}{2}} \cdot \sqrt{B}\right)}{B}\right) \]

       4. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{{2}^{\frac{1}{2}} \cdot \left({\left(\mathsf{neg}\left(F\right)\right)}^{\frac{1}{2}} \cdot \sqrt{B}\right)}{B}\right) \]

       5. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{{2}^{\frac{1}{2}} \cdot \left({\left(\mathsf{neg}\left(F\right)\right)}^{\frac{1}{2}} \cdot {B}^{\frac{1}{2}}\right)}{B}\right) \]

       6. unpow-prod-down N/A

          \[\leadsto \mathsf{neg}\left(\frac{{2}^{\frac{1}{2}} \cdot {\left(\left(\mathsf{neg}\left(F\right)\right) \cdot B\right)}^{\frac{1}{2}}}{B}\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\frac{{2}^{\frac{1}{2}} \cdot {\left(B \cdot \left(\mathsf{neg}\left(F\right)\right)\right)}^{\frac{1}{2}}}{B}\right) \]

       8. unpow-prod-down N/A

          \[\leadsto \mathsf{neg}\left(\frac{{\left(2 \cdot \left(B \cdot \left(\mathsf{neg}\left(F\right)\right)\right)\right)}^{\frac{1}{2}}}{B}\right) \]

       9. distribute-neg-frac2 N/A

          \[\leadsto \frac{{\left(2 \cdot \left(B \cdot \left(\mathsf{neg}\left(F\right)\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\mathsf{neg}\left(B\right)}} \]

       10. div-inv N/A

           \[\leadsto {\left(2 \cdot \left(B \cdot \left(\mathsf{neg}\left(F\right)\right)\right)\right)}^{\frac{1}{2}} \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(B\right)}} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left({\left(2 \cdot \left(B \cdot \left(\mathsf{neg}\left(F\right)\right)\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{\mathsf{neg}\left(B\right)}\right)}\right) \]

   12. Applied egg-rr 24.6%

       \[\leadsto \color{blue}{\sqrt{-\left(F \cdot B\right) \cdot 2} \cdot \frac{-1}{B}} \]

   if -3.8e-302 < F

   1. Initial program 26.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 34.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 37.3%

      \[\leadsto \color{blue}{\frac{1}{A \cdot \left(C \cdot 4\right) - B \cdot B} \cdot \sqrt{2 \cdot \left(\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}} \]

   7. Taylor expanded in C around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{\frac{1}{16} \cdot \frac{{B}^{2}}{{A}^{2} \cdot C} + \frac{1}{4} \cdot \frac{1}{A}}{C}\right)}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{16} \cdot \frac{{B}^{2}}{{A}^{2} \cdot C} + \frac{1}{4} \cdot \frac{1}{A}\right), C\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{16} \cdot \frac{{B}^{2}}{{A}^{2} \cdot C}\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{16} \cdot {B}^{2}}{{A}^{2} \cdot C}\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{16} \cdot {B}^{2}\right), \left({A}^{2} \cdot C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \left({B}^{2}\right)\right), \left({A}^{2} \cdot C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \left(B \cdot B\right)\right), \left({A}^{2} \cdot C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \left({A}^{2} \cdot C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\left({A}^{2}\right), C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\left(A \cdot A\right), C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right)\right), \left(\frac{1}{4} \cdot \frac{1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right)\right), \left(\frac{\frac{1}{4} \cdot 1}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right)\right), \left(\frac{\frac{1}{4}}{A}\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 34.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right)\right), \mathsf{/.f64}\left(\frac{1}{4}, A\right)\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 34.5%

      \[\leadsto \color{blue}{\frac{\frac{0.0625 \cdot \left(B \cdot B\right)}{\left(A \cdot A\right) \cdot C} + \frac{0.25}{A}}{C}} \cdot \sqrt{2 \cdot \left(\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \]

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{1}{4}}{A}\right)}, C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 37.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), F\right), \mathsf{+.f64}\left(A, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 37.3%

       \[\leadsto \frac{\color{blue}{\frac{0.25}{A}}}{C} \cdot \sqrt{2 \cdot \left(\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \]

   13. Taylor expanded in A around -inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right)\right) \]
   14. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right)\right) \]

       6. *-lowering-*.f64 13.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right)\right) \]

   15. Simplified 13.9%

       \[\leadsto \frac{\frac{0.25}{A}}{C} \cdot \sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{+15}:\\ \;\;\;\;0 - \sqrt{\frac{F \cdot -2 + \frac{2 \cdot \left(A \cdot F\right)}{B}}{B}}\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{2 \cdot \left(B \cdot \left(0 - F\right)\right)} \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{A}}{C} \cdot \sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 32.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;0 - \sqrt{\frac{F \cdot -2 + \frac{2 \cdot \left(A \cdot F\right)}{B\_m}}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(B\_m \cdot \left(0 - F\right)\right)} \cdot \frac{-1}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -80000000000000.0)
   (- 0.0 (sqrt (/ (+ (* F -2.0) (/ (* 2.0 (* A F)) B_m)) B_m)))
   (* (sqrt (* 2.0 (* B_m (- 0.0 F)))) (/ -1.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = 0.0 - sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m));
	} else {
		tmp = sqrt((2.0 * (B_m * (0.0 - F)))) * (-1.0 / 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 (f <= (-80000000000000.0d0)) then
        tmp = 0.0d0 - sqrt((((f * (-2.0d0)) + ((2.0d0 * (a * f)) / b_m)) / b_m))
    else
        tmp = sqrt((2.0d0 * (b_m * (0.0d0 - f)))) * ((-1.0d0) / 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 (F <= -80000000000000.0) {
		tmp = 0.0 - Math.sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m));
	} else {
		tmp = Math.sqrt((2.0 * (B_m * (0.0 - F)))) * (-1.0 / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= -80000000000000.0:
		tmp = 0.0 - math.sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m))
	else:
		tmp = math.sqrt((2.0 * (B_m * (0.0 - F)))) * (-1.0 / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -80000000000000.0)
		tmp = Float64(0.0 - sqrt(Float64(Float64(Float64(F * -2.0) + Float64(Float64(2.0 * Float64(A * F)) / B_m)) / B_m)));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(B_m * Float64(0.0 - F)))) * Float64(-1.0 / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -80000000000000.0)
		tmp = 0.0 - sqrt((((F * -2.0) + ((2.0 * (A * F)) / B_m)) / B_m));
	else
		tmp = sqrt((2.0 * (B_m * (0.0 - F)))) * (-1.0 / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -80000000000000.0], N[(0.0 - N[Sqrt[N[(N[(N[(F * -2.0), $MachinePrecision] + N[(N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(B$95$m * N[(0.0 - F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -8e13

   1. Initial program 15.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Simplified 17.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({B}^{3} \cdot \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{3}\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(-2 \cdot F + 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\left(-2 \cdot F\right), \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(\frac{2 \cdot \left(F \cdot \left(A + C\right)\right)}{B}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(2 \cdot \left(F \cdot \left(A + C\right)\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(\left(2 \cdot F\right) \cdot \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot F\right), \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(A + C\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(C + A\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      15. +-lowering-+.f64 3.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, F\right), \mathsf{+.f64}\left(C, A\right)\right), B\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 3.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(-2 \cdot F + \frac{\left(2 \cdot F\right) \cdot \left(C + A\right)}{B}\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}}{B}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot F + 2 \cdot \frac{A \cdot F}{B}\right), B\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot F\right), \left(2 \cdot \frac{A \cdot F}{B}\right)\right), B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(2 \cdot \frac{A \cdot F}{B}\right)\right), B\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \left(\frac{2 \cdot \left(A \cdot F\right)}{B}\right)\right), B\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\left(2 \cdot \left(A \cdot F\right)\right), B\right)\right), B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(A \cdot F\right)\right), B\right)\right), B\right)\right)\right) \]

      10. *-lowering-*.f64 20.7%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, F\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right), B\right)\right), B\right)\right)\right) \]

   10. Simplified 20.7%

       \[\leadsto \color{blue}{-\sqrt{\frac{-2 \cdot F + \frac{2 \cdot \left(A \cdot F\right)}{B}}{B}}} \]

   if -8e13 < F

   1. Initial program 20.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 19.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 19.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]

   6. Taylor expanded in C around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(B \cdot F\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-1, \left(B \cdot F\right)\right)\right)\right) \]

      2. *-lowering-*.f64 19.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, F\right)\right)\right)\right) \]

   8. Simplified 19.4%

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{-1 \cdot \left(B \cdot F\right)}} \]
   9. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(-1 \cdot \left(B \cdot F\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(\mathsf{neg}\left(B \cdot F\right)\right)}^{\frac{1}{2}}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(\mathsf{neg}\left(F \cdot B\right)\right)}^{\frac{1}{2}}\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(\left(\mathsf{neg}\left(F\right)\right) \cdot B\right)}^{\frac{1}{2}}\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(\mathsf{neg}\left(F\right)\right)}^{\frac{1}{2}} \cdot \color{blue}{{B}^{\frac{1}{2}}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\left({\left(\mathsf{neg}\left(F\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left({B}^{\frac{1}{2}}\right)}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(F\right)\right), \frac{1}{2}\right), \left({\color{blue}{B}}^{\frac{1}{2}}\right)\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(F\right), \frac{1}{2}\right), \left({B}^{\frac{1}{2}}\right)\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(F\right), \frac{1}{2}\right), \left(\sqrt{B}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 18.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(F\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(B\right)\right)\right) \]

   10. Applied egg-rr 18.1%

       \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\left({\left(-F\right)}^{0.5} \cdot \sqrt{B}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \left(\color{blue}{{\left(\mathsf{neg}\left(F\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{B}\right) \]

       2. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \left({\left(\mathsf{neg}\left(F\right)\right)}^{\frac{1}{2}} \cdot \sqrt{B}\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2} \cdot \left({\left(\mathsf{neg}\left(F\right)\right)}^{\frac{1}{2}} \cdot \sqrt{B}\right)}{B}\right) \]

       4. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{{2}^{\frac{1}{2}} \cdot \left({\left(\mathsf{neg}\left(F\right)\right)}^{\frac{1}{2}} \cdot \sqrt{B}\right)}{B}\right) \]

       5. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{{2}^{\frac{1}{2}} \cdot \left({\left(\mathsf{neg}\left(F\right)\right)}^{\frac{1}{2}} \cdot {B}^{\frac{1}{2}}\right)}{B}\right) \]

       6. unpow-prod-down N/A

          \[\leadsto \mathsf{neg}\left(\frac{{2}^{\frac{1}{2}} \cdot {\left(\left(\mathsf{neg}\left(F\right)\right) \cdot B\right)}^{\frac{1}{2}}}{B}\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\frac{{2}^{\frac{1}{2}} \cdot {\left(B \cdot \left(\mathsf{neg}\left(F\right)\right)\right)}^{\frac{1}{2}}}{B}\right) \]

       8. unpow-prod-down N/A

          \[\leadsto \mathsf{neg}\left(\frac{{\left(2 \cdot \left(B \cdot \left(\mathsf{neg}\left(F\right)\right)\right)\right)}^{\frac{1}{2}}}{B}\right) \]

       9. distribute-neg-frac2 N/A

          \[\leadsto \frac{{\left(2 \cdot \left(B \cdot \left(\mathsf{neg}\left(F\right)\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\mathsf{neg}\left(B\right)}} \]

       10. div-inv N/A

           \[\leadsto {\left(2 \cdot \left(B \cdot \left(\mathsf{neg}\left(F\right)\right)\right)\right)}^{\frac{1}{2}} \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(B\right)}} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left({\left(2 \cdot \left(B \cdot \left(\mathsf{neg}\left(F\right)\right)\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{\mathsf{neg}\left(B\right)}\right)}\right) \]

   12. Applied egg-rr 19.4%

       \[\leadsto \color{blue}{\sqrt{-\left(F \cdot B\right) \cdot 2} \cdot \frac{-1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;0 - \sqrt{\frac{F \cdot -2 + \frac{2 \cdot \left(A \cdot F\right)}{B}}{B}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(B \cdot \left(0 - F\right)\right)} \cdot \frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \frac{{\left(-2 \cdot \left(B\_m \cdot F\right)\right)}^{0.5}}{0 - B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (/ (pow (* -2.0 (* B_m F)) 0.5) (- 0.0 B_m)))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return pow((-2.0 * (B_m * F)), 0.5) / (0.0 - 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 = (((-2.0d0) * (b_m * f)) ** 0.5d0) / (0.0d0 - b_m)
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m)

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64((Float64(-2.0 * Float64(B_m * F)) ^ 0.5) / Float64(0.0 - B_m))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = ((-2.0 * (B_m * F)) ^ 0.5) / (0.0 - B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Power[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.6%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in A around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

   11. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

   12. hypot-lowering-hypot.f64 18.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

5. Simplified 18.3%

   \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]

6. Taylor expanded in C around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(B \cdot F\right)\right)}\right)\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-1, \left(B \cdot F\right)\right)\right)\right) \]

   2. *-lowering-*.f64 16.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, F\right)\right)\right)\right) \]

8. Simplified 16.4%

   \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{-1 \cdot \left(B \cdot F\right)}} \]
9. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right) \]

   3. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}}{B}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right), B\right)\right) \]

   6. pow1/2 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({2}^{\frac{1}{2}} \cdot \sqrt{-1 \cdot \left(B \cdot F\right)}\right), B\right)\right) \]

   7. pow1/2 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({2}^{\frac{1}{2}} \cdot {\left(-1 \cdot \left(B \cdot F\right)\right)}^{\frac{1}{2}}\right), B\right)\right) \]

   8. pow-prod-down N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({\left(2 \cdot \left(-1 \cdot \left(B \cdot F\right)\right)\right)}^{\frac{1}{2}}\right), B\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot \left(-1 \cdot \left(B \cdot F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(-1 \cdot \left(B \cdot F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{neg}\left(B \cdot F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   12. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(B \cdot \left(\mathsf{neg}\left(F\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, \left(\mathsf{neg}\left(F\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   14. neg-lowering-neg.f64 16.4%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, \mathsf{neg.f64}\left(F\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

10. Applied egg-rr 16.4%

    \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(B \cdot \left(-F\right)\right)\right)}^{0.5}}{B}} \]

11. Taylor expanded in B around 0

    \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}, \frac{1}{2}\right), B\right)\right) \]
12. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right), \frac{1}{2}\right), B\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \left(F \cdot B\right)\right), \frac{1}{2}\right), B\right)\right) \]

    3. *-lowering-*.f64 16.4%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(F, B\right)\right), \frac{1}{2}\right), B\right)\right) \]

13. Simplified 16.4%

    \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(F \cdot B\right)\right)}}^{0.5}}{B} \]

14. Final simplification 16.4%

    \[\leadsto \frac{{\left(-2 \cdot \left(B \cdot F\right)\right)}^{0.5}}{0 - B} \]
15. Add Preprocessing

Alternative 15: 1.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \sqrt{2 \cdot \frac{F}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (sqrt (* 2.0 (/ F B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return 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 = 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 Math.sqrt((2.0 * (F / B_m)));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt((2.0 * (F / B_m)))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return sqrt(Float64(2.0 * Float64(F / B_m)))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt((2.0 * (F / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[Sqrt[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 18.6%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)} \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)}\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot \sqrt{2}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left({\color{blue}{\left(\sqrt{-1}\right)}}^{2} \cdot \sqrt{2}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{2}}\right)\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(-1 \cdot \sqrt{\color{blue}{2}}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{*.f64}\left(-1, \color{blue}{\left(\sqrt{2}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 1.7%

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{*.f64}\left(-1, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 1.7%

   \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \left(-1 \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto 0 - \left(-1 \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   2. mul-1-neg N/A

      \[\leadsto 0 - \left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   3. cancel-sign-sub N/A

      \[\leadsto 0 + \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(0, \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

   5. sqrt-unprod N/A

      \[\leadsto \mathsf{+.f64}\left(0, \left(\sqrt{2 \cdot \frac{F}{B}}\right)\right) \]

   6. rem-square-sqrt N/A

      \[\leadsto \mathsf{+.f64}\left(0, \left(\sqrt{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \frac{F}{B}}\right)\right) \]

   7. sqr-neg N/A

      \[\leadsto \mathsf{+.f64}\left(0, \left(\sqrt{\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right) \cdot \frac{F}{B}}\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(0, \left(\sqrt{\left(\left(-1 \cdot \sqrt{2}\right) \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right) \cdot \frac{F}{B}}\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(0, \left(\sqrt{\left(\left(-1 \cdot \sqrt{2}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \cdot \frac{F}{B}}\right)\right) \]

   10. rem-square-sqrt N/A

       \[\leadsto \mathsf{+.f64}\left(0, \left(\sqrt{\left(\left(-1 \cdot \sqrt{2}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{\frac{F}{B}}\right)}\right)\right) \]

   11. swap-sqr N/A

       \[\leadsto \mathsf{+.f64}\left(0, \left(\sqrt{\left(\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\right)}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(0, \left(\sqrt{\left(\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(\sqrt{\frac{F}{B}} \cdot \left(-1 \cdot \sqrt{2}\right)\right)}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(0, \left(\sqrt{\left(\sqrt{\frac{F}{B}} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \cdot \left(\sqrt{\frac{F}{B}} \cdot \left(-1 \cdot \sqrt{2}\right)\right)}\right)\right) \]

   14. +-lft-identity N/A

       \[\leadsto \mathsf{+.f64}\left(0, \left(\sqrt{\left(\sqrt{\frac{F}{B}} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \cdot \left(0 + \sqrt{\frac{F}{B}} \cdot \left(-1 \cdot \sqrt{2}\right)\right)}\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(0, \left(\sqrt{\left(\sqrt{\frac{F}{B}} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \cdot \left(\sqrt{\frac{F}{B}} \cdot \left(-1 \cdot \sqrt{2}\right) + 0\right)}\right)\right) \]

   16. distribute-rgt-out N/A

       \[\leadsto \mathsf{+.f64}\left(0, \left(\sqrt{\left(\sqrt{\frac{F}{B}} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \cdot \left(\sqrt{\frac{F}{B}} \cdot \left(-1 \cdot \sqrt{2}\right)\right) + 0 \cdot \left(\sqrt{\frac{F}{B}} \cdot \left(-1 \cdot \sqrt{2}\right)\right)}\right)\right) \]

7. Applied egg-rr 1.7%

   \[\leadsto \color{blue}{0 + \sqrt{2 \cdot \frac{F}{B}}} \]

8. Final simplification 1.7%

   \[\leadsto \sqrt{2 \cdot \frac{F}{B}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))